Lab Manual.pdf

Elementary Inorganic Chemistry CHEM 112 Laboratory Manual

Chemistry Department Morehouse College

Spring 2017

This laboratory manual is the product of adaptation, modification, and new experiments initiated by the grant from U.S. Department of Education award number P120A030117. Partial support of W. M. Keck Foundation Grant number 031875 is also acknowledged. The authors have tried to follow the guided-inquiry approach in compiling these experiments.

Subhash C. Bhatia Chemistry Department Morehouse College & Natarajan Ravi Physics Department Spelman College

Copyrighted Fall 2004 Revised 12/14/2010 Troy L. Story

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Table of Contents

A. Laboratory Safety and Housekeeping

Page number

B. Safety Agreement C. Guidelines for the Laboratory Report 4 1. Investigation of the Effect of Solute on the Physical Properties of a Solvent 9 7 2. Determination of Molar Mass of a Compound using a Colligative Property 13 8 3. Enthalpy of a Chemical Reaction 4. Enthalpy and Entropy 5. Rate and Order of a Chemical Reaction

32

6. Determination of an Equilibrium Constant

38

7. Acids and Buffers

46

8. Determination of the Concentration of an Acid

51

9. Electrochemistry: Voltaic Cells, Determination of Equilibrium Constant

57

APPENDIXES A. B. C. D.

Common Instruments Logger Pro Instructions Calibration procedures for different sensors Frequently Asked Questions about Data Analysis

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16 67 79 84 88 22

MOREHOUSE COLLEGE Department of Chemistry Laboratory Safety and Housekeeping Students are asked to be particularly mindful of the following as they conduct chemistry experiments in Morehouse College instructional laboratories.

Safety EYE PROTECTION IS MANDATORY WHILE WORKING IN THE LABORATORY. NO STUDENT WILL BE PERMITTED TO WORK WITHOUT EYE PROTECTION. Be aware of the hazardous nature of the chemical being used. Consult your instructor or teaching assistant if there are questions regarding the use of any chemical. Do not substitute chemicals in a procedure unless instructed to do so. Become familiar with the procedures to be employed prior to their performance. Exercise extra care when necessary. Be aware of potentially hazardous situations such as open flames, spills, etc. Alert other students to the problem then correct and/or seek advice from your instructor or teaching assistant. Food and beverages are not allowed in the laboratory. Housekeeping Ask permission to remove equipment or chemicals from cabinets or shelves. Unauthorized removal of equipment or chemicals will result in a failing grade for the experiment being performed. Keep your work area free from chemicals and equipment not being used in the procedure. Be careful when obtaining chemicals from bottles and jars. Do not take more than what is needed. Replace the cap on the bottle or jar immediately after obtaining the chemical therein. Inform the professor or lab instructor of any spills immediately. Dispose of broken glass in the proper receptacle (not the trash can).

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CHEMICAL LABORATORY SAFETY INSTRUCTIONS Read the following required policies/procedures carefully before entering any laboratory. Failure to following these rules and regulations will result in expulsion from the laboratory. 1. In case of fire or accident, notify the instructor immediately 2. Approved eye protection must be worn at all times in the laboratory where chemicals are stored or handled. The approved type of eye protection is safety glasses. Contact lenses are prohibited in the laboratory. A student will not be allowed to work in the lab without eye protection. 3. During the first laboratory period, become familiar with the location of the safety features of the laboratory. Note the locations of fire extinguishers, fire blanket, safety shower, eyewash, the nearest telephone and the list of important telephone numbers. 4. Confine long hair when in the laboratory. Shoes must be worn. Shorts, miniskirts and sandals are not allowed in the laboratory. Aprons or lab coats are highly recommended and are provided in the bookstore. 5. Unauthorized experiments are prohibited. 6. Working alone in the laboratory is prohibited. 7. Horseplay or other acts of carelessness are prohibited. 8. Laboratory areas are not to be used as eating or drinking areas; do not eat or drink from lab glassware. 9. Mouth suction is not to be used to fill pipettes. Rubber suction bulbs will be provided. 10. Exercise great care in noting the odor of fumes; avoid breathing fumes of any kind. (Inhalation of high concentrations of certain chemicals, particularly volatile solvents, can cause respiratory failure; other chemicals are chronic poisons after inhalation over time.) 11. Avoid skin or eye contact with any chemical. If you receive a chemical burn from acid or alkali, immediately wash the burned area with copious amounts of water and call an instructor immediately.

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12. Broken glass, flammable materials, paper and solid must be deposited in the provided containers, not in the sinks. 13. Cuts and burns can be prevented by following a few simple rules: a) When inserting glass tubing or thermometers into rubber stoppers, always protect your hands by wrapping the glass tubing with a towel. In addition, place a drop of glycerin on both the glass and the hole of the stopper. b) Fire polish all sharp edges of broken glass. c) Discard cracked or broken glass immediately. d) NEVER heat a graduated cylinder with a burner flame. 14.Never pour water into concentrated acid. Concentrated acids and bases may be diluted by pouring the reagent into water while stirring it carefully and continuously. Also, never add concentrated acid to concentrated base or vice versa. 15. When heating a test tube, never point the test tube toward your laboratory partner, neighbor, or yourself. 16.The Bunsen burner should be burning only when it is being used. Before lighting a burner, make sure that flammable reagents such as acetone, benzene, ether, alcohol, etc. are not in the vicinity. 17.Do not carry reagent bottles to your bench. This is a matter of safety and courtesy to the other students in the class, and it minimizes the possibility of contamination of the reagent. 18. Do not insert spatulas or pipets into reagent bottles. 19. At the end of the laboratory period, wash and wipe off your bench area. Be sure to return all equipment to the proper place.

20. WASH HANDS BEFORE LEAVING THE LABORATORY.

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SAFETY AGREEMENT I have studied, I understand, and I agree to follow the safety regulations required for this course. I have located all emergency equipment and now know how to use them. I understand that I may be dismissed from the laboratory for failure to comply with stated safety regulations.

Signature Print Name _____________________________________________________________________

Date

Course Section Instructor Person(s) who should be notified in the event of an accident:

I grant permission to post my grades in this course at mid-semester and again at the end of the semester, if my grades are identified by only a portion of my I.D. number. ∼ yes

∼ no

Signature

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GUIDELINES FOR THE LABORATORY REPORT Each laboratory report should be written individually, not as a group; duplications are not accepted. In this way, students develop experience translating experimental data and calculations into written form. This process also supports the College’s writing skills program. The laboratory report should be written in the past progressive tense (passive voice), using an essay style. Avoid the use of pronouns (e.g., I, we, they, us). For example, write “the use of pronouns should be avoided” rather than “you should avoid the use of pronouns.” The lab format consists of the following five topics, with topics 1 and 2 prepared before class. All reports should be typed. 1. Introduction {(10 + 5 + 5) points} • State clearly the raw data measured and name the systems, reactions, compounds, etc., for which these measurements are performed. Example: In this experiment the volume of nitrogen is measured as a function of temperature at constant pressure. • What major instruments are used? • What properties are calculated from the raw data? 2. Theory {(10 + 10) points} • List the fundamental equations used to calculate the properties referred to in the introduction; define all symbols. • Write a detailed explanation for using the fundamental equations to convert raw data into calculated data. 3. Experiment {(5 + 10 + 5) points} • Draw diagram(s) of the overall experimental set-up. Label the important elements; do not diagram individual pieces. • Using your diagram, describe the lab procedure followed. Do not itemize; write an essay. You’ll know when you’ve written a suitable procedure if you can repeat the experiment using your own write-up. • Construct tables to display raw data. Attach computer print-outs of raw data. 4. Calculations {(10 + 10) points} • Use the equations from the theory section for all calculations. Provide a detailed example of each type of calculation; explain. Calculate uncertainties. • Construct tables to display calculated data and uncertainties. 5. Conclusion {(10 + 10) points} • Summarize the experimental and calculated data, and state whether these results are compatible with standards. • Discuss possible sources of error. • Solve all questions and problems associated with the experiment. 8

Experiment 1 Investigation for the Effect of Solute on the Physical Properties of a Solvent Objectives: a) to learn how to establish relationships between experimentally determined variables. b) to learn how to search the literature for accepted values of physical quantities. c) to determine what is needed to calculate a defined concentration unit. d) to calculate a defined quantity from the experimental data. e) to solve problems such as “The freezing point of water is 00C and butanol freezes at –89.50C. Predict the freezing point of a solution prepared by mixing 10 mL of water and 1 mL of butanol. The molal freezing point depression constant of water is 1.86 K kg/mol.” I. Introduction Consider a binary solution containing components 1 and 2, with partial pressures P1 and P2 , and mole fractions X1 and X2 . These mole fractions are defined by X1 = n1 /n

and X2 = n2 /n

where n = n1 + n2 = the sum of the number of moles of components 1 and 2. If X1 > X2, then component 1 is defined as the solvent since it is present in the largest concentration; component 2 is the solute. The partial pressures are just the vapor pressures of the individual components over the solution. Assume that the vapor pressure of the pure solute is very much less that that of the pure solvent, then the solute is called non-volatile. The overall effect of adding a non-volatile solute to a solvent is to decrease the vapor pressure of the resulting solution. Such an effect is called a colligative property, since this property depends only on the relative number of particles present. In this experiment you will investigate the freezing point of a binary solution consisting of a solvent (water) and a solute (butanol). The materials required for this experiment are as follows: Computer utility clamp Vernier Computer interface 18 × 150 mm test tube Logger Pro distilled water 2 Vernier Temperature Probes butanol ring stand sodium chloride Thermometer ice 400 mL beaker

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II. Theory The mathematical model for quantitative description of a freezing point depression problem is ΔTfp = −i m Kf

where

ΔTfp = Tfp (solution) – Tfp (solvent) m = molality of solution = moles of solute/kilograms of solvent Kf = molal freezing point depression constant i = number of particles produced when solute dissolves in water, e.g., NaCl produces 2 particles, Na+ and Cl−, and so i = 2. For butanol in water, i = 1, since butanol does not decompose into ions in aqueous media. In this experiment plots of ΔTfp vs m will yield a straight line with a negative slope equal to − Kf . III. Experiment 1. Calibrate the Temperature Probe (see instructions – Appendix C, p.68). 2.

Prepare the Computer for data collection.

3.

Freezing point of water: a. Transfer 10 mL of water into a test tube. b. Immerse Temperature Sensor connected to channel 1 of the interface into the water in the test tube. c. Lower the test tube with sensor into the ice bath which is prepared by adding ice to 400 mL beaker. Make sure that the meniscus of water in the test tube is below the ice level. d. Immerse the Temperature sensor connected to channel 2 of the interface into an ice bath.

4.

Click on

to begin data collection.

5.

With a very slight up and down motion of the Temperature probe in the test tube, continuously stir the water during cooling. Hold the top of the probe and not its wire. Make sure that the water is frozen. If not, what can do to freeze water?

6.

Continue with the experiment until data collection has stopped.

7.

Take the test tube out of the ice bath and let ice turn into liquid water.

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8.

To determine the freezing temperature of pure water you need to determine the mean (or average) temperature in the portion of graph with nearly constant temperature. Move the mouse pointer to the beginning of the graph’s flat part. Press the mouse button and hold it down as you drag across the flat part of the curve, selecting only the points in the plateau. Click on the Statistics button, . The mean temperature value for the selected data is listed in the statistics box on the graph. Record this value as the freezing temperature of pure water. Click on the upper-right corner of the statistics box to remove it from the graph.

9.

Add 0.5 mL of butanol to the test tube in step 7 and repeat steps 4 to 8. Make sure that the mixture is uniform and freezes. Then go to steps 10, 11, and 12.

10. Add 0.5 mL more of butanol to the test tube from step 9 and repeat steps 4 to 8. 11. Add 0.5 mL more of butanol to the test tube in step 10 and repeat steps 4 to 8. 12. Add 0.5 mL more of butanol to the test tube in step 11 and repeat steps 4 to 8. 13. Steps 8 through 12 will give you the freezing points of four solutions of water and butanol.

point of pure water, and freezing

14. Find the freezing points of pure water and butanol from the CRC Handbook of Chemistry and Physics. Compare the freezing point of pure water and butanol with your experimentally determined values. Can you make any conclusion based on the literature and experimental values? 15. What should you do to establish a relationship between the two measured quantities? Ask your instructor if you do not know what to do. 16. Based on the lecture and recitation, what measured and calculated quantities should you graph to obtain a linear relationship? 17. Calculate the quantities you decided to graph from the experimental data. 18. Enter the data into a spreadsheet. Experiment 2.

Follow the direction given in step 10 of

19. What is the mathematical relationship? What constant is related to the slope of the graph? 20. Prepare a table which lists the amount of water, butanol and the freezing point for each part of the experiment. Attach all graphs.

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Pure water #1 #2 #3 #4

Vbutanol 0

Vwater # kg water Freezing Pt 10 mL 0.010 kg

0.5 1.0 1.5 2.0

10 10 10 10

mL mL mL mL

mL mL mL mL

0.010 0.010 0.010 0.010

kg kg kg kg

IV. Calculations Show calculations for the number of moles of butanol and the molality of each of the four butanol-water mixtures. Place this information in a data table along with the ΔTfp values. Run nbutanol #1 #2 #3 #4

molality

ΔTfp

Plot ΔTfp vs m and compute the slope to obtain − Kf. Calculate the % error of your Kf when compared to the literature value. % error = 100%|Kf (lit) – Kf (exp)|/Kf (lit) V. Conclusion Refer to the lab write-up procedure.

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Experiment 2 Determination of the Molar Mass of a Compound Using a Colligative Property Objectives: a) to learn how to determine the boiling point of a liquid. b) to learn how to determine the molar mass of an unknown compound. c) to learn the differences between accuracy and reproducibility. d) to learn why scientists repeat experiments under the same conditions. e) to learn how to solve problems such as “The normal boiling point of cyclohexane is 80.70C; 1.00 mL of benzene (D = 0.8736 g/mL) is dissolved in 20.00 mL of cyclohexane (D = 0.7785 g/mL). If the molal boiling point elevation constant for cyclohexane is 2.790C/m and the molar mass of benzene is 78.0 g/mol, then what is the boiling point for the mixture of benzene and cyclohexane.” I. Introduction In experiment 8, you should have observed that the addition of a solute to a solvent lowers (decreases) the freezing point of the solvent. The relationship established is that ΔTf = −m Kf , where ΔTf is the change in the freezing point of the solvent, m is molality of the solution and Kf is a proportionality constant known as the molal freezing point depression constant. The present experiment investigates the phenomena that addition of a non-volatile solute to a solvent also changes the boiling point of the solvent. In addition, it will be shown that a closer observation of both freezing point depression (exp 8) and boiling point elevation (exp 9) reveals a technique for determining the molar mass of an unknown compound when dissolved in liquid cyclohexane. The materials required for this experiment are as follows: Computer cyclohexane Vernier Computer Interface 500 mL or 1 liter beaker Logger Pro Pipette, test tube Temperature Probe Unknowns 1,2, and 3 II. Theory The mathematical model for quantitative description of a boiling point elevation problem takes the same form as for the freezing point depression problem, namely, where

ΔTbp = i m Kb ΔTbp = Tbp (solution) – Tbp (solvent) m = molality of solution = moles of solute/kilograms of solvent nunknown = #moles of unknown sample 13

(1)

W = mass of unknown sample M = molar mass of unknown sample Kb = molal freezing point depression constant i = number of particles produced when solute dissolves in water, e.g., NaCl produces 2 particles, Na+ and Cl−, and so i = 2. For the unknowns used, i = 1. Using this information, the equation ΔTbp = m Kb becomes

(2) In this experiment the molar mass is computed by rearranging the above equation to get

(3) III. Experiment 1. Use 500-mL beaker to prepare a water bath. 2. Calibrate the temperature sensors (see the instructions – Appendix C, p.68). 3. Determine the mass of clean and dry test tube. 4. Take 10.00 mL of cyclohexane into the pre-weighed test tube and insert the test tube into the water bath kept on the hot plate. Gently heat the water bath and measure the boiling point of cyclohexane using the calibrated temperature probe. 5. Clean and dry the test tube. Add a definite amount (0.25 to 0.5 g) of an unknown 1 or 2 or 3, to 10.00 mL of cyclohexane. Make sure that the solution is homogenous. Repeat step 4 to determine the boiling point of the solute-solvent mixture. Repeat this procedure at least three times with the same amounts of the solute and solvent. Record the pressure.

