1. Physics 231 Topic 12: Temperature, Thermal Expansion, and Ideal Gases
Alex Brown
Nov

2. homework
3rd midterm
final Thursday 8-10 pm
makeup Friday final 9-11 am

3. Key Concepts: Temperature, Thermal Expansion, and Ideal Gases
Temperature and Thermometers
Thermal Energy & Temperature
Thermal Expansion
Coefficient of thermal expansion
Ideal Gases
State Variables
Ideal gas law
Kinetic Theory of Gases
Kinetic & thermal energy
Maxwell distribution
Covers chapter 12 in Rex & Wolfson

4. Conversions:
Tc = Tk
Tf = (9/5)Tc + 32
Helium
boils at Tk=4

5. Binding Forces Kinetic energy ~ T (temperature) Potential Energy -Emin
-Emin R
The curve depends on
the material, e.g. Emin is
different for water and
iron R
2 atom/molecules

6. Solid (low T) Kinetic energy ~ T Potential Energy Rmin R -EminR
-Emin
The temperature (and thus kinetic energy)
is so small that the atoms/molecules can only
oscillate around a fixed position Rmin

7. Liquid (medium T) Potential Energy Kinetic energy ~ T Rmin R -EminR
-Emin
On average, the atoms/molecules like to
stick together but sometimes escape and
can travel far.

8. Gas (high T) Kinetic energy ~ T Potential Energy Rmin -Emin R
-Emin R
The kinetic energy is much larger than
Emin and the atoms/molecules move around
randomly.

9. What happens if the temperature of a substance is increased?
Rmin=Rave(T=0)
Kinetic energy ~ T R
T=0: Average distance between
atoms/molecules: Rmin
-Emin

10. What happens if the temperature of a substance is increased?
Rmin=Rave(T=0)
Kinetic energy ~ T
Rave(T>0) > Rmin
T>To: The average distance
between atoms/molecules is
larger than Rmin:
the substance expands R
-Emin

11. Thermal expansion L= Lo T A =  Ao T  = 2 V =  Vo T  = 3
length L
L= Lo T
surface
A =  Ao T  = 2
volume
V =  Vo T  = 3 L0
: coefficient of linear expansion
different for each material
Some examples:
 = 24x /K Aluminum
 = 1.2x /K Alcohol
T=T0
T=T0+T

13. A Heated Ring A metal ring is heated. What is true:
The inside and outside radii become larger
The inside radius becomes larger, the outside radius becomes smaller
The inside radius becomes smaller, the outside radius becomes larger
The inside and outside radii become smaller
PHY 231

14. Demo: Bimetallic Strips
top
bottom
Application: contact in a refrigerator
top<bottom if the temperature increases,
The strip curls upward, makes contact and switches
on the cooling.

15. Water: a special case Coef. of expansion is negative: If T drops
the volume becomes
larger
Coef. Of expansion is
positive: if T drops the volume becomes smaller
Below this ice is formed (it floats on water)

16. Ice  (g/cm3) liquid 1 Phase transformation 0.917 ice
Ice takes a larger volume than water!
A frozen bottle of water might explode

17. Thermal equilibrium Thermal contact High temperature Low temperature
High kinetic energy
Particles move fast
Low temperature
Low kinetic energy
Particles move slowly
Transfer of kinetic energy
Thermal equilibrium: temperature is the same everywhere

18. Zeroth law of thermodynamics
If objects A and B are both in thermal equilibrium
with an object C, than A and B are also in thermal
equilibrium.
There is no transfer of energy between A, B and C

19. Ideal Gas: properties Collection of atoms/molecules that
Exert no force upon each other
The energy of a system of two atoms/molecules cannot be reduced by bringing
them close to each other
Take no volume
The volume taken by the atoms/molecules
is negligible compared to the volume they
are sitting in

20. Potential
Energy
Rmin
-Emin R
Ideal gas: we are neglecting the potential energy between
The atoms/molecules
Potential
Energy
Kinetic energy R

21. Properties of gases V = volume P = pressure
T = temperature in K (Kelvin)
n = number of moles
Example balloon

22. Molecular mass

23. Name
Number of electrons Z X A
molar mass
in grams

24. Weight of 1 mol of atoms
1 mol of atoms weighs A grams (A is the molar mass)
Examples:
1 mol of Hydrogen weighs g
1 mol of Carbon weighs g
1 mol of Oxygen weighs g
1 mol of Zinc weighs g
What about molecules?
H2O 1 mol of water molecules:
2 x 1.0 g (due to Hydrogen)
1 x g (due to Oxygen)
Total: 18.0 g

25. Number of atoms and moles

26. Example A cube of Silicon (molar mass 28.1 g) is 250 g.
How much Silicon atoms are in the cube?
B) What would be the mass for the same number of
gold atoms (molar mass 197 g)
Total number of moles n = M / Mmolar = 250/28.1 = 8.90
N = n NA = (8.9) (6.02x1023) = 5.4x1024 atoms
M = n Mmolar = (8.90) (197 g) = 1750 g

27. Question 1 mol of CO2 has a larger mass than 1 mol of CH2
1 mol of CO2 contains more molecules than 1 mol of CH2
1) true 2) true
1) true 2) false
1) false 2) true
1) false 2) false

