03 Gaseous State 2009

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THE GASEOUS STATE Gases The quantitative particle theory applied to gases (more properly called the Kinetic Theory of Gases) has shown to be extremely useful in Chemistry. One of the great things about it is that it explains some of the phenomena that you encounter in your everyday life. For example, have you ever inflated a pool float until it was firm, thrown it into a cold pool, and then wondered why the float then seemed like it was not fully inflated? What determines the time it takes to smell the perfume of a woman who walks past you? How do hot air balloons work? How can a small barbeque tank hold enough propane to cook with all summer long? Gases have special properties that liquids and solids don't have. The particles that make up the gas are free to move about, and a gas will take up the size and shape of its container. Knowing the volume of a gas tells you very little about the quantity of matter, because any sample of gas will “fill” its container. Consider how different a gas is from a solid. In a gas, the size of the sample has very little to do with the size of the actual particles that make up the gas itself. Even in relatively dense gas samples, the space in between the particles will be much larger than the particles themselves. For that reason gases are very easily (squeezed) so they show high compressibility. Ideal Gases Ideal gases are just an idea; or better, a model to face the study of real gases. Nevertheless, in a first approximation and for an incredible high number of situations, the ideal gas approximation leads to excellent results. In this ideal gas model, we treat the particles as point masses, or particles with mass but no volume. Ideal gases differ from real gases in another important way. In real gases, there will be an attraction between the particles in the gas as they come close to each other and between them and the particles of the container. These attractions (that are responsible for gases turning into liquids) are often minor and in this approach are neglected (considered to be zero).

It is important to remember the differences between real gases and ideal gases but it is also important to remember that real gases will act most like ideal gases at low pressure and high temperature, when the gas sample is less dense.

2 Pressure Suppose you pull from the ends of a spring: it gets tense. If you squeeze a sponge or stretch a rubber film they will also get tense. Tension is the degree of stiffness or tightness shown by the object because of the forces on it. Tensions are different in different directions in solid materials. Tension is many times (when direction is not important) called pressure. In liquids and gases tension is the same in all directions and pressure is the usual term to name it. Pressure and force are frequently mistaken but, although being closely related they are not the same thing. Pressure and force are related in the same way as population and population density. Generally speaking pressure is defined and calculated as the ratio between force and area.

Forces applied on a small surface will create a stronger deformation and tension (pressure). A relatively small force on a very small surface can produce a tremendous pressure. That is why a sharp knife cuts or a thumbnail can be punched in a wooden block. Can you explain why you can stand up on the snow if you have skis or snow shoes but you will sink if you don’t wear them? The unit of pressure is the Pascal (Pa) a very small unit. One Pascal is the pressure produced by a one Newton force (the weight of 100 g of ham) distributed on a one square metre. Pressure in gases How is the pressure at the walls of a container filled with a gas originated? According to the particle theory, particles are constantly moving at random until they crash against the walls of the container. The pressure on them is due to the continuous tiny collisions of the gas particles against the solid walls of the jar or bottle. In Chemistry, pressure is often measured in kilopascals (kPa), millimetres of mercury (mm of Hg, also called torr after Torricelli who created the first barometer), or atmospheres (atm). For convenience sake, a standard atmospheric pressure has been set 760 torr = 1.00 atm. Pressure is measured with barometers (the atmospheric pressure) or manometers (pressure gauges for gases in containers). The figures to the right show two different classes of manometers.

3 Mass, volume, pressure and temperature in gases If we have a block of wood, we know that a second block of the same class of wood that has the same volume as the first one, will have the same mass of the first block too. But having two equal volumes of the same gas does not necessarily mean their masses are equal! That is because most of the gas is empty space and the same container can accommodate lots of extra particles! Nevertheless there will be a difference in both systems: the pressure of the gases will not be the same because in case one of them has more particles there will be more collisions against the walls of the container. Furthermore, if the system (container) that has more particles and more pressure is cooled down, the collisions of the particles against the walls will be fewer (the particles move more slowly) and less energetic. Consequently, lowering the temperature will lower the pressure too and finally we can be hoaxed again because both systems have the same volume and now also the same pressure yet not the same mass! In the case of gases, masses and volumes behave as in solids and liquids provided the temperature and the pressure is kept constant. By keeping the pressure and the temperature at constant values, chemists operate measuring volumes of gases instead of masses which is, in these cases, rather tedious and inaccurate. There are simple laws that relate all these variables for a gas: let’s go deeper into them. Boyle’s law Imagine a gas sample trapped in a cylinder which allows you to adjust the volume. It can be a sealed syringe with a plunger (piston) that can be pressed down. As you push the plunger into the syringe’s body notice how the changes in volume cause the pressure to change. The smaller the volume, the higher the pressure will go. This change is not arbitrary: although both volume and pressure change at the same time, their product remains constant. Because of the relationship between the pressure and volume of a gas sample at constant temperature, if you double the value of one, you divide the other by two. If you make the volume 10 times smaller, the pressure will be 10 times greater.