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6. Complete the chart below for your raw data table.

W Pure cyclohexane #1 #2 #3 #4

0

Vcyclohexane

Boiling Pt of solution

10 mL 10 10 10 10

ΔTbp xxxxxxxx

mL mL mL mL

IV. Calculations For cyclohexane ( C6 H10 ): Kb = 2.79 K kg mol−1 ; TNbp = 356 K Calculate the molar mass of the unknown sample from the following equation (refer to theory section).

Run W #1 #2 #3 #4

ΔTbp

#kg solvent

M

% error

% error = 100%|M(lit) – M(exp)|/M(lit) Calculate the average value of the molar mass and the average % error of your unknown. suggests another way of evaluating M,

The theory

namely, use of graphs of ΔTbp vs W yields a straight line whose slope is given by slope = Kb / (#kg of solvent X M ) . In this way M = Kb / (#kg of solvent X slope ). V. Conclusion. Refer to the lab write-up procedure.

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Experiment 3 Enthalpy of a Chemical Reaction Objectives a) to measure the temperature change of the reaction between hydrochloric acid solution and magnesium turnings. b) to calculate the enthalpy, ΔH, of the formation of magnesium chloride. c) to compare your calculated enthalpy of the reaction with the accepted value.

I. Introduction Thermodynamics is the study of energy changes for macroscopic physical systems. This experiment is designed to introduce techniques and concepts associated with measuring energy changes of a chemical reaction. The energy change of interest is the enthalpy change ΔH, which is a representation of an energy change in a (T, P) – coordinate system, where T is the system temperature and P is the system pressure. At constant pressure (denoted by subscript P ), qP = ΔHP , where qP is the heat absorbed (or emitted) by a system at constant pressure. Fundamentally, heat is identical to electromagnetic radiation. In this experiment you will first determine the heat capacity of a calorimeter by measuring: (1) the initial temperature of hot water, (2) initial temperature of (cold water + calorimeter), and (3) the temperature of the mixture formed when hot water is poured into the cold water. You will then measure the initial and final temperatures of the chemical reaction of magnesium and hydrochloric acid, and use this information to calculate the enthalpy change for this reaction. Materials Vernier computer interface computer Temperature Probe Styrofoam cup calorimeter two 250 mL beakers Glass stirring rod

Magnesium metal 1.0 M Hydrochloric acid two 50 mL graduated cylinders ring stand utility clamp

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II. Theory A convenient way to measure ΔHP is by placing the reaction vessel in thermal contact (heat flows between the vessel and another object, in this case, water) with water and measuring the temperature change of the water. Since it is known from experiment that 75.38 Joules of heat will raise the temperature of one mol of water by one Kelvin (called the heat capacity of water = C P , H2O(liq) ), then the experiment is a simple matter of measuring the temperature change in the water after the reaction occurs. This technique is referred to as calorimetry and the object containing the water is called a calorimeter. Note that use of a calorimeter requires that the reaction vessel and the calorimeter are isolated from the surroundings, implying that all energy which is emitted by the chemical reaction is absorbed by the calorimeter; the converse statement is true. This is a statement of the first law of thermodynamics; a representation of the law of conservation of energy. Some of the foregoing statements can be expressed in mathematical form, namely, by the following equations for isolated reaction and calorimeter. According to the first law of thermodynamics,

ΔHreaction + ΔHH O + ΔHcalorimeter = 0

(1)

ΔHP = nC P (Tfinal −Tinitial )

(2)

2

But

where

C P = Heat capacity at constant pressure P(units = Jmol -­‐ 1K -­‐ 1 ) = Amount of energy at constant P required to raise the temperature of one mol of an object by one Kelvin

Tfinal = final Kelvin temperature, Tinitial = initial Kelvin temperature Hence, eqn. (1) becomes

ΔHreaction + nH O C P,H2O(liq) (Tfinal −Tinitial ) + Ccal (Tfinal −Tinitial ) = 0 2

(3a)

Then ΔHreaction can be calculated from the equation

ΔHreaction = −nH O C P,H2O(liq) (Tfinal −Tinitial )−Ccal (Tfinal −Tinitial ) 2

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(3b)

To find the heat capacity of the calorimeter Ccal , hot water at temperature Th is added to a calorimeter containing cold water at temperature TC , where the cold water and the calorimeter are at the same temperature. Hence,

ΔHH O,T + ΔHH O,T + ΔHcalorimeter = 0 2

h

2

c

(4a)

Substituting for ΔH reveals the following result:

nh C P,H2O (Tfinal −Th ) +nc C P,H2O (Tfinal −Tc )+ Ccalorimeter (Tfinal −Tc ) = 0

(4b)

where C P , H2O = 75.38 J mol-1K-1 = heat capacity of liquid water, nh and nc are symbols for the number of moles of hot and cold water respectively, and Ccalorimeter is the heat capacity of the calorimeter. After measuring the temperatures, the only unknown is Ccalorimeter , which can then be calculated from eqn. (4b), giving

Ccalorimeter

− ⎡⎢nh C P,H2O (Tfinal −Th ) +nc C P,H2O (Tfinal −Tc )⎤⎥ ⎦ = ⎣ (Tfinal −Tc )

(5)

NOTE: Heat capacity at constant pressure CP or constant volume CV is a more definitive term than specific heat.

III. Experiment A Styrofoam cup nested in a beaker is used as a calorimeter, as shown in Figure 1. In Part I of this experiment, the heat capacity of the calorimeter will be determined from temperature measurements and calculations.

Figure 1 In Part II, the reaction between magnesium and hydrochloric acid will be studied with calorimetry in order to determine the enthalpy change at constant pressure for this reaction.

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Part I: Determination of Heat Capacity of the Calorimeter 1. Determine the mass of a dry foam coffee cup. 2. Place about 30 mL of cold tap water into the cup. Add a little ice to the water so that when it is melted, the temperature will be below room temperature (try around 10 to 150C). Determine the total mass of the water and the coffee cup. 3. Calibrate the temperature sensor and using the temperature sensor, measure the temperature of the water. 4. Place approximately 30 mL of hot water (about 10 to 150C above room temperature) into a second coffee cup. It is necessary to know the exact mass of the hot water in the coffee cup, but this may be determined in different ways. Determine the mass of hot water. 5. Measure the temperature of the hot water using the temperature sensor. 6. Add the hot water to the cold water in the first coffee cup, stir it gently, and record the equilibrium temperature. You will need the masses of the hot and cold water and the temperature changes to later perform the data analysis.

7. Use the law of conservation of energy to calculate the heat capacity of the calorimeter.

Mass of hot water

Mass of cold water

Thot

Tcold

Tfinal

#1 #2 #3

Part II: Determination of Heat of a Reaction 1. Obtain and wear goggles. 2. Connect a Temperature Probe to Channel 1 of the Vernier computer interface. Connect the interface to the computer with the proper cable. 3. Nest a Styrofoam cup in a 400 mL beaker as shown in Figure 1. Pour 100 mL of 1.0 M HCl into the Styrofoam cup. Keep stirring this and adjust the stirring rate to vigorous but without splashing. Weigh 0.25 g of magnesium turnings on an analytical balance.

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4. Start the Logger Pro program on your computer. 5. Use a utility clamp to suspend the Temperature Probe from a ring stand (see Figure 1). Lower the Temperature Probe into the reaction mixture. 6. Conduct the experiment

a. Click to begin the data collection and obtain the initial temperature of the HCl solution. b. After you have recorded three or four readings at the same temperature, quickly add the magnesium turnings. Use a glass stirring rod to stir the reaction mixture gently and thoroughly. c. Data may be collected for 5 minutes. You may terminate the trial early by clicking , if the temperature readings are no longer changing. As the reaction occurs, you should observe the temperature climb. Continue taking data until a final, constant temperature plateau is well established. The experiment can be stopped at this point. d. Click the Statistics button, . The minimum and maximum temperatures are listed in the statistics box on the graph. If the lowest temperature is not a suitable initial temperature, examine the graph and determine the initial temperature. e. Record the initial and maximum temperatures, in your data table, for Trial 1. 7. Rinse and dry the Temperature Probe, Styrofoam cup, and stirring rod. Dispose of the solution as directed. 8. Repeat Steps 3 and 5-7 to conduct a second trial. If directed, conduct a third trial. Print a copy of the graph of the second trial to include with your data and analysis. Sample mass

Tinitial

Tfinal

#1 #2 #3

IV. Calculations Calculate the heat capacity of the calorimeter by using eqn. (5). Include a sample calculation in your report.

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Trial

#1

Trial #2

Trial

#3

Average Uncertainty

Ccalorimeter Use the average heat capacity of the calorimeter and eqn.(3b) to calculate the enthalpy change of the reaction, then divide this number by the number of mols of magnesium. The result is the molar enthalpy change for the reaction with respect to consumption of magnesium. Include a sample calculation in your report. Trial

#1

Trial #2

Trial

#3

ΔHreaction ΔH reaction =

ΔHreaction nMg

V. Conclusion: Refer to the lab write-up procedure.

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Average Uncertainty

Experiment 4 Enthalpy and Entropy Objectives (a) to use an electrochemical cell for determining ΔG (the change in Gibbs energy), ΔS (the change in entropy) and ΔH (the change in enthalpy), for the reaction of solid zinc with aqueous copper sulfate. (b) to determine ΔH for the same reaction as in (a), namely, the reaction of solid zinc with aqueous copper sulfate, but this time with the use of calorimetric methods. (c) to compare enthalpy values ΔH obtained by the two methods in (a) and (b). (d) to use ΔG for the reaction of solid zinc with aqueous copper sulfate to discuss the criteria for deciding whether a reaction is spontaneous or at equilibrium.

I. Introduction Thermodynamics is concerned with average energy changes and entropy changes of macroscopic physical systems. Changes of a system referred to enthalpy energy change ΔHT , entropy change ΔST and Gibbs free energy change ΔGT are related according to the equation ΔGT = ΔHT - TΔST

(1a)

where the subscript T implies eqn.(1) is valid only at constant T. Another thermodynamic equation for ΔGT is

ΔGT = −nFεT where

(1b)

εT is the cell voltage (the electromotive force = emf) for an electrochemical

reaction at temperature T, e.g., reaction (2a). Zn(s) + CuSO4(aq) → ZnSO4(aq) + Cu(s)

(2a)

This experiment first uses electrochemical methods to determine ΔGT , ΔST and ΔHT ; then, calorimetric methods are used to determine, once again, ΔHrxn . The results for ΔHT and , ΔHrxn obtained by these two methods are compared. Materials: Vernier computer interface computer Temperature Probe Styrofoam cup calorimeter two 250 mL beakers Glass stirring rod

Copper sulfate solution Zinc sulfate solution two 50 mL graduated cylinders ring stand utility clamp Potassium nitrate solution 22

II. Theory The electrochemical method: The electrochemical method offers simple and accurate means for the determination of thermodynamic quantities. Consider the electrochemical cell Cu(s)/CuSO4(aq) || Zn(s)/ZnSO4(aq) Cell voltage measurements are made in a Chem-Carrou-Cell, as will be shown and described by your instructor. The overall voltaic cell reaction incated by the symbols above is Zn(s) + Cu2+(aq) → Zn2+(aq) + Cu(s)

(2b)

The half-cell reactions are Oxidation (at anode) :

Zn(s) → Zn2+(aq) + 2 e-1

.........

+0.76 V

Reduction (at cathode):

Cu2+(aq) + 2 e-1 → Cu(s)

.........

+0.34 V

Overall reaction:

Zn(s) + Cu2+(aq) → Zn2+(aq) + Cu(s) ...

Note that the procedure for calculating the cell voltage reaction is simply to (a) write one half-cell reaction as an associated half-cell voltage, (b) write the other half-cell voltage and (c) add the reactants, products and half-cell only rule.

1.10 V

for an oxidation-reduction oxidation reaction with the reaction with its’ half-cell voltages. Addition is the

The quantity of electrical energy produced or consumed during a electrochemical reaction can be measured accurately by means of electrochemical cells, where the output measured is the cell voltage ε; this number is related to the energy change ΔG by the following equation:

ΔGT = −nFεT

(3)

where n = the number of moles of electrons transferred in a redox reaction.

F = Faraday’s constant of = 96,485 C /mole of electrons. Equating equations (1) and (3), then dividing both sides by nF, gives a linear relationship between the voltage ε and the temperature T, as given by

εT = −

ΔHT TΔST + nF nF

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(4a)

or

⎛ ΔS ⎞ ΔHT εT = ⎜⎜⎜ T ⎟⎟⎟T − ⎜⎝ nF ⎟⎠ nF

(4b)

Assuming Δ H and Δ S are independent of temperature over the temperature range Δ T of the experiment, then

⎛ ΔS ⎞⎟ ΔH ⎟⎟T − εT = ⎜⎜ ⎜⎝ nF ⎟⎠ nF

(4c)

To determine Δ S and Δ H , eqn.(4c) shows that if ε is plotted versus the absolute temperature T, the resulting graph is of the form of a straight line y = mx + b, where m is the slope and b is the y – intercept.

ΔS nF

(5a)

−ΔH nF

(5b)

Slope= y intercept = Hence,

(a) Calculation of Δ S : Measurement of slope yields Δ S ,since Δ S = nF (slope). (b) Calculation of Δ H : Measurement of the y – intercept yields Δ H , since Δ H = - nF (y intercept). (c) Calculation of Δ GT : ΔGT values at different temperatures T are calculated by inserting measured values of the cell voltage εT at T into eqn.(3), since D GT = -­‐ nF eT . Calculate ΔG = <ΔGT > = the average value of ΔGT , and compare to ΔG calculated from ΔG = ΔH - T ΔS (use average T ). (d) Calculation of equilibrium constant K : In contrast to experimental values of ε , values of ε can be calculated with the use of the Nernst equation, given by ν

ν

C D RT RT ⎡⎣⎢C ⎤⎦⎥ ⎡⎣⎢D ⎤⎦⎥ 0 0 ε =ε − lnQ = ε − ln nF nF ⎡ A⎤ νA ⎡B ⎤ νB ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦

(6a)

where Q is the reaction quotient for the prototype reaction given below, and ε 0 is the standard potential measured at 250C and 1 atm. 24

nA A + nBB → nCC + nD D For reaction (2b), eqn. (6a) is 1

⎡ 2+ ⎤ RT RT ⎣⎢ Zn ⎦⎥ 0 0 ε =ε − lnQ = ε − ln nF nF ⎡Cu 2+ ⎤1 ⎢⎣ ⎥⎦

(6b)

At thermodynamic equilibrium ΔG = 0 and Q = Kexp ; hence, eqn. (3) implies ε =0 at thermodynamic equilibrium. Therefore, for a reaction at thermodynamic equilibrium,

nA A + nBB ! nCC + nD D ν

ν

⎡ ⎤C⎡ ⎤D RT RT ⎢⎣C ⎥⎦ eq ⎢⎣D ⎥⎦ eq 0 ε = lnK = ln nF nF ⎡ A⎤ νA ⎡B ⎤ νB ⎢⎣ ⎥⎦ eq ⎢⎣ ⎥⎦ eq

(7)

Eqn, (7) implies that the equilibrium constant K can be determined by measurements of the standard cell voltage for the reaction, then using eqn.(7) in the form