28. Properties of gases V = volume P = pressure
T = temperature in K (Kelvin)
n = number of moles
Example balloon

29. Boyle’s Law (fixed n and T)
½P0 2V0
P0 V0
2P0 ½V0
At constant temperature: P ~ 1/V
implies that PV = constant

30. Charles’ law (fixed n and P)
2V0 2T0
V0 T0
If you want to maintain a constant pressure, the
temperature must be increased linearly with the volume
V ~ T
implies that (V/T) = constant

31. Gay-Lussac’s law (fixed n and V)
P0 T0
2P0 2T0
If, at constant volume, the temperature is increased,
the pressure will increase by the same factor
P ~ T
implies that (P/T) = constant

32. Brown’s law (fixed T and P)
2n0 2V0
n0 V0
If you double the number of particles the volume doubles
n ~ V
implies that (V/n) = constant

33. Boyle & Charles & Gay-Lussac IDEAL GAS LAW
Does not depend on what type or atom or molecule
n = number of moles
R = universal gas constant 8.31 J/mol·K
If the number of moles is fixed

34. Example An ideal gas occupies a volume of 1.0 cm3 at 200 C at 1 atm.
How many atoms are in the volume?
B) If the pressure is reduced to 1.0x10-11 Pa, while the
temperature drops to 00C, how many atoms remained
in the volume?
PV = nRT, so n = PV/(TR) with R=8.31 J/mol K
T=200C=293K, P=1atm=1.013x105 Pa, V=1.0cm3=1x10-6m3
n=4.2x10-5 mol
N = n NA = (4.2x10-5) NA=2.5x1019
T = 00C = 273K , P = 1.0x10-11 Pa, V = 1x10-6 m3
n=4.4x10-21 mol N=2.6x103 particles (almost vacuum)

35. And another!
An air bubble has a volume of 1.50 cm3 at 950 m depth (T=7oC). What is its volume when it reaches the surface (T=20oC).
(water=1.0x103 kg/m3)?
P950m=P0 + water g h
= x (1.0x103)(9.8)(950)
= 94.2 x 105 Pa
Vsurface=146 cm3
Expanded by a factor of 97

36. Quiz A volume with dimensions L x W x H is kept under
pressure P at temperature T. If the temperature is
raised by a factor of 2, and the height is made
5 times smaller, by what factor does the pressure change,
i.e. what is P2/P1? No gas leaks or is added.
a) b) c) d) e) 10
Use the fact PV/T is constant if no gas is added/leaked
P1V1 / T1 = P2V2 / T2
P1V1 / T1 = P2 (V1/5) / (2T1)
P2 = (5)(2)(P1 ) = 10 P a factor of 10.

37. “Standard temperature and pressure” (STP)

38. Moles

39. macroscopic to microscopic
macroscopic quantities
N = number of atoms or molecules (microscopic)

40. Quiz
Given P1 = 1 atm P2 = 2 atm
V1 = 2 m V2 = 10 m3
T1 = 100 K
N1 = NA N2 = 10 NA
T2 = ? K
200
500
2000
5000
100

41. Example How many air molecules at in the room with a volume of 1000 m3
(assume only molecular nitrogen is present N2)?
PV = N kB T
T = 293
P = 1.013x105 Pa
V = 1000 m3
N = 2.5x1028

42. microscopic description: kinetic theory of gases
1) The number of objects is large (statistical model)
2) Their average separation is large
3) The objects follow Newton’s laws
4) Any particular object can move in any direction with a
distribution of velocities
5) The objects undergo elastic collision with each other
6) The objects make elastic collisions with the walls
7) All objects are of the same type

43. Movie of gas in two dimensions

44. mean free path
d = average distance between collisions
air at P = 1 atm d = 68 nm = 68 x 10-9 m
high vacuum P = 10-5 Pa d = 1m
in space P = Pa d = 108 m

45. The Maxwell Distribution
However we can model the distribution of the velocities (& thus the kinetic energies) of the individual gas molecules. The result is the Maxwell Distribution.
The root-mean-square (rms)
velocity is

46. Energy of one object
Objects inside the container have a distribution of velocities
around an average – so each object has an average kinetic energy
given by
average translation kinetic energy
average squared velocity
mass of the object (atom or molecule)

48. Relationship to ideal gas law
The objects bounce off of each other and the walls
of the container (elastic). One can derive the following result
How the average kinetic energy of one
atom is related to temperature

49. root-mean-square (rms) velocity for
one atom or molecule

50. Example
What is the rms speed of air at 1 atm and room temperature (293 K)?
Assume it consist of molecular Nitrogen only (N2)?
R = 8.31 J/mol K
T = 293 K Mmolar = (2 x 14)x10-3 kg/mol
vrms = 511 m/s = 1140 mph !

51. Total thermal energy
d is the number of “degrees of freedom” for the motion
d = 3 for an atom (motion in x, y, z directions) like helium gas
d = 5 for a diatomic molecule (motion in x, y, z and two ways to rotate)
like nitrogen molecule N2 or hydrogen molecule H2
(Homework question for “one degree of freedom” use d = 1)

52. Example
What is the total thermal kinetic energy of the air molecules in the
lecture room (assume only molecular nitrogen is present N2)?
Eth = (d/2) PV = 2.5x108 J
d = 5
P = 1.013x105 Pa
V = 1000 m3
Using KE = (1/2) mv2 this is equivalent to 1000 cars
with m=1000 kg each moving with v = 22.3 m/s (50 mph)
Can we use that energy to do work?

53. Diffusion