4 This relationship, called Boyle's Law, is summarized by the statement: If temperature remains constant the volume of a fixed mass of gas is inversely proportional to its pressure. When two variables are inversely proportional, like pressure and volume in the example above, the product of the two variables will always remain constant. The formula that can be used to calculate the effects of pressure changes on the volume of a gas at constant temperature is shown below:

P1V1 = P2V2 (If T is constant) Charles’ and Gay-Lussac’s Laws Charles's law, states that at constant pressure, the volume of a fixed mass of gas is directly proportional to its Kelvin temperature. Gases expand as they are heated and they contract when they are cooled. In other words, as the temperature of a sample of gas at constant pressure increases, the volume increases. As the temperature goes down, the volume decreases as well.

The mathematical expression for Charles's law is shown below:

V1/T1 = V2/T2 (If P is constant) Gay-Lussac’s law is similar to Charles’s but involves changes in pressure as the gas is heated at constant volume. It states that at constant volume, the pressure of a fixed mass of gas is directly proportional to its Kelvin temperature.

P1/T1 = P2/T2

(If V is constant)

5 Remember that Charles' and Gay-Lussac’s laws calculations must be done in the Kelvin scale. Graphs The three laws studied can be represented graphically just like any other function. In Boyle’s law, as P T = a constant (call it kB) we can write P = kB/V. The plot of the function is one of the branches of a hyperbola. For the other two laws V/T and P/T are constants kC and kGL and solving both for V and P results in V = kCT and P = kGLT

Isotherms

isobars

isochors

The Ideal Gas Equation (Combined Laws) Suppose a gas is at temperature T1 and has certain pressure and volume. If we squeeze it keeping the same temperature, the new values of V and P will be somewhere on the same isotherm. If in a second step we keep the new pressure constant but heat the gas to a new temperature (we move from the red to the blue isotherm (T3). Now the three variables P, V and T have changed. But it can be shown with a little maths and knowing that the laws hold for each case we can get to the so called “Gas equation”

The meaning of this equation is that no matter what happens to a gas, as long as its mass (the number of particles) is constant the values of the three properties, P, V and T are linked and cannot be all three changed arbitrarily. Summing Up: The Particle Theory Explanation of the Ideal Gases Laws The pressure of a gas is caused by the continuous knocking of its particles against the walls of the container. If the gas is in a cubic container (just to make things easier) its particles will take some time to knock against the opposite walls of it. Now if the container is squeezed to half its length (the volume is halved), but the particles move at the same speed, they will make two trips in the same time they needed to make just one. Consequently, they will collide with the walls twice as much and the pressure will be doubled. This is what Boyle’s law states!

6 If the box instead of being squeezed is heated, the particles will move faster and will rebound more frequently and more energetically against the container’s walls. That is why pressure increases with temperature (Gay-Lussac’s law). Similar reasoning leads to Charles’ law. STP : Standard Temperature and Pressure For convenience, standard temperature has been set at 273 Kelvin, which is equal to 0oC and standard pressure at 1.00 atmospheres (760 torr). Standard temperature and pressure is abbreviated as STP. STP

T = 273 K

P = 1 atm

QUESTIONS AND PROBLEMS 1- A brick weighs 20 N. It is 5 cm high, 10 cm wide and 20 cm long. aCalculate the area of the three different surfaces (sides) bCalculate the pressures produced by the brick on a tight rubber sheet when it rests on each of its three different surfaces cIn which case the tension of the rubber sheet will be the greatest? 2- A mass of 3.2 mg of oxygen is kept in a closed syringe at 20 ºC. The volume of this mass is 24 cm3 and the pressure of the system is 1 atm (760 torr). What happens to the gas’ pressure if: aThe temperature is raised with the plunger kept in place. bThe plunger is pushed down without changing the temperature. cYou allow part of the gas to escape. 3- For the same mass of oxygen as in problem (2) find the pressure inside the syringe if (at the same temperature) the plunger is moved to allow a volume of a- 36 cm3 b- 48 cm3 c- 12 cm3 d- 2,4 cm3 4- The same mass of oxygen is immersed in a water bath at 40 ºC. If the volume is kept constant (the plunger in original position): will the pressure rise to 2 atm? Why? If your answer is negative find the value of the new pressure 5- After a chemical process in a closed container, the pressure of the gases inside is 650 torr. If the admission valve is open: will the gases escape or will air enter the container? 6- Is the pressure of the air inside an inflated balloon equal to the external atmospheric pressure? Give reasons for your answer. 7- A bubble of air is released by a deep-sea diver at a depth where the pressure is 4 atm. Assuming that its temperature remains constant: how much bigger its volume will be just before reaching the surface where the pressure is 1 atm.?

7 8- The pressure of the air in a car tyres 2.7 atm when the temperature is -3 °C. If the temperature rises to 27 °C what is the pressure then? Assume the volume doesn’t change.) 9- A mass of gas has a volume of 380 cm 3 at a pressure of 560 torr and at a temperature of 7 °C. What is its volume at STP? 10- A metal tank in a garage normally contained compressed air at 4 atm. And 7 °C. In a fire the tank exploded although it was known to be safe up to a pressure of 14 atm. If you were the detective of the insurance company, what value would you calculate for the minimum temperature of the fire near the tank?