K = exp (nFε 0/RT ) 0

Note that ε is obtained experimentally by using an experimental arrangement where [Zn2+] = [Cu2+], implying the logarithmic term of the Nernst eqn.(6b) is zero. Under these conditions, the standard voltage of the cell is equal to 0

the measured voltage ε , i.e., ε = ε . The calorimetric method: Although calorimeters can be constructed to operate at constant pressure or constant volume, constant pressure calorimeters are most convenient since they can operate at the constant pressure of the atmosphere. A constant pressure calorimeter is a perfectly insulated vessel which contains a large known mass of calorimeter solution in perfect thermal contact with an accurate thermometer and a small reaction tube (Figure 1). When measured quantities of reactants are introduced into the reaction tube, the energy emitted by the reaction changes the temperature of the calorimeter solution. Enthalpy changes are then calculated by inserting these temperature changes into certain fundamental equations of thermodynamics. The most important equation is the first law of thermodynamics, which states that, for constant pressure calorimeters,

25

ΔHrxn +ΔHcalorimeter + ΔHcalorimeter = 0

(8)

solution

Since at constant pressure P,

ΔHP = nC P (Tfinal −Tinitial ) ! P T −T = mC ( final initial )

, constant P , constant P

(9)

! P and where nC P = mC

n = moles C P = heat capacity in units of Jmol −1K −1 m = mass in units of grams, ! P = heat capacity in units of Jg −1K −1 C Then eqn. (8) becomes,

ΔHrxn = − ΔHcalorimeter − ΔHcalorimeter solution

= −ncalorimeter C P , calorimeter (Tfinal −Tinitial ) −Ccalorimeter (Tfinal −Tinitial ) solution, solution

where

ncalorimeter = number of moles of solution in the calorimeter solution

C P , calorimeter = heat capacity of the solution in the calorimeter solution, Ccalorimeter = heat capacity of the calorimeter Tfinal = final temperature (K ) of the solution in the calorimeter Tinitial = initial temperature (K ) of the solution in the calorimeter For this experiment the relevant heat capacities are

! P , calorimeter = m C 3.8Jg −1K −1) solution ( solution ! P, calorimeter " 30JK −1 C Hence, eqn. (10) becomes

26

(10)

ΔHrxn = −ncalorimeter C P , calorimeter (Tfinal −Tinitial ) −Ccalorimeter (Tfinal −Tinitial ) solution, solution

! P , calorimeter T −T = −mcalorimeter C ( final initial ) −Ccalorimeter (Tfinal −Tinitial ) solution

(11)

solution

= − ⎡⎢msolution (3.8Jg −1K −1) + 30JK −1 ⎤⎥ (Tfinal −Tinitial ) ⎦ ⎣

III. Experiment In Part I of this exercise, you will be measuring voltages and temperature, displaying them on the screen, and sending stable voltage readings and the corresponding temperatures to the spreadsheet. In part II, you will be collecting temperature readings as a function of time. This program should have an on-screen graphing temperature range between 150C and 300C and on-screen graphing time running from 0 to 60 min. Part I: Electrochemistry

Figure 1 1.

Calibrate your temperature sensor with ice-cold and hot tap water.

2. Fill a 600 mL beaker three-quarters full of ice. Fill the rest of the beaker with cold tap water until the beaker is filled to one cm from the top. Stir the ice/water mixture to distribute the ice evenly. Using a rubber band, assemble three vials to form a Chem-Carrou cell. 3. Place the beaker on a hot-plate – don’t turn the heat on yet! Rest the ChemCarrou-Cell on top of the beaker so that the bottoms of the wells of the cell are immersed in the ice/water mixture. Let the set-up stand for 15 min until the temperature stabilizes. 4. Pour about 5 mL of 0.1 KNO3 into the center well. Pour enough of 0.5 M ZnSO4 into well #1 to touch the salt bridge strip (soaked with KNO3). Into well #2, pour enough of 0.50 M CuSO4 to touch the salt bridge strip (soaked with KNO3). 27

5. With clean tweezers, take a strip of filter paper and dip one end into the central well (where immersion in the KNO3 solution will hold one end); dip the other end into well #1. Repeat this procedure with another strip of filter paper, dipping the other end into well #2. This creates the salt bridge for your galvanic cell. 6. With clean tweezers take a zinc metal strip and sand it to remove any oxide coating. Bend 2 cm of one end of the strop and immerse it in the ZnSO4 solution (well #1). The rest of the metal strip (3 cm) extends out to the edge of the cell and should be bent over the rim. Repeat the same procedure with the copper metal strip and place it in well #2. Later the electrical leads (alligator clips) from the interface will be attached to the metal strips. 7. Fasten your temperature sensor to the ring stand with a clamp and adjust it such that its tip can be immersed in the central well (KNO3 solution) of the cell. It is assumed that the temperature of the ZnSO4 solution and CuSO4 solution well be very close to the temperature of the KNO3 solution throughout the experiment. 8.

Start the computer program to monitor the temperature.

9. Read and record the temperature. Attach the alligator clips to the metal strips and take the voltage reading. If a negative number appears on the screen, reverse the wires. Disconnect the wires immediately after the reading is recorded. 10. Turn on the hot plate to high and take voltage and temperature readings every 100C up to 400C. Connect the wires only while actually reading the voltage. Prolonged connection of the wires will cause the electric current to flow through the cell. The discharge will result in changes in concentrations of the solutions. Since the measured voltage depends on these concentrations, there will be an error in the voltage readings. 11.

28

[CuSO4] ---

Volume --[KNO3]

---

Volume --Temperature T Cell Voltage ε Part II: Calorimetry: 1. Weigh a clean and dry 150 mL beaker on the top loading balance, then add 50 mL of 0.5 M CuSO4 solution. Weigh the beaker a second time to determine the mass of the solution in the calorimeter. Nest the 150 mL beaker inside a 400 mL and 600 mL beaker, thread the stirring rod and temperature sensor through the Styrofoam lid and position the lid on the inner beaker. The temperature sensor should be as deep into the solution as possible but without strain. Let the apparatus stand so that the components attain the same temperature. 2. Using a top loading balance, weigh 0.5 g of zinc powder into a plastic weighing boat, then weigh the boat plus its contents on the analytical balance.

2. Start the Logger Pro program on your computer. 3. Conduct the experiment. a)

Click to begin the data collection and obtain the initial temperature of the copper sulfate solution.

b)

After you have recorded three or four readings at the same temperature, quickly add the zinc powder. Use a glass stirring rod to stir the reaction mixture gently and thoroughly. Save the boat and any Zn stuck to it for weighing at the end of run. Stir thoroughly and plot temperature vs time until a well defined cooling trend is established. It is important that a rubber policeman be used continually to ensure that any residue produced (copper coated zinc powder) is broken up.

29

c)

Data may be collected for 5 minutes. You may terminate the trial early by clicking , if the temperature readings are no longer changing. As the reaction occurs, you should observe the temperature climb. Continue taking data until a final, constant temperature plateau is well established. The experiment can be stopped at this point.

d)

Click the Statistics button, . The minimum and maximum temperatures are listed in the statistics box on the graph. If the lowest temperature is not a suitable initial temperature, examine the graph and determine the initial temperature. Record the initial and maximum temperatures, in your data table, for Trial 1.

e)

4. Weigh the weighing boat on the analytical balance and obtain the amount of Zn added by difference. 5. Use the spreadsheet file to plot and print the graph of your data. 6. From your plot of temperature vs time, determine the initial and final temperatures and calculate the heat of reaction per mole of zinc. Compare this value with ΔH obtained in Part I.

#1

#2

Mass (beaker) Mass (beaker + CuSO4) Mass (CuSO4 )

Volume (CuSO4 ) Mass (boat) Mass (boat + Zn) Mass (Zn)

30

#3

#4

IV. Calculations Recall the equations

⎛ ΔS ⎞ ΔHT ε = ⎜⎜⎜ T ⎟⎟⎟T − ⎜⎝ nF ⎟⎠ nF ΔST ΔHT ; y intercept = − nF nF ΔGT = ΔHT −TΔST Slope =

Plot the ε vs T data and calculate ΔS from the slope and ΔH from the y – intercept. Calculate ΔGT from cell voltage measurements εT and average to get ΔG = <ΔGT> ; compare to ΔG from ΔG = ΔH - ΔS , where is the average temperature. Finally calculate ΔHrxn from the equation

ΔHrxn = − ⎡⎢msolution (3.8Jg −1K −1) + 30JK −1 ⎤⎥ (Tfinal −Tinitial ) ⎣ ⎦ ΔS = (nF )(slope) ΔH = −nF ( y intercept) ΔG = ΔGT ΔG = ΔH − T ΔS

ΔHrxn = − ⎡⎢msolution (3.8Jg −1K −1) + 30JK −1 ⎤⎥ (Tfinal −Tinitial ) ⎣ ⎦

V. Conclusion: Refer to the lab write-up procedure.

31

Experiment 5 Rate and Order of a Chemical Reaction Objectives

(a) to learn, to define, and understand terms such as rate, order, mechanism, activation energy, activated complex, etc., which are often encountered in kinetics. (b) to plan and design an experiment to determine parameters of the kinetics of a reaction. (c) to calculate the specific rate constant and the rate law. (d) to gain experience with the oscillation between experiment and theory.

Background material

In chemical kinetics, experiments are performed to determine the rate (speed), order and mechanism of chemical reactions, and the results are compared to mathematical models of these reactions. The interaction between experiments of this type and mathematical models of the rate, order and mechanism of a chemical reaction defines the study of chemical kinetics. When comparing this discipline to thermodynamics, it is noted that the second law of thermodynamics provides a criteria for determining whether reactions are spontaneous or not, but the law does not provide information about the speed, order and mechanisms of reactions. Chemical kinetics provides this information. To define reaction rate, order and mechanism, consider the reaction 2A + B → P. Several symbols are used for an expression of the speed (magnitude of the velocity) or rate for this reaction, as given by the following equation:

⎛ −1 ⎞ d [ A] −d [B ] +d [P ] Rate = ⎜ ⎟ = = dt dt ⎝ 2 ⎠ dt

(1)

−1 d [ A] / dt is the rate of consumption of reactant A , −d [B] / dt is the 2 rate of consumption of reactant B, and +d [P ] / dt is the rate of formation of product The quantity

±1 d [ X ] / dt , where nX is the nX stoichiometric coefficient in the balanced chemical equation. In order to make certain all rates have a positive sign, a negative sign is attached to rates of consumption since d [ A] < 0 and d [B] < 0 ; however, a positive sign is attached to the rate of formation

P. In general, the rate of any component X is given by

since d [P ] > 0 . Either symbol for the rate can be used, but for some computations, one

may be more convenient. Experiments in kinetics involve measuring changes in A, B

32

and P as functions of time, e.g., changes in concentration, partial pressure, pH , conductance, or absorbance of electromagnetic radiation as a function of time. Suppose experimentation on the reaction 2A + B → P has led to the following equation for describing the average reaction rate:

⎛ −1 ⎞ d [ A ] = k [ A ][B ] ⎜ ⎟ ⎝ 2 ⎠ dt

(2)

where the rate of consumption of reactant A is expressed as a constant k times the product of the concentrations of reactants A and B; k is called the specific rate constant and eqn.(2) is called the rate law for the reaction. On the right side of eqn.(2), since [A] is raised to the first power, then the reaction is called first-order in A; similarly, the reactant is first-order in B since [B] is raised to the first power. Addition of the exponents gives the number 2, and so the reaction is said to be second-order overall. According to chemical kinetics, the fact that the exponents in the rate law are one, implies that the slowest step in the reaction is (3) where (AB)‡ is called the activated complex. In general, the exponents of the reactant concentrations in the rate law are identical to the coefficients of the reactants for the slowest reaction step. This slowest step is referred to as the rate-determining step because the overall reaction cannot move faster than the slowest step. A rate-determining step is called unimolecular if one reactant particle is present, bimolecular if two reactant particles are present, and termolecular if three reactant particles are present, etc. This concept of molecularity is used for the description of all elementary steps in a multi-step reaction, including the rate-determining step. The rate determining reaction (4) is one of the reactions referred to as the mechanism for the overall reaction 2A + B → P. To illustrate this concept assume that, not only is the reaction first-order in A and first-order in B, but that it is also a reaction which occurs in two steps. It is emphasized that information on the rate, order and mechanism of any chemical reaction is obtained only from experiments. Moving forward, note that the remaining reaction in this hypothetical mechanism is obtained by subtracting the rate-determining step from the overall reaction, as illustrated in the following development:

33

I. Introduction To illustrate basic principles in chemical kinetics, the reaction of solid calcium carbonate with the strong acid HCl, is investigated. This is a familiar reaction observed in everyday life (over time), since many historic monuments are constructed from marble (a form of calcium carbonate) which slowly reacts with the acid content in rain, and decomposes according to the reaction

CaCO3(s) + 2H+(aq)

Ca2+(aq) + CO2(g) + H2O(liq)

(5)

Using a gas pressure probe, the pressure of carbon dioxide is measured as a function of time, allowing computation of the specific rate constant for this reaction. Using the specific rate constant and the Arrhenius equation, the activation energy is calculated.

II. Theory In this experiment the kinetics of reaction #5 is followed by setting the number of moles of one of the reactants (H+) to be very large in comparison to the number of moles of the other reactant (CaCO3), a kinetics technique called the isolation method; it implies that some information about the kinetics of a reaction can be obtained by following the exceptionally small component (CaCO3 in this case). The order obtained for CaCO3, using the isolation method is called the “pseudo-order’” for CaCO3,. Following this process, the isolation method is then employed for the remaining component H+ by using a large number of moles of CaCO3. in comparison to the number of moles of H+. If the rate law is pseudo x-order in CaCO3 and pseudo y-order in H+, then this data implies that the reaction is pseudo ( x + y ) - order overall. Hence, the overall rate law is of the form

−dnCaCO3 dt

x = knCaCO nHy + 3

(6a)

Note that nA is used rather than [A] because calcium carbonate is a solid. For this experiment it is only required to find the specific rate constant k, since the value x ( = 1) is given and the number of moles of CaCO3 is exceptionally small; hence, the time dependence of [H+] is ignored. According to the stoichiometry of

34

reaction (5) the number of moles of CaCO3 consumed is the number of moles of CO2 formed; hence, experimental measurements of the pressure of CO2(g) as a function of time can be used to study the kinetics of reaction (5). With these considerations, the following differential form of the pseudo-rate law is the correct model:

−dnCaCO3 dt

= knCaCO3

(6b)

where nCaCO3 = number of moles of calcium carbonate. The integrated form of the rate law is

⎛ nCaCO ln ⎜ 0 3 ⎜ nCaCO 3 ⎝

⎞ ⎟ = −kt ⎟ ⎠

final ⎛ nCO − nCO2 2 ln ⎜ final ⎜ nCO2 ⎝

⇒

⎞ ⎟ = −kt ⎟ ⎠

(7)

where the stoichiometry of reaction (5) is given by

2H + ( aq )

CaCO3 (s )

+

0 nCaCO3 = nCaCO −x ; 3

nH + = nH0 + − 2 x

→ Ca 2+ (aq ) + CO2 (g ) + H2O ( liq ) nCa2+ = x ;

nCO2 = x ; nH2O = x

0 = nCaCO − nCO2 3 final = nCO − nCO2 2

The substitution in eqn. (7) is valid since at the end of the reaction, i.e., when t is very large, the stoichiometry of the reaction predicts that 0 0 final 0 nCaCO3 = nCaCO − x final = 0 , hence x final = nCaCO and nCO = x final = nCaCO 3 3 2 3

Using the ideal gas equation (PV = nRT) in eqn.(7) gives

⎛ n − nCO2 ln ⎜ final ⎜ nCO 2 ⎝ final CO2

⎞ ⎟ = −kt ⎟ ⎠

⇒

Hence, final ⎛ PCO − PCO2 2 ln ⎜ final ⎜ PCO 2 ⎝

35

final ⎛ PCO V PCO2V 2 − ⎜ RT RT ln ⎜ final ⎜ PCO2 V ⎜ RT ⎝

⎞ ⎟ = −k t ⎟ ⎠

⎞ ⎟ ⎟ = −kt ⎟ ⎟ ⎠ (8)

Focusing on eqn.(8), it is noted that it is the equation of a straight line (y = mx +b); final ⎛ PCO − PCO2 ⎞ 2 hence a plot of ln ⎜ ⎟ versus t is predicted to give a straight line with a slope final ⎜ ⎟ P CO2 ⎝ ⎠ –k. In this manner the specific rate constant k for the reaction is determined. Once the rate constant k is known, it is possible to calculate an approximate value of the activation energy Ea by use of the Arrhenius equation, given by

k = A exp (-Ea/RT)

(9)

where A is a constant called the pre-exponential constant, Ea is the activation energy, T is the absolute temperature and R is the gas constant. Materials: analytical balance Vernier computer interface Vernier Gas Pressure Sensor thermometer rubber stopper

125 mL Erlenmayer flask 1 liter beaker ~800 mL water tubing

III. Experiment 1. Set up the experiment as shown below.

Figure 1 2. Weigh a known quantity (0.01 – 0.05 g) of calcium carbonate and place it in the flask.

