Expansion

Temperature and Heat -------------------------------

Joseph F. Alward, PhD Department of Physics University of the Pacific

Lord Kelvin

Fahrenheit and Celsius F = (9 / 5) C + 32

----------------------------------------------------------------Example 1: What is 100 C in Fahrenheit ? (9 / 5)100 + 32 = 180 + 32 = 212 F ----------------------------------------------------------------Example 2: At what temperature do the Fahrenheit and Celsius scales read the same? x = (9 / 5) x + 32 (- 4 / 5) x = 32 x = - 40 - 40 C is the same temperature as - 40 F

Celsius and Kelvin

Example 1: What is 100 C in Kelvin? 100 + 273.15 = 373.15 K -------------------------------------------------------Example 2: What is 0 K in Centigrade? C = K - 273.15 = 0 - 273.15 = -273.15

William Thomson (Lord Kelvin) 1824-1907

K = C + 273

Absolute Zero

No motion or pressure at -273.15 C Absolute zero : O K

Example Problem

The equation of a straight line: y = mx + b P = mT + P0 4000 = m(0) + P0 (drop units) P0 = 4000 Pa ----------------------------------------0 = m(-273.15) + 4000

What is the pressure when the temperature is -100 C? (1 Pascal = 1 N/m2)

m = 4000/273.15 = 1.46 Pa/C ----------------------------------------P = 1.46 T + 4000 P = 1.46(-100) + 4000 = 3854 Pa

Thermal Expansion

Bridge expansion joint. In August the teeth will mesh together.

Concrete patio buckles. No expansion gap was included.

Linear Expansion

This track was laid in February; the picture was taken in August.

Space between slabs of concrete in sidewalk allows expansion in the summer.

Oil Pipe Line

Hole Expansion

Railroad track expansion joint

Hole Expansion Explained

Coefficient of Linear Expansion Example: Steel rod L = 5 meters α = 11 x 10-6 C-1 T0 = 25 C T = 55 C -------------------------------∆Τ = 55 C - 25 C = 30 C ∆L = αL∆Τ = 11 x 10-6 C-1 x 5 m (30 C) = 1.65 x 10-3 m = 1.65 mm

∆L = αL∆Τ α = coefficient of linear expansion [C-1]

Linear Expansion Example ∆L = αL∆Τ ∆LBrass + ∆LAluminum = 0.002 m 19 x 10-6 (1.5)∆Τ + 23 x 10-6 (0.8)∆Τ = 0.002 T0 = 28 C At what temperature will the gap be closed? -----------------------------------------------------------Brass: 19 x 10-6 C-1 Aluminum: 23 x 10-6 C-1

Volume Expansion

∆Τ = 42.64 C T = 28 + 42.64 = 70.64 C

Volume Expansion

Atoms have small amplitude of vibration at low temperature.

At higher temperature, atoms have larger amplitude of vibration.

Volume Expansion Problem: What is the change in volume of an aluminum sphere of radius r = 20 mm whose

β = 72 x10-6 C-1 ∆V = βV0∆T V0 = (4/3)πr3

temperature changes from 50 C to -40 C?

= 33,500 mm3 ∆T = - 90 C (negative) ∆V = -217 mm3

Coefficients of Area and Volume Expansion Rectangle L0 = initial length W0 = initial width ∆Τ = change in temp ∆L = change in length ∆W = change in width L = L0 + ∆L = L0 + αL0∆Τ W = W0 + ∆W = W0 + αW0∆Τ

∆A = A - A0 = (L0 + αL0∆Τ)( W0 + αW0∆Τ) - L0W0 = L0W0 + 2α L0W0∆Τ + α2 L0W0(∆Τ)2 - L0W0 = (1 + α∆Τ/2)2α L0W0∆Τ Note: α = 10-6 C-1 typically ∆Τ < 50 C typically Ignore α ∆Τ/2 compared to 1: ∆A = 2α L0W0∆Τ = 2α A0∆Τ = γA0∆Τ γ = coefficient of area expansion = 2α β = coefficient of volume expansion = 3α What are the units of these coefficients?

Loosening a Tight Nut

A nut is very tight on a screw.

The nut expands, the screw expands,

How shall it be loosened? By heating, or by cooling?

and the space expands. Shrink-fit iron rims on wooden wheels.

Expansion of Glass Boiling Water Breaks Ordinary Glass Cup

Expansion Coefficients of Selected Materials Material

Linear Expansion Coefficient α x 10-6 C-1

Pyrex has one-third the coefficient of expansion of ordinary glass.

Bimetallic Strips

A bi-metallic strip two layers of two different metals.

Aluminum Brass Concrete Steel Glass (ordinary) Glass (Pyrex)

24 19 12 11 9 3.2

Brass: Steel:

α = 19 x 10-6 C-1 α = 11 x 10-6 C-1

Brass side expands more than steel when heated, and contracts more when cooled.

Bimetallic Strip Application

The heat generated by the current passing through the steel and brass strips raises their temperature, causing uneven expansion.

Radiator Overflow Reservoir Radiator is full (15 quarts) when T = 6 C. How much overflow will occur at T = 92 C ? ----------------------------------------------------------∆ T = 92 - 6 = 86 C Coolant: ∆ V = βV0 ∆ T = (410 x 10-6 )(15)(86) = 0.529 quart Radiator: ∆ V = βV0 ∆ T = (51 x 10-6)(15)(86) = 0.066 quart Overflow: 0.529 - 0.066 = 0.463 quart βcoolant = 410 x 10-6 C-1 βradiator = 51 x 10-6 C-1

Ice is Less Dense than Water

Water molecules in crystal form have an open-structured hexagonal arrangement, so water expands upon freezing (water becomes less dense). (From Conceptual Physics, by Paul G. Hewitt)

Why Do Lakes Freeze on the Top First? Freezing of Lakes As air above lake cools, the water at the top contracts and becomes more dense than the water below. This cooler water sinks and warmer water takes its place at the top. When the lake is at 4 C, further lowering of the air temperature causes an expansion of the water on top. This top water is now less dense than the water below, so it stays on top, where it freezes.