36

3. Place the flask in a water bath and connect a Gas Pressure sensor to Channel 1 of the Vernier Computer interface. Connect the interface to the computer using the proper cable. 4. Through a second hole in the stopper, add 1.5 mL of 1.0 – 2.0 M hydrochloric acid. 5. Start the Logger Pro program on your computer. 6. Prepare to run the reaction and collect pressure data. 7. Gently shake the flask and click data till the pressure does not change.

to begin the data collection. Gather

8. Repeat the experiment at least three more times with different amounts of calcium carbonate. Note that the number of moles of hydrogen ion is held large and constant, according to the use of the isolation method. 9. The data should contain pressure-time graphs for four different amounts of calcium carbonate used and the temperature at which the reaction was performed. 10. Include in this section all graphs of raw data, and construct a table to display raw data and experimental uncertainties.

IV. Calculations

Using theoretical considerations discussed in the theory, plot final ⎛ PCO − PCO2 2 ln ⎜ final ⎜ PCO2 ⎝

⎞ ⎟ vs. t ⎟ ⎠

and determine the rate constant k from the slope. Calculate the activation energy Ea from the Arrhenius equation. Tabulate values of k, Ea and the uncertainties in these values, in a table. Assuming reaction (5) is pseudo first-order with respect to hydrogen ion concentration, write the pseudo-rate law expression for the reaction.

V. Conclusion

Refer to the lab write-up procedure. Reference: Adapted from Choi, Martin, M.F.; Wong, P.S., Experiment titled “Using a Datalooger To Determine First-Order Kinetics and Calcium Carbonate in Egg Shells”, J. Chem. Ed. 81, 859, 2004.

37

Experiment 6 Determination of an Equilibrium Constant Objectives a) to experimentally investigate an equilibrium reaction and to determine the equilibrium constant of a reaction. b) to learn to define and use terms such as chemical equilibrium, equilibrium constant. (c) to plan and design an experiment to determine the equilibrium constant, (d) to gain experience with the oscillation between experiment and theory.

I. Introduction In this experiment the absorbance of light by FeSCN 2+ is measured with a Colorimeter when FeSCN 2+ is produced in the following equilibrium chemical reaction:

Fe 3+ (aq) + light brown

SCN − (aq) colorless

!

FeSCN 2+ (aq) dark red

(1)

Measurements of several (A, C ) pairs, where A is the Absorbance and C is the concentration of FeSCN 2+ , are used to construct a calibration curve of Absorbance versus known concentrations of FeSCN 2+ . This data is used • to calculate the equilibrium constant of reaction (1) and • to calculate the concentration of FeSCN 2+ from Absorbance measurements in an unknown reaction equilibrium. In addition, qualitative studies are conducted for the effect of adding certain chemicals to reaction (1). When a chemical reaction proceeds to completion, it can be assumed that the reactants are quantitatively converted to products. However, there are many chemical reactions that stop far short of completion. In these situations, it is concluded that although the concentrations of the reactants and products changed as a function of time, after a certain length of time no change in the concentrations of either the reactant or the product is detected. Once this state is attained known, as the “chemical equilibrium state”, an equilibrium constant can be calculated. Obviously, reactive collisions of atoms and molecules which go to completion have an infinite equilibrium constant, while for non-reactive collisions of atoms and molecules the equilibrium constant is zero; there are a range of possibilities between these two extremes. From the reaction kinetics point of view, the equilibrium position can be understood as a point in time where the rate of the forward reaction is exactly equal to the rate of the reverse reaction. Consider a chemical reaction of the type, 38

nA A + nB B ! nCC + nD D where A, B, C, and D represent atoms, molecules and/or ions and the ni are the coefficients in the balanced chemical equation. The equilibrium constant K is represented by the following equation: nC

nD

A

B

[C ] [D ] K = n n [ A ] [B ]

where the square brackets indicate the concentrations of chemical species at equilibrium. Materials: Vernier computer interface

0.200 M Fe(NO3)3 solution in 1.0 M HNO3 solution 0.0020 M Fe(NO3)3 solution in 1.0 M HNO3 solution 0.0020 M SCN– solution in 0.10 M HNO3 KSCN solution of unknown concentration in 0.10 M HNO3 solution eight 100 mL beakers test tube rack plastic Beral pipets 0.1 M AgNO3, 1M Na3PO4, solid NaF 0.1 M Hg(NO3)2, 0.1 M Na2C2O4 solution

computer Vernier Colorimeter Temperature Probe (optional) plastic cuvette four 10.0 mL pipettes pipet pump or bulb six 20 × 150 mm test tubes 50 mL volumetric flask Concentrated HCl

II. Theory Standard solutions; Concentrations of reactants at time t = 0: Given a mixture of an aqueous solution of A, an aqueous solution of B, and water, the resulting initial concentrations [A]0 and [B]0 are given by

[ A]0 =

nA Vsolution

;

nA and VA + VB + VH2O

[B ]0 =

nB Vsolution

;

nB VA + VB + VH2O

(2)

Colorimeter, Transmittance T, Absorbance A: When light of intensity I0 travels a path length b through a cuvette containing a liquid solution of concentration C, Beer’s law predicts that the light is attenuated, with the fraction of light transmitted as I/I0 . According to Beer’s law,

⎛ I ⎞⎟ ⎜ log ⎜⎜ ⎟⎟⎟ α ⎜⎝ I ⎟⎠ 0

(

−b C

)

so

⎛ I ⎞⎟ I ⎜ log ⎜⎜ ⎟⎟⎟ = −ε b C and = 10−ε b C ⎜⎝ I ⎟⎠ I0 0

39

(3)

where the proportionality sign α is replaced by an equal sign and a constant ε, and where the constant ε is called the molar absorptivity of the solution. The quantity I/I0 is called the Transmittance T = I/I0 ; the Absorbance is defined by A = log (1/T ). It follows that,

⎛ I ⎞⎟ ⎜ A = log 1 / T = log ⎜⎜ 0 ⎟⎟⎟ = log 10+ε b C ⎜⎝ I ⎟⎠

(

(

)

)

(4)

Hence, the absorbance is

A = εb C

(Beer’s law )

(5)

In this experiment, the stoichiometry of the following reaction is monitored by measuring the absorbance of electromagnetic radiation (light in this case) by FeSCN 2+ at different concentrations of FeSCN 2+ .

Fe 3+ (aq) + SCN − (aq) ! FeSCN 2+ (aq) C−x

x

x

(6)

3+

Note that since Fe is a light brown color, SCN-­‐ is colorless and FeSCN 2+ is a dark red color, the most intense absorption lines are due to FeSCN 2+ . The procedure requires an initial concentration of Fe3+ which is very much greater than the concentration of SCN – ; SCN – is the limiting reactant. From the stoichiometry,

x = ⎡⎢FeSCN 2+ ⎤⎥ = ⎡⎢ SCN − ⎤⎥ ⎣ ⎦ ⎣ ⎦ A plot of Absorbance versus [FeSCN positive slope, for example,

2+

]

is expected to yield a straight line with a

40

This curve can serve as a calibration curve to experimentally determine the concentration of an unknown solution of the same reaction components. It is only required to measure the Absorbance of FeSCN 2+ in the equilibrium reaction of the unknown KSCN solution, then use the fitted curve A = m C + b to calculate C of the unknown KSCN solution.

III. Experiment Part I: Qualitative Into a clean 400 mL beaker add 250 mL of distilled water, 1 mL of 1 M KSCN and 1 mL of 1 M Fe(NO3)3 solution. This stock solution will have an intense red color due to the formation of the complex FeSCN2+. Obtain 10 clean medium-sized test tubes and label them 1 through 9. Add 10 mL of this stock solution into each one of the test tubes. Keep test tube 1 as a control and add the following reagents into test tubes 2 through 9 respectively: 2 3 4 5 6 7 8

2 mL 1 M Fe(NO3)3 solution 1 mL 1 M KSCN solution 0.5 mL (10 drops) 0.1 M AgNO3 solution 2 mL Conc. HCl solution 1 mL 1 M Na3PO4 solution 1 mL 0.1 M Na2C2O4 solution several crystals of solid NaF

Cover each test tube with paraffin film and shake to mix. Compare the color intensity of each of the test tubes with that of the control test tube 1. In each test record your observations and make a table. Based on your observations, suggest a way to determine the concentrations of the reactants and products at equilibrium. Part II: Prepare and Test Standard Solutions 1. Obtain and wear goggles. 7. Label five 100 mL beakers (or other glassware) 1-5. Obtain 25 mL of 0.200M Fe(NO3)3, 30 mL of 0.0020 M KSCN, and 220 mL of distilled water. CAUTION: Fe(NO3)3 solutions in this experiment are prepared in 1.0 M HNO3 and should be handled with care. Prepare four solutions according to the chart below. Use a 10.0 mL pipet and a pipet pump or bulb to transfer each solution to a 50 mL volumetric flask. Mix each solution thoroughly. Measure and record the temperature of one of the above solutions to use as the temperature for the equilibrium constant, Keq. Complete the following table before proceeding: Beaker number 1 2 3 4 5

0.200 M Fe(NO3)3 (mL)

0.0020 M SCN– (mL)

H 2O (mL)

5.0 5.0 5.0 5.0 5.0

0.0 2.0 3.0 4.0 5.0

45.0 43.0 42.0 41.0 40.0

41

Initial Conc. Of Fe(NO3)3

Initial Conc. Of KSCN

3. Connect a Colorimeter to Channel 1 of the Vernier computer interface. Connect the interface to the computer with the proper cable. 4. Start the Logger Pro program on your computer. 5. Calibrate the Colorimeter. a. Prepare a blank by filling an empty cuvette ¾ full with distilled water. Place the blank in the cuvette slot of the Colorimeter and close the lid. b. Click on EXPT, then set the wavelength on the Colorimeter to 470 nm, press the CAL button, and proceed directly to Step 6. If your Colorimeter does not have a CAL button, continue with this step to calibrate your Colorimeter. c. Choose Calibrate } CH1: Colorimeter (%T) from the Experiment menu, then click “one point calibration.” . d. Turn the wavelength knob on the Colorimeter to the “0% T” position. e. Type “0” in the edit box. f. When the displayed voltage reading for Input 1 stabilizes, click . g. Turn the knob of the Colorimeter to the Blue LED position (470 nm). h. Type “100” in the edit box. i. When the voltage reading for Input 1 stabilizes, click , then click . 6. You are now ready to collect absorbance data for the standard solutions. Click to begin data collection. Note: Take readings within 4 minutes of preparing the mixtures. a. Empty the solution from the cuvette. Using the solution in Beaker 1, rinse the cuvette twice with ~1 mL amounts and then fill it ¾ full. Wipe the outside with a tissue, place it in the Colorimeter, and close the lid. Wait for the absorbance value displayed in the Meter window to stabilize. Click , type the concentration of FeSCN2+(obtain from table) in the edit box, and press the ENTER key b. Discard the cuvette contents as directed. Rinse and fill the cuvette with the solution in Beaker 3. Follow the procedure in Part a of this step to measure the absorbance, and enter the concentration of this solution. c. Repeat Part b of this step to measure the absorbance of the solutions in Beakers 4 and 5. d. Click when you have finished collecting data to view a graph of absorbance vs. concentration. Click the examine button, , and record the absorbance values for each data pair. 7. Click the Linear Fit button, . A best-fit linear regression line will be shown for your five data points. This line should pass near or through the data points and the origin of the graph. (Note: Another option is to choose Curve Fit from the Analyze menu, and then select Proportional. The Proportional fit has a y-intercept value equal to 0; therefore, this regression line will always pass through the origin of the graph). Leave the graph and best fit line displayed and proceed to Step 8. Part III: Test an Unknown Solution of KSCN 8. Obtain about 10 mL of the unknown KSCN solution. Use a pipet to measure out 5.0 mL of the unknown into a clean and dry 100 mL beaker. Add precisely 5.0 mL of

42

0.200 M Fe(NO3)3 and 40.0 mL of distilled water to the beaker. Stir the mixture thoroughly. 9. Using the solution in the beaker, rinse a cuvette twice with ~1 mL amounts and then fill it ¾ full. Wipe the outside with a tissue, place it in the Colorimeter, and close the lid. Watch the absorbance readings in the Meter window. When the readings stabilize, record the absorbance value for your unknown in your data table. Remove and clean the cuvette. 10. Determine the concentration of the SCN– in unknown solution. j. With the linear-regression curve still displayed on your graph, choose Interpolate from the Analyze menu. k. A vertical cursor now appears on the graph. The cursor’s concentration and absorbance coordinates are displayed in the floating box. l. Move the cursor along the regression line until the absorbance value is approximately the same as the absorbance value you recorded in Step 9. The corresponding concentration value is the concentration of the unknown solution, in mol/L. Record this value in your data table. Part IV: Prepare and Test Equilibrium Systems

11. Prepare five 50 mL Erlenmeyer flasks of solutions, according to the chart below. Follow the necessary steps from Part I to test the absorbance values of each mixture. Record the test results in your data table. Note: You are using 0.0020 M Fe(NO3)3 in this test.

Erlenmeyer Flask number

0.0020 M Fe(NO3)3 (mL)

0.0020 M SCN– (mL)

H2 O (mL)

1 2 3 4 5

3.00 3.00 3.00 3.00 3.00

0.00 2.00 3.00 4.00 5.00

7.00 5.00 4.00 3.00 2.00

12. To get suitable data for the calculation of K, determine the net absorbance of the solutions in Test Tubes 2-5. To do this, subtract the absorbance reading for Test Tube 1 from the absorbance readings of Test Tubes 2-5, and record these values as net absorbance in your data table. This section should contain the table from part I, the graphs from parts II, and III, and observations from part IV.

43

IV. Calculations Concentrations: As an example for calculating the initial concentrations of Fe(NO3)3 and KSCN, consider the following portion of the chart in Part II-2: Beaker number

0.200 M Fe(NO3)3 (mL)

0.0020 M SCN– (mL)

H 2O (mL)

Initial Conc. Of Fe(NO3)3

Initial Conc. Of KSCN

5.0 5.0

0.0 2.0

45.0 43.0

0.02 mol/L

8 x 10-5 mol/L

1 2

nFe(NO )

⎡Fe (NO ) ⎤ ! 3 3 ⎦⎥ ⎣⎢ 2 V

3 3

Fe(NO3 )

3

+VKSCN +VH O 2

=

(0.200mol / L)(0.005L) 0.005L + 0.002 + 0.043L

= 0.02mol / L

(0.002mol / L)(0.002L) = 8×10−5 mol / L nKSCN = +VKSCN +VH O 0.005L + 0.002L + 0.043L Fe(NO3 ) 2 3

⎡KSCN ⎤ ! ⎢⎣ ⎥⎦ 2 V

Qualitative considerations: a) Write the net ionic equations of any reactions that occur in test tubes 2 to 9 when chemicals 2-9 are added to the test tubes (Use the table below). Complexes and Precipitates Chemicals added to reaction (1) : Ag+, Hg2+ , Cl- , PO43 , F- , C2O42_________________________________________________________________ Possible products formed when the above chemicals are added to reaction (1): FeCl4-

FePO4

FeF63-

Fe(C2O4)3-

AgSCN(s)

Hg(SCN)42-

Ag2C2O4(s) ________________________________________________________________

44

b) When equilibrium is re-established in the eight identical stock solutions after addition of chemicals 2-9 (i.e., add Fe3+ to test tube #2, add SCN- to test tube #3, etc.), use le Chatelier’s principle to predict whether the changes Δ[Fe3+] , Δ[SCN-], and Δ[FeSCN2+] are (+), (-) or (0). Write (+), (-) or (0) in the chart. 3+

Fe SCNFeSCN2+

Fe3+ SCN- Ag+ Cl- Hg2+ PO43- C2O42- NaF(s)

Equilibrium constant: Use data from part IV-11 to calculate the equilibrium constant for reaction(6). Compute the average K and the uncertainty. Concentration of unknown from graph: Use the measured Absorbance for the unknown and the equation, A = m C + b , from the calibration curve, to calculate the concentration C of the unknown.

V. Conclusion

Refer to the lab write-up procedure.

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Experiment 7 Acids and Buffers Objectives (a) to explore the properties and role of buffers in daily life. (b) to investigate the acid contents of common beverages. (c) to prepare and test two acid buffer solutions. (d) to determine the buffer capacity of prepared buffers.

I. Introduction A buffer is a mixture of a weak acid and its conjugate base, or a weak base and its conjugate acid. A buffer’s function is to absorb acids (H+ or H3O+ ions) or bases (OHions) so that the change in pH of the system is very small. The pH in which a buffer solution is effective is generally considered to be in the range ( pK a -­‐ 1) to ( pK a + 1), where pKa is the negative of the logarithm of the equilibrium constant for an acid. In many biological systems, buffers are critical. Blood plasma, a natural buffer in humans, is a bicarbonate buffer that keeps the pH of blood between 7.2 and 7.6. The pH of the intercellular fluid in human body is around 7.2 and the pH of the fluid in the stomach is very acidic ~1.0. With the intake of food, the pH of the stomach fluid changes and can become more acidic. Antacid medication can then be taken which contains bicarbonate and brings the pH close to the initial value. In this experiment, a phosphate buffer will be prepared and the buffering capacity of the prepared buffer will be tested. Materials Vernier computer interface computer Vernier pH Sensor magnetic stirrer and stir bar three 250 mL beakers 100 mL graduated cylinder 25 mL graduated cylinder two 50 mL burets and two burets Balance

0.1 M sodium hydroxide, NaOH, solution H3PO4 (aq) NaH2PO4(s) Antacid solution Ring stand Utility clamp Distilled water clamps

46

II. Theory By design, a buffer is an equilibrium system, for example, a buffer can be prepared with nitrous acid (HNO2) and its conjugate base (NO2-). This weak acid alone establishes the equilibrium shown below.

HNO2 (aq) ! H + (aq) + NO2− (aq)

(1)

with equilibrium constant expression given by,

Ka =

[H + ][NO-2 ] [HNO 2 ]

(2)

When NO2– is added to HNO2 the concentration of each component in reaction (1) changes, but the equilibrium constant Ka remains constant. To prepare a buffer system with nitrous acid, a conjugate base such as sodium nitrite (NaNO2) is added. The resulting system is a mixture of HNO2, NO2–, and H+ ions. If a strong base such as NaOH is added to a non-buffer (HNO2 alone), the pH changes dramatically, but when added to the buffer (HNO2 + NaNO2), the following reaction serves to resist a change in pH of the buffer:

OH − + H + (buffer ) ! H2O If a strong acid such as HCl is added to a non-buffer (HNO2 alone), the pH changes dramatically, but when added to the buffer, the following reaction serves to resist a change in pH of the buffer:

H + + NO2− (buffer ) ! HNO2 Another form of eqn. (2) is obtained by first taking the logarithm of each side of this equation to give

[NO-2 ] log K a = log [H ]+log [HNO2 ] +

[NO-2 ] or log [H ] = log K a −log [HNO2 ] +

By multiplying the latter equation by a minus sign, it becomes

−log [H+ ] = −log K a + log

[NO-2 ] [HNO2 ]

For any chemical concentration (e.g., H+ ) or equilibrium constant (e.g., Ka ), the “p” of that quantity is defined by pH ≡ −log ⎡⎢H + ⎤⎥ , pOH ≡ −log ⎣ ⎦ hence 47

⎡OH − ⎤ and pK ≡ −logK , a a ⎢⎣ ⎥⎦

pH = pK a +log

[NO-2 ] [HNO2 ]

(3)

The above equation for pH of the solution is called the HendersonHasselbach equation; it is useful for calculating the pH of a buffer solution.

III. Experiment Part I: Qualitative (Exploring Buffers) 1. Take 25.0 mL of three different beverages in three separate beakers. Fill the burette with 0.1 M sodium hydroxide. 2. Connect the pH sensor with the interface and the computer and calibrate the pH probe. Measure the initial pH of beverage A. 3. Add sodium hydroxide step wise to beverage A, and find out the volume of sodium hydroxide required to bring the pH of beverage A to 7.0. 4. Repeat the experiment for beverages B and C. 5. Record your observations in the table below.

Beverage A Beverage B Beverage C (pH)initial Volume of NaOH when pH = 7 Part II: Prepare and Test Buffer Solution A 1. Obtain and wear goggles. 2. Write down the procedure to prepare 100 mL of the phosphate Buffer with a pH of 2.0. 3. Set up two burettes, buret clamp, and ring stand (see Figure 1). Rinse and fill one buret with cola product. Rinse and fill the second buret with anti-acid solution. 4. Use a pipet to measure out 10.00 mL of the Buffer solution into a 250 mL beaker and add 15 mL of distilled water. Place the beaker on a magnetic stirrer, beneath the buret of cola, and add a stirring bar. If no magnetic stirrer is available, you will stir with a stirring rod during the testing.

48

5. Connect a pH Sensor to Channel 1 of the Vernier computer interface. Connect the interface to the computer using the proper interface cable. Suspend the pH Sensor in the pH =2 buffer solution, as shown in Figure 1. Make sure that the sensor is not struck by the stirring bar. 6. Start the Logger Pro program on your computer.

Figure 1 7.You are now ready to test Buffer. You will slowly and carefully add cola product to the buffer solution. a. Take an initial pH reading of the buffer solution. Click and monitor pH for 5-10 seconds. Once the displayed pH reading has stabilized, click . In the edit box, type “0” (for 0 mL added). Press the ENTER key to store the first data pair. Record the initial pH value in your data table. b. Add a small amount of the cola product, up to 0.50 mL. When the pH stabilizes click . Enter the current buret reading and press ENTER to store the second data pair. c. Continue adding the cola product in small increments that raise/lower the pH consistently and enter the buret reading after each increment. Your goal is to raise the pH of the buffer by precisely 2 pH units. d. When the pH of the buffer solution is precisely 1 unit smaller/larger than the initial reading, stop further addition of cola product. 7. Repeat Step 7, using the antacid solution in the other buret. (Do not discard excess anti-acid). 8. Click

. Print a copy of the first trial.

9. Repeat this experiment with another beverage and record your results. 10. Record your observations in the table below. 11.

49

Cola

Anti-acid Unknown Beverage

(pH)initial 0 mL

V1

0 mL

0 mL

(pH)2 V2 (pH)3 V3 ---

---

---

---

(pH)final Vfinal where ΔpH = (pH)finial – (pH)initial =2

IV. Calculations Beverage A Beverage B Beverage C (pH)initial (from data) Δ(pH) = 7 - (pH)initial [H+]initial (from pHinitial) [H+]final

10-7M

10-7M

Δ[H+] Relative buffer capacity (Scale: 1,2,3, with 3 implying the greatest Buffer capacity

V. Conclusion: Refer to the lab write-up procedure.

50

10-7M

Experiment 8 Determination of the Concentration of an Acid Objectives a) b) c) d) e)

to learn how to prepare a standard solution. to distinguish primary and secondary standards. to use the pH probe to monitor a chemical reaction. to calculate a defined quantity from the experimental data. to learn to work problems such as: “0.5 mole of hydrogen chloride gas is bubbled through and dissolved in 0.25 L of water; calculate the molarity of the aqueous hydrochloric acid.”

I. Introduction Many acids are prepared by passing the corresponding gas through water. In this process accurate determination of the concentration of the solution is not possible; it can only be estimated. For example, to prepare aqueous hydrochloric acid, hydrogen chloride gas is passed through water. Can you think of the possible scientific law that can used to estimate the concentration of the hydrochloric acid? One of the ways to determine the concentration of an acid is to react it with a base of known concentration. This neutralization reaction lets you calculate the concentration of the acid. Unfortunately, most bases are hygroscopic. Therefore, it is not possible to prepare a solution of a base whose concentration is accurately known. To determine the concentration of a base, an acid must be found whose solution can be made with accurate concentration. This acid is known as a primary standard while the base is known as the secondary standard. A titration is a process to determine the volume of a solution needed to react with a given amount of another substance. For example, a strong acid such as hydrochloric acid and a strong base such as sodium hydroxide react to form an aqueous solution of sodium chloride and water, with a pH of 7.0. When the number of equivalents of acid equals the number of equivalents of base, the solution is said to be at the equivalence point, this being the definition of the equivalent point. Any further addition of the base causes a further increase in the pH (decrease in H + concentration) solely due to the addition of base. Generally, the pH values at the equivalence point differ depending upon the type or nature of acid and the base. In this experiment, you will explore a chemical reaction between (1) a weak acid and a strong base and (2) a strong acid and a strong base. From the equivalence point, you will calculate the concentration of the base knowing the concentration of the acid. The concentration of an unknown acid or base can be determined by the titration method using the known concentration of the primary

51

standard. A hygroscopic substance can not be used as a primary standard (Why?). A typical pH-Volume titration curve will look as shown below.

pH

Volume NaOH (mL)

Materials required for this experiment are as follows: Computer Logger Pro Vernier pH Sensor Ring stand 2, utility clamps Burette (50 mL) Oxalic acid salt (powder) Pipet pump

Vernier Computer interface Wash bottle Distilled water Two 250 mL beaker Pipette (20 mL) Sodium hydroxide(pellets or solution) Hydrochloric acid (unknown)

This experiment is designed to study a particular type of chemical reaction (acid-base) by monitoring the change in pH as the reaction proceeds. II. Theory A very seldom used concentration unit, the normality, is particularly useful for defining the equivalence point in an acid-base titration. Before discussion of normality, however, the definition of molarity is presented. The molarity M is the ratio of the number of moles of solute divided by the volume of the solution in liters, as given by

M=n

solute

/Vsolution

n solute is the number of moles of solute and Vsolution is the volume of solution in units of liters. where

The normality N is defined in a similar fashion, namely by

N = n eqsolute / Vsolution

52

eq

where n solute = number of equivalents of solute. The number of equivalents is defined in terms of the number of moles as follows:

n

eq

=νn

where, for simple acids and bases, ν = # of replaceable H+ or OH− groups. For example, HCl ν =1 H2SO4 ν =2

H3PO4

HNO3 HC2H3O2 NaOH Ca(OH)2 NH3 (aq) Ca(OH)2 CaCl2 Al(OH)3 AlCl3 Hence,

ν ν ν ν ν ν ν

=3 =1 =1 =1 =2 =1 =2

ν=3

N = n eqsolute / Vsolution =ν n solute /Vsolutio = ν M.

Therefore, once the molarity is known, it is a simple matter of multiplying the molarity by ν in order to obtain the normality (units = equivalents per mole). In titration reactions of acids and bases, the equivalence point is defined as the point where

n eqacid = n eqbase

or, upon rearranging N = n

eq

solute

/ Vsolution,

NA VA = NB VB or ν AMA VA = νB MB VB III. Experiment 1. Prepare 100 mL of standard solution of oxalic acid which is 0.05 M. (For the procedure, you may refer to your text book). Oxalic acid is used as the primary standard. 2. Prepare 200 mL of approximately 0.1 M solution of sodium hydroxide. 53

3. Calibrate the pH Sensor (see instructions – Appendix C, p.70) 4. Place 20 mL of 0.1 M oxalic acid solution into a 250 mL beaker. Add 3 drops of phenolphthalein acid-base indicator. 5. Use a utility clamp to suspend a pH sensor on a ring stand as shown in Figure 1. Situate the pH sensor in the acid solution and make sure that the tip of the pH sensor is fully covered by acid. 6. Obtain a 50 mL burette and rinse the burette with a few mL of the ~ 0.1 M sodium hydroxide solution. Fill the burette to about the 0 mL mark. CAUTION: Sodium hydroxide solution is caustic. Avoid spilling it on your skin or clothing. Use diagram in previous experiment. Complete the box for the “volume” column. Figure 1

7. Prepare the computer for data collection by clicking on the clock icon. Choose events with entry under mode option. The pH reading should be between 1.0 and 2.0 for the acid solution. 8. You are ready to begin collecting data. Before adding the titrant, Click on and monitor pH for 5-10 seconds. Once the displayed pH reading has stabilized, enter the values of pH and the volume of the titrant (0 mL) in the spreadsheet. 9. This process goes faster if one person manipulates and reads the burette while another person operates the computer and enters volumes. a. Add the next increment of sodium hydroxide titrant (about 1 mL). When the pH stabilizes, enter the corresponding values in the spreadsheet. You have now saved the second data pair for the experiment. b. Continue adding sodium hydroxide solution in increments of 1 mL and enter the burette reading after each increment. Proceed in this manner until the pH is 5.0.

54

c. When the pH value of approximately 5.0 is reached, change to increments of a few drops. Enter a new burette reading after each increment. Note: It is important that all increment volumes in this part of the titration be equal; that is, one-drop increments. d. After a pH value of approximately 10 is reached, again add larger increments (1 mL), and enter the burette level after each increment. e. Continue adding sodium hydroxide solution until the pH value is constant (around pH = 12). 11. When you have finished collecting data, click contents as directed by your teacher.

. Dispose of the beaker

12. Print copies of the table and the graph. 13. Take 20 mL of hydrochloric acid instead of oxalic acid as indicated in step 2 and repeat steps 3 through 10 to determine the concentration of hydrochloric acid. Equivalence Point Determination: One way of determining the precise equivalence point of the titration is to take the first and second derivatives of the pH-volume data. The equivalence point volume corresponds to the peak (maximum) value of the first derivative plot, and to the volume where the second derivative equals zero on the second derivative plot. Derivative on Logger Pro 3.8.4.2 To take the derivative on Logger Pro, the data acquisition must be stopped. Under the heading “Data”, select “New Calculated Column.” Toward the bottom left of the new window that pops up, select “Functions”, then “Calculus”, then “Derivative.” In the middle button of this window, select “Variable” and “pH.” Click “Done” to complete the process. To verify that Logger Pro did calculate the derivative, there should be a third data column (most likely blue text) labeled “CC” (for calculated column) displayed in Logger Pro to the right of your titration curve data. To plot this new data, right click on the column and select “Graph Options” followed by “Y-Axis Column”, then select “Calculated Column”, then hit “Done.” Repeat the process for the second derivative (CC2). Print the data and perform calculations for the experiment. 14. Your raw data table for this experiment consists of the computer printout which has been cut and pasted onto your report. Attach the graphs after the data table. IV. Calculations Provide detailed calculations of all molarities and molarities for sodium hydroxide and hydrochloric acid. Include answers to the following questions in this section. 1. Write chemical and the net-ionic equations for reactions in this experiment. 2. Why is the pH increasing upon addition of sodium hydroxide and reaches a constant value towards the end of the experiment? Explain.

55

3. 4.

Why is it required to take the derivative of the data to get the equivalence point? Why not use sodium hydroxide to standardize hydrochloric acid?

V. Conclusion Refer to the lab write-up procedure.

56

Experiment 9 Electrochemistry: Voltaic Cells, Determination of an Equilibrium Constant Objectives (a) to measure the redox potential of a chemical reaction using an electrochemical cell. (b) to learn the method of constructing voltaic cells and develop an electrochemical series based on potential differences between half-cells. (c) to learn the use of the Nernst Equation; calculate equilibrium constant. (d) to gain experience with the oscillation between experiment and theory.

I. Introduction In Parts I and II of this experiment, a Voltage Probe is used to measure the potential of a voltaic cell whose electrodes are copper and lead. Two different voltaic cells each with unknown metal electrodes are then used, and through careful measurements of the cell potentials, the unknown metals are identified. In Part III of the experiment, measurements are made of the potential of a special type of voltaic cell with identical electrodes; it is called a concentration cell. In the first concentration cell, the cell consists of two copper (Cu) electrodes, one in each well of the cell, with one well having a concentration c1 of CuSO4 solution (reduction occurs) and the other cell also containing CuSO4 solution but with a concentration c2 (oxidation occurs). In the second concentration cell, the cell consists of two lead (Pb) electrodes, one in each well of the cell, with one well having a concentration c1 of Pb(NO3)2 solution (reduction occurs) and the other cell containing a mixture of Pb(NO3)2 (concentration c2) solution and KI (oxidation occurs). Most of the Pb2+ from Pb(NO3)2 and I- from KI, form the precipitate lead iodide PbI2 , with the equilibrium between PbI2 (sol ) , Pb2+ and I- ions given by

PbI2 (sol) ! Pb 2+ (aq) + 2I − (aq)

In this experiment, the Nernst equation is used to calculate the solubility product constant, Ksp, for PbI2.

Materials

Vernier computer interface computer Voltage Probe three 10 mL graduated cylinders 24-well test plate string Cu and Pb electrodes two unknown electrodes, labeled X and Y 150 mL beaker

0.10 M Cu(NO3)2 solution 0.10 M Pb(NO3)2 solution 1.0 M CuSO4 solution 0.050 M KI solution 1 M KNO3 solution 0.10 M X nitrate solution 0.10 M Y nitrate solution steel wool plastic Beral pipets

57

II. Theory

Any chemical reaction involving the transfer of electrons from one substance to another is an oxidation-reduction (redox) reaction. If the experimental arrangement allows the electrons to flow through an external circuit, electrical work is done. In an oxidation half-cell reaction, the reactant loses electrons and is said to be oxidized, but is a reducing agent. In a reduction half-cell reaction, the reactant gains electrons and is said to be reduced, but is an oxidizing agent. Consider the following redox reaction: Zn(s) + Pb2+(aq) → Zn2+(aq) + Pb(s)

(1)

This redox reaction can be divided into oxidation and reduction half-reaction, namely Zn(s) → Zn2+(aq) + 2e− : oxidation half-rxn; Zn is oxidized; Zn is a reducing agent. Pb2+(aq) + 2e− → Pb(s): reduction half-rxn; Pb2+ is reduced; Pb2+ is an oxidizing agent

Figure 1. A voltaic cell (Figure 1) is a device used to separate a redox reaction into its two component half-reactions in such a way that the electrons are transferred through an external circuit rather than by direct contact of the oxidizing agent and the reducing agent. This transfer of electrons through an external circuit produces an electric current. Each side of the voltaic cell is known as a half-cell. For the redox reaction (1), each half-cell consists of an electrode (the metal of the half-reaction) and a solution containing the corresponding cation of the half-reactions. The electrodes of the halfcells are connected by a wire along which the electrons flow through an external circuit. In the cell, oxidation takes place at the zinc electrode, liberating electrons to the external circuit. Reduction takes place at the lead electrode, consuming electrons coming from the external circuit. By definition, - the electrode at which oxidation occurs is called the anode; it has a negative sign. - the electrode at which reduction occurs is called the cathode; it has a positive sign.

58

When all the concentrations in the zinc/lead systems are 1 molar and the temperature is 250C, the cell voltage is 0.63 volts. It would be a monumental task to assemble a list of all possible cells and report their voltage. Instead potentials are tabulated for half-reactions. Since half-cell potentials cannot be measured directly, one half-cell reaction serves as a standard and all other half-cell potentials are measured relative to the standard. The process is to construct a cell with one electrode as the standard electrode, then the cell voltage is measured. This voltage serves as the halfcell voltage for the remaining cell. The standard chosen by convention is: 2H+ + 2e− → H2(g)

ε 0 = 0.00 V

The notation ε 0 is called the standard electrode potential and is the assigned potential of the standard hydrogen electrode when the concentration of H+ is 1 M and the pressure of the hydrogen gas is 1 atm. The measured cell voltage using the standard hydrogen electrode is therefore the half-cell potential of the other half reaction. In the zinc/lead cell the measured potential of 0.63 volts can be deduced from the sum of the potentials of the two half-reactions. Zn → Zn2+

+ 2e−

Pb2+ + 2e− → Pb ______________________________ Zn + Pb2+ → Zn2+ + Pb

ε 0 = +0.76 V ε 0 = -0.13 V ε 0 = +0.63 V

Note: The sign of the standard reduction potential for the zinc half-reaction is reversed to give the potential of the oxidation half reaction. The Nernst Equation The voltage difference between electrodes, the cell voltage, is also called the electromotive force, or emf ( ε or ε cell ). Under standard conditions (25oC, 1M solution concentration, a atm gas pressure), these voltages are known as Standard emfs ( ε 0 or 0 ). ε cell In reality, standard conditions are often difficult if not impossible to obtain. The Nernst Equation allows cell voltages to be predicted when the conditions are not standard. Nernst developed the following equation while studying the thermodynamics of electrolyte solutions:

59

⎛ 2.303RT ⎞⎟ ⎟ logQ εcell = ε0cell − ⎜⎜ ⎜⎝ nF ⎟⎟⎠

(2)

In equation (2), R is the gas constant (8.314 J mole-1 K-1), T is the temperature (Kelvin), F is Faraday’s constant (96,485 coulombs/mole), n is the number of electrons transferred in the balanced oxidation/reduction reaction, and n

n

⎡C ⎤ C ⎡D ⎤ D ⎢ ⎥ ⎢ ⎥ Q = ⎣ ⎦ n ⎣ ⎦n ⎡ A⎤ A ⎡B ⎤ B ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥

for reaction nA A + nBB → nCC + nD D

(3)

At room temperature (250C, i.e., 298.15 K ), the Nernst equation becomes

⎛ 0.0591⎞⎟ ⎟ logQ εcell = ε0cell − ⎜⎜ ⎜⎝ n ⎟⎟⎠

(4)

0 In equations (2) and (4), note that if Q = 1, then ε cell = ε cell .

At equilibrium, εcell = 0 then Q→ K eq , and eqn(3) becomes

⎛ 0.0591⎞⎟ ⎟ logK eq 0= ε0cell − ⎜⎜ ⎜⎝ n ⎟⎟⎠ Hence, measurement of where

0 allows computation of the equilibrium constant Keq, ε cell

K eq = 10

n ε0cell 0.0591

(5)

Concentration Cell For a concentration cell both electrodes are made of the same metal, e.g., copper. In this case the half-cell and cell reactions are

Oxidation

Cu (s ) → Cu 2+ (c 2 ) + 2e−1

0 ; EOX =− 0.34V

Reduction

Cu 2+ (c1) + 2e−1 → Cu (s )

0 ; ERED =+ 0.34V

_____________________________________________________ Cell

Cu 2+ (c1) → Cu 2+ (c 2 ) 60

0 ; Ecell =0

Hence

⎛ 0.0591⎞⎟ ⎛ 0.0591⎞⎟ c 2 0 ⎟⎟logQ = 0 −⎜⎜ ⎟log Ecell = Ecell − ⎜⎜ ⎜⎝ n ⎟⎠ ⎜⎝ n ⎟⎟⎠ c1 ⎛ 0.0591⎞⎟ c 2 ⎟log = −⎜⎜ ⎜⎝ n ⎟⎟⎠ c1

where c1 and c2 are concentrations of the indicated ions. For a known concentration c1 and cell voltage measurement E 0, concentration c2 can be calculated. In addition to a concentration cell with two copper electrodes, in this experiment a concentration cell is also constructed with two lead Pb electrodes. The electrolyte solutions in this case have Pb 2+ ions at concentration c1 being reduced and Pb 2+ ions at concentration c2 being oxidized.

III. Experiment

Part I: Determine the Eo for a Cu-Pb Voltaic Cell 1. Obtain and wear goggles. 2. Use a 2-vial set-up as your voltaic cell. Use Beral pipets to transfer small amounts of 0.10 M Cu(NO3)2 and 0.10 M Pb(NO3)2 solution to two neighboring wells in the test plate. CAUTION: Handle these solutions with care. If a spill occurs, ask your instructor how to clean up safely. 3. Obtain one Cu and one Pb metal strip to act as electrodes. Polish each strip with sand paper or steel wool. Place the Cu strip in the well of Cu(NO3)2 solution and place the Pb strip in the well of Pb(NO)3 solution. These are the half cells of your Cu-Pb voltaic cell. Wash hands! 4. Make a salt bridge by soaking a short length of string in a beaker than contains a small amount of 1 M KNO3 solution. Connect the Cu and Pb half cells with the string. 5. Connect a Voltage Probe to Channel 1 of the Vernier computer interface. Connect the interface to the computer with the proper cable. 6. Start the Logger Pro program on your computer. Open the file “20 Electrochemistry” from the Advanced Chemistry with Vernier folder. 7. Measure the potential of the Cu-Pb voltaic cell. Complete the steps quickly to get the best data. a. Click

to start the data collection.

b. Connect the leads from the Voltage Probe to the Cu and Pb electrodes to get a positive potential reading. Click immediately after making the connection with the Voltage Probe.

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c. Remove both electrodes from the solutions. Clean and polish each electrode. d. Put the Cu and Pb electrodes back in place to set up the voltaic cell. Connect the Voltage Probe to the electrodes, as before. Click immediately after making the connection with the Voltage Probe. e. Remove the electrodes. Clean and polish each electrode again. f. Set up the voltaic cell a third, and final, time. Click immediately after making the connection with the Voltage Probe. g. Click the Statistics button, . Record the mean in your data table as the average potential. Part II: Determine the Eo for Two Voltaic Cells Using Pb and Unknown Metals 8. Obtain a small amount of the unknown electrolyte solution labeled “0.10 M X” and the corresponding metal strip, X. 9. Use a Beral pipet to transfer a small amount of 0.10 M X solution to a well adjacent to the 0.10 M Pb(NO3)2 solution in the test plate. 10. Make a new salt bridge by soaking a short length of string in the beaker of 1 M KNO3 solution. Connect the X and Pb half cells with the string. 11. Measure the potential of the X-Pb voltaic cell. Complete this step quickly. a. Click to start the data collection. b. Connect the leads from the Voltage Probe to the X and Pb electrodes to get a positive potential reading. Click immediately after making the connection with the Voltage Probe. c. Remove both electrodes from the solutions. Clean and polish each electrode. d. Set up the voltaic cell again. Connect the Voltage Probe as before. Click immediately after making the connection with the Voltage Probe. e. Remove the electrodes. Clean and polish each electrode again. f. Test the voltaic cell a third time. Click immediately after making the connection with the Voltage Probe. g. Click the Statistics button, . Record the mean in your data table as the average potential. 12. Repeat Steps 8-11 using the unknown and its corresponding electrolyte solution labeled “Y”. Cu/Pb Voltaic cell

Cell voltage Concentrations

Concentrations

of species in cell of species in cell for oxidation

#1 #2 #3

62

for reduction

X/Pb Voltaic cell

Cell voltage Concentrations

Concentrations

of species in cell of species in cell for oxidation

for reduction

#1

#2

#3

Y/Pb Voltaic cell

Cell voltage Concentrations

Concentrations

of species in cell of species in cell for oxidation

#1

#2

#3

63

for reduction

Part III: Prepare and Test Two Concentration Cells 13. Set up and test a copper concentration cell. a. Prepare 40 mL of 0.050 M CuSO4 solution by mixing 2 mL of 0.5 M CuSO4 solution with 38 mL of distilled water. b. Set up a concentration cell in two vials by adding enough (to touch the salt bridge) 0.050 M CuSO4 solution to one well and enough (to touch the salt bridge) 0.5 M CuSO4 solution to a neighboring well. Use Cu metal electrodes in each well. Use a KNO3-soaked string or paper strip as the salt bridge, as in Parts I and II. c. Click to start the data collection. d. Test and record the potential of the concentration cell in the same manner that you tested the voltaic cells in Parts I and II. Cu/Cu

Cell voltage Concentration c2 Concentration c1

concentration Voltaic cell #1

#2

#3

14. Set up a concentration cell to determine the solubility product constant, Ksp, of PbI2. (a) Prepare 10 mL of 0.050 M Pb(NO3)2 solution by mixing 5 mL of 0.10 M Pb(NO3)2 solution with 5 mL of distilled water. (b) Mix 9 mL of 0.050 M KI solution with 3 mL of 0.050 M Pb(NO3)2 solution in a small beaker. In this reaction, most of the Pb2+ and I– will form the precipitate PbI2, but a small amount of the ions will remain dissolved. (c) Set up the half cells in neighboring wells of the 24-well test plate. Place 5 mL of 0.050 M Pb(NO3)2 solution in one half cell, and 5 mL of the PbI2 mixture, from the small beaker, into an adjacent half cell. Use Pb electrodes in each half cell. Use a KNO3-soaked string as the salt bridge. (d) Test and record the potential of the cell in the same manner that you tested the voltaic cells and the copper concentration cell. 15. Handle the electrodes and the electrolyte solutions as directed. Rinse and clean the vials. CAUTION: Handle these solutions with care. If a spill occurs, ask your instructor how to clean up safely.

64

Cell voltage Concentration c2 Concentration c1

Pb/PbI2 concentration Voltaic cell #1

To be calculated #2 To be calculated #3 To be calculated

IV. Calculations - Complete the following tables: Average cell Uncertainty in voltage cell voltage Cu/Pb cell

Name of unknown (reference table) xxxxxxxxxxxxxx

X/Pb cell Y/Pb cell Cu/Cu cell

xxxxxxxxxxxxxx

Pb/PbI2 cell

c2 =

Cell Cu/Pb

Oxidation half-cell reaction

X/Pb Y/Pb Cu/Cu Pb/PbI2

65

±

Reduction half-cell reaction

Cell Cu/Pb

Balanced Net Ionic Cell Reaction…………………

X/Pb Y/Pb Cu/Cu Pb/PbI2

Include in the section for calculations answers for the following questions: a) 1. Draw a diagram for the Pb/PbI2 cell. 2. Name electrodes and label with a plus or minus sign. 3. Indicate the electrode where oxidation occurs and the electrode where reduction occurs. 4. Label the wire with an arrow to specify the direction of electron flow. 5. Write the names of ions and precipitates in solution. 6. Include a salt bridge and describe its’ purpose. b) Perform the following calculations: 1. Note: PbI2 ! Pb2+ + 2Ic2 2 c2 so [I-] = 2 c2 = ? 2. Ksp = [Pb2+] [I-]2 = (c2 ) (2 c2 )2 = 4 (c2 )3 3. The accepted value of the Ksp of PbI2 is 9.8 x 10-9. Calculate the percent error for your experimental Ksp of PbI2 . c) What is the difference between a voltaic cell and an electrolytic cell?

V. Conclusion: Refer to the lab write-up procedure. 66

Appendix A Common Laboratory Equipment Some common laboratory equipment are depicted below. Some of these are used for measurements, and these are described in the section on instruments. However, some of these are not used for measurements, and these will be described here. Bunsen Burner A Bunsen burner is shown in the figure below. It is used for heating at higher temperatures than that which can be achieved by a hot plate. The rubber tube supplies gas, where the screw at the bottom can be turned to control the rate of flow of gas into the burner. The other end of the rubber tube is connected to the gas supply in the laboratory. This gas supply is equipped with a lever. If the lever is perpendicular to the gas supply pipe, then it is closed; and if the lever is parallel to the gas supply line, it is maximally open. The Bunsen burner allows air as well as gas to enter to supply the flame. The amount of air is controlled by turning the top part of the burner above where the tube is connected. The amount of air will affect the temperature of the flame. The blue flame is the hottest. To light the burner, use a device that produces a spark. Do not use matches or a lighter. Turn the lever on the gas supply in the lab so that a small amount of gas flows. Immediately use the spark to light the burner. You may then turn the lever for more gas and change the settings on the burner. Crucible The crucible and lid are made of a material that can be heated to high temperatures. Use this when you want to heat something to a high temperature. The lid should be slightly tilted when heating.

67

68

69

Meter Stick/Ruler Use:

Length measurements for ordinary size objects or distances

Scale:

Metric –1 mm between smallest lines (see figure below), 1 cm between largest lines

Uncertainty: ±0.05 cm Calibration: Unless the meter stick is deformed or warped, assume that the manufacturer calibrated it properly. However, for metal rulers, this is more of a problem than for wood or plastic. Metals tend to expand or contract more depending on the temperature. Proper Technique: The end of the meter stick is usually worn or not clear. Therefore, it is a good practice to measure the length starting from the 1 cm mark, and to subtract 1 cm from the value you read from the other end. Place the meter stick and the object on a firm surface. To read the meter stick, cover one of your eyes and position your other eye so that a line drawn from this eye to the point that you are reading is approximately perpendicular to the meter stick. Estimate the last digit as shown in the example below.

The reading is 47.64 ± 0.05 cm. The second decimal digit of 4 is estimated.

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Graduated Cylinder Use:

To measure the volume of a fluid

Scale:

The smallest divisions are 0.5 ml for the graduated cylinder shown in the Figure. Other graduated cylinders with different diameters or lengths may have different scale divisions.

Uncertainty: ± 0.25 ml for the graduated cylinder shown in the Figure. Check the graduated cylinder you use and take one-half of the smallest division for the uncertainty. Calibration: Unless the graduated cylinder is obviously defective, assume that it is calibrated by the manufacturer. Proper Technique: Check the liquid in the graduated cylinder to make sure that there are no bubbles; you may tap the cylinder to remove the bubbles. The level of the liquid should have a parabolic shape, which is called the meniscus. Read the volume at the bottom of the meniscus. The reading in the Figure is 20.3 ± 0.5 ml. The last digit of 3 is estimated, but any error is accounted for by the uncertainty.

meniscus

71

Vernier Caliper Use: Length measurements for small ordinary objects and distances less than about 10 cm Scale:

Metric – 1 cm for the largest division 1 mm for the smallest division Vernier – 0.9 cm equally spaced divisions located on the sliding part just below the main metric scale

Uncertainty: ± 0.01 cm Calibration: Assume that the manufacturer calibrated the caliper for accurate reading at room temperature, unless it is deformed or visibly damaged Proper Technique: The caliper has two sets of jaws – the main jaws and the inside jaws (see the Figure on the next page). Use the inside jaws to measure inner diameters of hollow tubes; otherwise, use the main jaws. Move the wheel to grasp the object, as shown. Do not excessively squeeze the object. The first mark on the vernier scale indicates the reading on the metric scale. The Figure shows a reading of between 4.5 and 4.6 cm. You obtain the second (hundredths) decimal digit by finding the number on the vernier scale where the marks of the vernier scale and the metric scale coincide. The reading below is 4.56 cm. Read the scale with one eye closed, and where a line drawn from your other eye to the reading is perpendicular to the caliper. You must also obtain a zero reading for the caliper. This is the reading where the jaws are as close together as possible. If the first mark on the vernier scale is not aligned with the zero on the metric scale, then you must obtain a reading that you subtract each time that you use the caliper. This reading may be positive or negative, depending on whether the first mark on the vernier scale is to the right or left of the zero on the metric scale, respectively. If it is positive, then read the instrument normally. If it is negative, then find the number on the vernier scale that coincides with a mark on the metric scale. The negative reading is (10 – this number)/10 cm.

72

inside jaws

vernier scale metric

scale

wheel main jaws The reading is between 4.5 and 4.6 cm. The sixth number coincides for a reading of 4.56 ± 0.01 cm.

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Analytical Balance Use:

For mass measurements up to 210 grams. The advantage that the analytical balance has over other balances is its precision. Use this balance if you require great precision. Sometimes, this precision is not possible, as is the case of measuring the mass of a chemical that is very volatile.

Scale: The scale is digital and the output is to the ten-thousandths of a gram (see figure below) Uncertainty: ± 0.0003 g Calibration: You may calibrate the balance using an object whose mass is known very accurately. You may also test whether the balance gives reasonable results by measuring the mass of a standard object, whose mass is approximately known. Otherwise, assume that the manufacturer calibrated it. Proper Technique: The analytical balance is a delicate instrument; do not handle it roughly. It should rest on a level surface. Turn it on by pressing on the lower left or right on the front panel. Wait until the reading stabilizes to zero. Gently open the door on the left or right side and place the object in the center of the pan. It is best to avoid touching the object so as not to transfer any oil or dirt; use tongs instead. Do not press down on the pan. Wait until the reading stabilizes. The example in the figure shows a reading of 0.2336 ± 0.0003 g. Close the door after use. The instrument will turn itself off automatically.

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Top-Loading Balance Use:

For mass measurements up to 510 grams. This balance has less precision than the analytical balance, but it is still precise. Use it when you require intermediate precision

Scale:

The scale is digital and the output is up to the hundredths of a gram (see figure below)

Uncertainty: ± 0.01 g Calibration: You may calibrate the balance using an object whose mass is known very accurately. You may also test whether the balance gives reasonable results by measuring the mass of a standard object, whose mass is approximately known. Otherwise, assume that the manufacturer calibrated it. Proper Technique: The balance should rest on a level firm surface. Turn on the balance by pressing on the lower left or right on the front panel. Wait until the reading stabilizes to zero. Gently place the object on the center of the balance. Do not press down on the balance. It is best to avoid touching the object so as not to transfer any oil or dirt; use tongs instead. Wait until the reading stabilizes. The figure shows a reading of 3.04 ± 0.01 g. The instrument will turn itself off automatically.

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Use:

Scale:

Triple-Beam Balance For mass measurements up to 610 grams. This is the least precise of the three balances. Use this based on your precision requirements, and for objects greater than 100 grams. There are three scales – one from 0 to 10 grams, one from 0 to 500 grams in steps of 100 grams, and one from 0 to 100 grams in steps of 10 grams. After you balance the object, you add the reading from each scale to obtain the mass of the object.

Uncertainty: ± 0.05 g Calibration: You may calibrate the balance using an object whose mass is known very accurately. You may also test whether the balance gives reasonable results by measuring the mass of a standard object, whose mass is approximately known. Otherwise, assume that the manufacturer calibrated it. Proper Technique: Position the balance so that it is level. Check that it reads zero when it is empty and all of the sliding masses are in the zero position. When the lever comes to a stop, the horizontal lines must be aligned. If this is not the case, then turn the screw on the left below the pan so that the horizontal lines are aligned. Place the object in the center of the pan. It is best to avoid touching the object so as not to transfer any oil or dirt; use tongs instead. Move the sliding masses so that the lever is in a position where the horizontal lines are aligned; you must wait until the lever comes to a stop. Make sure that the tens and hundreds sliding masses are securely placed in the grooves. Add the masses from each beam to obtain the mass of the object. Estimate the digit corresponding to the hundredths of a gram by observing the distance between the smallest divisions where the reading occurs.

screw to zero the balance

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horizontal lines

Barometer Use:

Pressure measurements

Scale:

mm of Hg or inches of Hg

Uncertainty: ± 0.1 mm of Hg or ± 0.01 inches of Hg Calibration: Assume that the instrument is calibrated by the manufacturer. However, you can measure atmospheric pressure and it should be in the vicinity of 760 mm of Hg. Proper Technique: The picture on the top right shows the full barometer. There is a screw at the bottom of the barometer to level the mercury – see the picture in the bottom left. Turn the screw so that the small white pointer (cannot be seen in the picture) just barely touches the mercury (cannot be seen in the picture). This is called zeroing the barometer. The mercury level forms a parabolic shape called a meniscus. Position your eye level to the bottom of the meniscus, where you will read the scale. Read inches on the right and mm on the left, as shown. For the mm side, note that the top number in the figure is 800, so that the 90 actually means 790 mm.

pointer leveling Screw

meniscus inches

77

mm

The reading on the main mm scale is between 734 and 735 mm of Hg. The amount in between is found on the vernier scale. Find the number on the vernier scale that aligns with a marking on the main scale. In this case it is 6, so the reading is 734.6 ± 0.1 mm of Hg. Do the same for the inches scale. The reading is 28.80 ± 0.01 inches of Hg.

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Appendix B LOGGER PRO/LAB PRO

Instructions for Logger Pro • • • • •

Plotting and analyzing data points Launching Logger Pro with the interface Calculations Saving your work Formatting the Axes Plotting and Analyzing Data Points

Suppose that you have collected data without the use of a sensor or with the use of a sensor connected to the computer. You can use Logger Pro to plot and analyze that data. Follow these instructions. Launch Software

Launch the Logger Pro software by double-clicking on the icon on the desktop. Close the Tip of the Day dialogue box, if it appears. Select Continue without Interface on the Connect to LabPro dialogue box, then click OK. You should see two blank columns on the left, labeled X and Y, and a set of coordinate axes on the right.

Label Columns

You may re-label the column X by selecting, from the Menu bar, Data -> Column Options --> X. Change the Name and Short Name, and enter the appropriate units. You may do the same for column Y.

Enter Data

You can now click on a cell in one of the columns and start entering data. As you enter a data point, it is automatically plotted. Data points are automatically connected by lines. To remove this option, double-click anywhere on the graph and de-select Connect Points from the dialogue box, then select Done. To circle each data point, doubleclick anywhere on the graph and select Point Protectors.

Format Axes

Double-click anywhere on the graph. Select Axes Option to change label or to choose autoscale or to enter your own scale. Select Done when you are finished. You may also choose which column to plot on which axes. Click on the label on the x- or y-axis and select the column that you wish to plot on that axis. 79

Calculations

You may create a new column to calculate a quantity using the values in the other columns. From the Menu bar, choose Data --> New Calculated Column. Enter the Name, Short Name, and Units for the new column. Enter the formula for the new calculated quantity on the Equation line; use the pop-down Variables when you want to insert the value from another column and use *, /, +, -, or an operation from the Functions button in your formula. Click on Done when you are finished. The new column should now appear to the right of the other columns with the calculated values. To change the formula, go to Data --> Column Options --> Name of Column.

Statistics

The Stat button allows you to perform statistics on selected data points. Click and drag to select the data points on the graph, and press the Stat button. A box appears with information on maximum, minimum, mean, median, standard deviation, and number of points. This feature is useful, for example, in finding the average of measurements after the value has stabilized to an equilibrium, such as measuring the freezing point.

Linear Fit

You may want to try a linear fit to some or all of the data. Click and drag on the graph to select the data points that you want to include in the fit; they will become shaded. From the Menu bar, choose Analyze --> Linear Fit. The software uses the Method of Least Squares to determine the best fit line, which is drawn. A box appears with the values of the slope, intercept, and correlation. Click on the x in the upper left corner of this box to delete the line and the fit.

Curve Fit

You may want to try to fit the points to a curve. From the Menu bar, select Analyze --> Curve Fit. Choose a type function from the General Equation box, or define your own function by clicking on the Define Function button (use the Short Name for the variable when you define your own function). Press the Try Fit button. The best-fit values for the parameters appear on the right with the RMSE value in the lower right. The RMSE value is a number that relates to how close the data points are to the curve – the higher the number, the further away the data points are. Use this number to compare different fits. You may continue to try other functions; select the function and press the Try Fit button. Select OK when you are finished. The fitted curve and a box with the values for the parameters appear on the graph. You may click on the x in the upper left of this box to delete the fit and the box. Launching Logger Pro with an Interface

The interface box is a device that is shaped like a calculator. It is used as an interface between the computer and a sensor. Sensors are used for specific measurements. For example, you use a pressure sensor to measure pressure. Other 80

types of sensors detector, among computer. It is computer. Follow Connections

include temperature sensor, pH sensor, force probe, and motion others. However, the sensor does not connect directly to the first connected to the interface box, which is connected to the the general procedures below to set up the system.

Connect the interface to the computer using a cable with a USB connection at one end and a round 8-pin connection at the other end. The USB end connects to USB port on the computer. Plug the round end of the connection cord into the interface box – there is only one place that it will fit (you may need to slide a plastic cover out of the way to make the connection). Connect the AC Adapter to the interface box and to a power outlet. You must use the AC Adapter that has the word Vernier on it with a picture of a vernier caliper. If you use any other, you may cause the interface box to burn out and not function properly. You may now connect a sensor to the interface box. There are many slots on the interface where sensors are connected. They are labeled DIG/SONIC 1, DIG/SONIC 2, CH1, etc. Use a connection cord with ends that resemble a telephone connector. Insert one end in the appropriate slot in the interface box (it usually only fits into one type of slot), and the other end in the sensor itself, if needed

Launch Software Launch the Logger Pro software by double-clicking on the icon on the desktop. Close the Tip of the Day dialogue box, if it appears. You need to tell the computer the port at the back or front of the computer where the interface is connected. This port is usually com1. Select this. If that does not work, try com2. You need to inform the software which sensors are connected. Click the ‘Lab Pro’ button. A picture of the interface box will appear. Click the button that represents the appropriate slot, and select the type of sensor that is connected. Setting Up

Under Experiment à Data Collection, you may set the length of time to collect data, and the sampling rate (the number of data points collected per unit of time). If you want to remove measuring as a function of time, you may click on the button with an image of a clock and select by events. When the data is collected it is stored in columns that are displayed in a data table on the left of the screen. The data is also usually displayed graphically on the right of the screen.

81

Calculations You may create a new column to calculate a quantity using the values in the other columns. From the Menu bar, choose Data --> New Calculated Column. Enter the Name, Short Name, and Units for the new column. Enter the formula for the new calculated quantity on the Equation line; use the pop-down Variables when you want to insert the value from another column and use *, /, +, -, or an operation from the Functions button in your formula. Click on Done when you are finished. The new column should now appear to the right of the other columns with the calculated values. To change the formula, go to Data --> Column Options --> Name of Column.

Saving Your Work Do not save any work on the hard drive of the computer. It may not be available the next time you look for it. You may save work as a Logger Pro 3 File. Use the File menu, and select Save or Save As. You may double click on previously saved file to start the Logger Pro software with previous work. The appropriate sensor does not have to be connected when you work with previously saved files where you will use the software for data analysis as opposed to data collection. If not, click on Ignore Sensors when a dialogue box pops up. You can even work without the Lab Pro interface attached. You may also save your work as a text file that can be opened using Excel, for example. Select File à Export as Text. To open the text file in Excel, first launch Excel. Open the text file in Excel. Click Next when a dialogue box pops up that says that Delimiters have been found. Keep clicking Next, then Finish. The data will appear in Excel, along with some preliminary stuff that can be deleted or edited.

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Formatting the Axes To format the axes, right click anywhere on the coordinate system to get a pull-down menu with a list of options. The most important is Coordinate Graph Options and Autoscale Graph. There are several options under these; some are self-explanatory, such as the option not to connect the points. Play with the others to see what happens. Under Coordinate Graph Options, you also have Axes Options, where you may set the scale, and maximum and minimum. You may also double-click anywhere on the numbers on an axis to get a dialogue box to set the scale. You may set it to autoscale or you can set the scale manually. You can change the variable that is on one of the axis. Click on the axis to see a list of variables. Select the one that you want.

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Appendix C Calibrations Pressure Sensor The pressure sensor with an attached syringe is illustrated in the figure below. For the situation shown, the sensor is measuring the pressure of the gas in the syringe, which has a volume of 5.0 ± 0.5 ml, as measured from the position of the front edge of the inside black ring.

Connections

Attach the cord to an appropriate slot in the interface box (probably CH1 or CH2); it is appropriate when it fits. By clicking on the Lab Pro button after you have launched the Logger Pro software, check to make sure that the Logger Pro software is informed that a pressure sensor is connected and in which slot it is connected. See the section Launching Logger Pro with an Interface.

Calibration • • • • • • •

•

Click on the Lab Pro button. When the image of the interface box appears, click on the button where the pressure sensor is connected to get a pull-down menu, where you select Calibrate. A Calibrate dialogue box appears; click on the Calibrate Now button. Enter the appropriate units, e.g., mmHg Expose the sensor box to the atmosphere (remove the syringe, if it attached) and enter the reading from a barometer. Click on Keep. Go to Reading 2 Push the plunger of the syringe to the 0-ml mark and connect it to the pressure sensor. Pull the plunger all the way to the 20-ml mark (you will have to hold it). This should correspond to zero-pressure. Enter 0 for the pressure and click on Keep. Click on Done

84

Collecting Data • •

• • • • •

If you want to measure the pressure as a function of time, then press Collect to begin, and Stop to end; see the section on formatting the graph to change the display If you do not want to measure the pressure as a function of time, then turn off the timing mechanism by clicking on the button that has the image of a clock; if you want to enter other data such as volume, select Events with Entry and one column to represent volume (type name, short name, and units); go to the remaining steps Click on the Collect button to begin data collection When the pressure reading has stabilized, click Keep Enter the volume or other data, and click OK Repeat for different volume or whatever variable Press Stop when you are finished collecting data

Uncertainty: ± 0.40 mmHg Temperature Probe The temperature probe is illustrated below. thermometer for experiments.

It is used as you would use a

Connections: The long needle end is inserted into the system where you want to measure the temperature. The other end is inserted into the Lab Pro interface, either CH1 or CH2. By clicking on the Lab Pro button after you have launched the Logger Pro software, check to make sure that the Logger Pro software is informed that a temperature probe is connected and in which slot it is connected. See the section Launching Logger Pro with an Interface. Calibration: • • •

Click on the Lab Pro button When the image of the interface box appears, click on the button where the temperature probe is connected to get a pull-down menu, where you select Calibrate A Calibrate dialogue box appears; click on the Calibrate Now button 85

• • • • •

Enter the appropriate units, e.g., 0C Insert the temperature probe into the center of an ice bath. Measure the temperature of the ice bath with a thermometer and enter that value in the dialogue box. Press Keep For Reading 2, insert the temperature probe into the center of a beaker containing tap water. Measure the temperature with a thermometer and enter this value in the dialogue box. Press Keep Repeat the procedure for another temperature, such as a beaker of hot water. With this, you have a three-point temperature calibration Click on Done

Collecting Data • • •

• • • • •

Place the temperature probe in the center of the system for which you are measuring the temperature; for a liquid, do not allow the probe to touch the walls of the container If you want to measure the temperature as a function of time, then press Collect to begin, and Stop to end; see the section on formatting the graph to change the display If you do not want to measure the temperature as a function of time, then turn off the timing mechanism by clicking on the button that has the image of a clock; if you want to enter other data such as volume or pressure, select Events with Entry and one column to represent volume or pressure (type name, short name, and units); go to the remaining steps Click on the Collect button to begin data collection When the temperature reading has stabilized, click Keep If you have other data to enter, enter the them and click OK; if not, press Stop. Repeat for other data points for whatever variable you are collecting Press Stop when you are finished collecting data

Uncertainty:

0 – 400 C: ± 0.050 C 40 – 1000C: ± 0.10 C

PH sensor Connect pH sensor to the channel 1 of the interface. a. Click on Lab Pro icon on the screen. b. Click on the pH Sensor and click on Calibrate icon. Click on Calibrate Now. c. Insert the pH sensor into a solution of known pH and the voltage will be shown on the window. Enter the known value of the pH in the box and press .

86

d. Go to Reading 2. Now insert the pH sensor into a beaker containing a a second solution of known pH. Once the voltage reading stabilizes as seen on the window, enter the value and press . Click on Done. Repeat the procedure at another solution of known pH. With this you have obtained a three-point pH calibration. Uncertainty: ± 0.005 units

Conductivity Probe The Conductivity Probe can be easily calibrated at two known levels, using any of the Vernier data-collection programs. The calibration units can be µS/cm (microsiemens per centimeter). • • •

Connect the Conductivity Probe to the interface (channel 1). Click on Lab Pro icon on the screen. Click on the Conductivity Probe and click on calibrate icon. Click on Calibrate Now. • Enter appropriate units of conductivity (e.g., µS/cm) and Select the conductivity range setting on the probe box: low = 0 – 200 µS, medium = 0 – 2000 µS, and high = 0 – 20,000 µS. Note: If you are not sure which setting to use, you may first want to start with a highest range and go down depending on the value shown). • Zero Calibration Point: Simply perform this calibration point with the probe out of any liquid solution (e.g., in the air). A very small voltage reading will be displayed on the computer. Call this value 0 µS. Standard Solution Calibration Point: Place the Conductivity Probe into a standard solution (solution of known concentration), such as the sodium chloride standard that is supplied with your probe. Be sure the entire elongated hole with the electrode surface is submerged in the solution. Wait for the displayed voltage to stabilize. Enter the value of the standard solution (e.g., 1000 µS). Click on Done. With this method you have calibrated the Conductivity Probe.

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Appendix D Frequently Asked Questions about Data Analysis 1.

When I use my calculator or computer to calculate a quantity, how do I know how many digits to keep from the result as shown on the display of the calculator or computer?

2.

How do I know if my results or measurements are good?

3.

Do I need to repeat my measurement?

4.

How does Logger Pro determine the best-fit line or curve?

5.

How do I determine the uncertainty of a calculated quantity from the uncertainties of the numbers used in the calculation?

6.

How do I determine the best-fit line or curve to my data points?

7.

What do I graph to get a straight line when it is obvious that the data points do not lie on a straight line?

1.

How do I compare numbers in science?

1. When I use my calculator or computer to calculate a quantity, how do I know how many digits to keep from the result as shown on the display of the calculator or computer? This is essentially a question about the number of significant figures and uncertainty. The principle here is that any calculated value cannot have less uncertainty or more significant figures than the quantities used in the calculation. Let us illustrate this idea with an example. Suppose you measured the diameter, D, and length, L, of an object that is modeled as a cylinder, and you measurements are

D = 0.85 ± 0.05 cm , L = 4.33 ± 0.05 cm . Using the calculator, you now determine the volume from the known formula for the volume of a cylinder, as follows,

Volume =

πD 2 L 4

=

π (0.85 cm )2 ⋅ 4.33 cm 4 88

= 2.457059249 cm 3 .

Excel normally displays 7 figures, and will show 2.457059. You may set the number of decimal digits that are displayed in Excel by selecting Format à Cells à Number, and changing the number in the appropriate box. In Excel, the syntax for π is PI( ). If your calculator does not have a button for π, then type 3.141592654 or calculate cos-1(-1) with your calculator in the radian mode. To answer the original question, note that you measured the diameter and length to 2 and 3 significant figures, respectively. Therefore, the number of significant figures of the calculated result is 2, which is the smaller of the significant figures of the measured values. Your calculated volume is 2.5 cm3. If you will use the volume for further calculations, then keep at least 1 more significant figure. Therefore, for further calculations, use at least 2.46 cm3. Another problem, in addition to determining the number of significant figures of the calculated result, is to determine the uncertainty of the calculated result. Is the uncertainty on the volume ± 0.05, the same as for the diameter and length. The answer is no! One method to determine the uncertainty of the calculated result is to determine the maximum and minimum. This method works when it is clear, from the formula, how to get the maximum and minimum. For example, you obtain the maximum that the volume could be by using the maximum diameter and maximum length, as follows

Volume max =

π (0.85 + 0.05)2 ⋅ (4.33 + 0.05) 4

= 2.79 cm 3 ,

where I have kept only 3 figures because of my previous analysis. Similarly, the minimum that the volume could be is

Volume min =

π (0.85 − 0.05 )2 ⋅ (4.33 − 0.05 ) 4

= 2.15 cm 3 .

These calculations show that the volume is between 2.15 and 2.79 cm3, where the best value is 2.5 cm3. The situation is pictured below: 2.15

0.35

0.29

2.5

2.79

where 0.35 and 0.29 represent the differences between the best value and the minimum and maximum, respectively. The uncertainty is 0.32, the average value of these differences. The volume is 2.5 ± 0.3 cm3. 2.

How do I know if my results or measurements are good? This is a question about accuracy or precision. The words accuracy and precision, although similar in meaning in everyday English, have different meanings 89

in a scientific context. Before answering the question, you must decide whether you are concerned about the accuracy or precision of the results or measurements. In measuring physical quantities, one assumes that there is a true value. The accuracy of the measurement is how close you came to the true value. Of course, one does not a priori know the true value. Therefore, at times, a scientist will use a value that is accepted by the scientific community to gauge the accuracy of a measurement. This accepted value arises from the agreement of various measurements by many scientists. If you are comparing your measurement with an accepted value, you may say that your measurement is accurate if the accepted value lies within the range determined by your uncertainty. For example, if the accepted value is 4.75, a measurement of 5.1 ± 0.5 is more accurate than a measurement of 4.635 ± 0.001. The example above illustrates the scientific meaning of the term precision. The second measurement is more precise because it is measured to more significant figures. Thus, the uncertainty determines the level of precision. The uncertainty can be determined by considering the instrumental uncertainty and/or by repeating the measurement and calculating the standard deviation. Repeating the measurement is a good idea when there are factors that affect the measurement, which are beyond instrumental uncertainty. The situation, however, is not as simple as stated above. How can the first measurement in the example be considered accurate when it is not precise? Is the result meaningful? One must strive for both accuracy and precision. An important job of an experimentalist is to honestly determine the accuracy and precision of a measurement. You should not overstate either in order to make your work seem better than it is. You should find the level of accuracy and precision based on your instruments and techniques. 3. Do I need to repeat my measurement? To answer this question, you must consider the purpose of repeating a measurement. In measuring physical quantities, one assumes that there is a true value and the measurement process is to discover this true value. However, it is not assumed that this process is perfect; there are factors that influence the result of a measurement. Repeat the measurement in order to see how these factors affect the outcome of the measurement. The variation of different measurements will inform you about the precision of the measurement. The instrumental uncertainty is one factor that affects the outcome of a measurement. If other factors influence the measurement smaller than the instrumental uncertainty, then repeating the measurement will not yield any variation. In this case, there is no need to repeat the measurement, and the instrumental uncertainty is the uncertainty of the measurement. This is usually the 90

case in measuring the mass using a balance. However, it is a good idea to check this by repeating the measurement once. On the other hand, if there are factors that affect the measurement greater than the instrumental uncertainty, then repeated measurements will give a variation of results. The best value of the measurement (not to be confused with true value) is the average of the repeated measurements, and the uncertainty is a statistical quantity called the standard deviation. If there are N measurements labeled, x1 , x2 ,…, xN , then the formula for the standard deviation, labeled σ, is

σ≡

N

∑

(x − xi )2

i =1

N −1

,

where x is the symbol for the average. 4. How does Logger Pro determine the best-fit line or curve? Logger Pro uses a method called the Method of Least Squares. To determine the best-fit line or curve, one needs a criterion. The Method of Least Squares assumes the following criterion: out of all possible lines or curves, the best-fit line or curve is the one that comes closest to all the points. This closeness is calculated by summing up the square of the distances from each point to the line or curve. The line or curve with the least sum of square of distances is the best-fit line or curve. You may wonder why the method uses the square of the distances rather than the distance itself. One answer is that the distance is calculated by finding the difference between the y-coordinate of the point and the line or curve. You may imagine that some points are above the line or curve and some are below. This gives rise to distances, some of which are positive and some of which are negative. Using the square of the distances makes everything positive and gives a better idea about the closeness of the line or curve. The mathematical procedure for implementing this criterion involves calculus for finding the minimum sum of squares of distances. It also involves algebra. The results are complex formulae for the best-fit slope and intercept, in the case of a line. Logger Pro programs these formulae to calculate the best-fit slope and intercept for you. 5. How do I determine the uncertainty of a calculated quantity from the uncertainties of the numbers used in the calculation? The procedure for doing this is called propagation of uncertainty. This procedure is outlined in the answer to question 1. Refer to the answer to question 1. 91

6. How do I determine the best-fit line or curve to my data points? The criterion for the best-fit curve or line is the curve or line that comes “closest” to your data points. How is this closeness determined? It is determined from the distances of the data points to the curve (by curve here I mean curve or line). However, some of the data points will be above the curve (positive distance) and some will be below the curve (negative “distance”). It is better to determine the closeness by summing the square of the distances. Thus, if we define a quantity

∑ [y

− f ( xi ) ]

2

i

,

where (xi , yi ) is the ith data point and f is the curve that you are trying to fit to the data, then the smaller this quantity, the better the fit. Note that the expression in the square brackets is the distance between the ith data point and the curve. A method of this type to determine the best-fit curve is called Method of Least Squares. How do you carry out the method of least squares in practice? Let us take the example of a straight line. The idea is to examine all possible lines and to pick the one that minimizes the expression above. The line is describes by its slope and intercept. These quantities are called parameters. Thus the expression above can be thought of as a function of the slope and intercept of the line. One can then use calculus to find the values of the slope and intercept that minimizes the expression. For those who are familiar with calculus, this means finding where the derivatives with respect to the slope and the intercept are zero. You can do the same procedure for other types of functions, once you define the parameters. A polynomial of degree two has three parameters – the coefficient of the three terms. The software – Excel or Logger Pro – are set up to perform the above procedure and to determine the values of the parameters for the best-fit in the type of function that you specify. They will not compare different types of function. For example, you may use Logger Pro to determine the best-fit straight line. However, Logger Pro will not tell you whether an exponential function is a better fit. To compare fits in Logger Pro, use the R2 value for a straight line or the RMSE value functions that are not straight lines. These values are related to the expression above with some slight differences. For R2, the values that are closer to 1, the better; and for RMSE, the smaller the better. 7. What do I graph to get a straight line when it is obvious that the data points do not lie on a straight line? Suppose that it is obvious that a graph of y vs. x is not a straight line. You may graph other quantities, such as y vs. 1/x, or y vs. x2, or 1/y vs. x , etc. Each time, you may try a linear fit either in Excel or using Logger Pro. The closeness of the 92

value of the correlation or R2 to 1 indicates the goodness of the fit. The idea of the correlation is a concept from statistics. To graph the other quantities, you will have to compute them. In Excel, you can write a formula preceded by an equal (=) sign to compute another quantity. In Logger Pro, you can get another column by choosing Data --> New Calculated Column, and writing a formula (see the section of Plotting and Analyzing Data Points not Obtained from a Sensor). 8. How do I compare numbers in science? This depends on which numbers you are comparing and why. Possible numbers for comparison include a measurement you made, a number that you calculated from measurements or obtained from data analysis, a number you predicted from a theory, a number that you obtained from the literature as an accepted value, or a number from another person’s experiment. Reasons for comparing numbers include checking to see whether your instruments are operating properly, testing the predictions of a theory, testing to see if you can reproduce the results of an experiment, and checking whether the technique you used is valid. You can compare two experimental results for agreement or disagreement. (You are not trying to determine who is right or who is wrong.) To do this, check whether the ranges of the numbers, as determined by their perspective uncertainties, overlap. If they overlap, then you have agreement. You may also get more quantitative in the comparison by computing a percent difference (not error), as follows

% difference =

number 1 - number 2 × 100 . average of the two numbers

You may also compute percent difference to compare your number to the accepted value. In general, percent difference is a way to compare numbers without specifying which is correct. Another reason for comparing numbers is to test whether your instrument or technique is working properly. The idea here is that you have a number from another source (e.g., accepted value or theoretical value) that you assume is correct. Your instrument or technique is set up to reproduce this result. Again, if the ranges overlap, as determined from the uncertainties, then your instrument or technique is assumed to be valid. A quantitative measure of the accuracy of your technique or instrument is the percent error, defined as

% error =

your value - value assumed correct × 100 . value assumed correct 93

You may also want to compare your number to the prediction of a theory, in order to test the theory. Here, it is unknown whether the theory is valid, unlike the situation above, where the theory is assumed valid and you are testing the technique or instrument. To test the theory, you want to determine if the prediction of the theory falls within the range determined by the experimental uncertainty. You may also want to calculate a percent error to indicate the level of disagreement. If you are satisfied that the experiment is correct, then the error is on the theory, not on the experiment. In that case, the theory needs to be modified.

94