Physics 231
Topic 12: Temperature,
Thermal Expansion, and Ideal
Gases
Alex Brown
Nov
18-23
MSU Physics
231 Fall2015
2015
1
homework
3rd midterm
final Thursday 8-10 pm
makeup Friday final 9-11 am
MSU Physics 231 Fall 2015
2
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
MSU Physics 231 Fall 2015
3
Conversions:
Tc = Tk - 273.15
Tf = (9/5)Tc + 32
Helium
boils at
Tk=4
MSU Physics 231 Fall 2015
4
Binding Forces
Potential Energy
Kinetic energy ~ T (temperature)
0
-Emin
R
The curve depends on
the material, e.g. Emin is
different for water and
iron
R
2 atom/molecules
MSU Physics 231 Fall 2015
5
Solid (low T)
Potential Energy
0
Kinetic energy ~ T
Rmin
R
-Emin
The temperature (and thus kinetic energy)
is so small that the atoms/molecules can only
oscillate around a fixed position Rmin
MSU Physics 231 Fall 2015
6
Liquid (medium T)
Potential Energy
Kinetic energy ~ T
Rmin
0
R
-Emin
On average, the atoms/molecules like to
stick together but sometimes escape and
can travel far.
MSU Physics 231 Fall 2015
7
Gas (high T)
Kinetic energy ~ T
Potential
Energy
Rmin
0
-Emin
R
The kinetic energy is much larger than
Emin and the atoms/molecules move around
randomly.
MSU Physics 231 Fall 2015
8
What happens if the temperature
of a substance is increased?
Rmin=Rave(T=0)
Kinetic energy ~ T
0
-Emin
R
T=0: Average distance between
atoms/molecules: Rmin
MSU Physics 231 Fall 2015
9
What happens if the temperature
of a substance is increased?
Rmin=Rave(T=0)
Kinetic energy ~ T
Rave(T>0) > Rmin
0
R
-Emin
T>To: The average distance
between atoms/molecules is
larger than Rmin:
the substance expands
MSU Physics 231 Fall 2015
10
Thermal expansion
surface
volume
L
L= Lo T
length
A =  Ao T
V =  Vo T
 = 2
 = 3
L0
: coefficient of linear expansion
different for each material
Some examples:
 = 24x10-6 1/K Aluminum
 = 1.2x10-4 1/K Alcohol
MSU Physics 231 Fall 2015
T=T0
T=T0+T
11
MSU Physics 231 Fall 2015
12
A Heated Ring
A metal ring is heated. What is true:
a) The inside and outside radii become larger
b) The inside radius becomes larger, the outside radius
becomes smaller
c) The inside radius becomes smaller, the outside radius
becomes larger
d) The inside and outside radii become smaller
PHY
231
MSU
Physics 231 Fall 2015
13
13
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.
MSU Physics 231 Fall 2015
14
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)
MSU Physics 231 Fall 2015
15
Ice
 (g/cm3)
liquid
1
Phase transformation
0.917
ice
Ice takes a larger volume than water!
A frozen bottle of water might explode
MSU Physics 231 Fall 2015
16
Thermal equilibrium
Thermal contact
Low temperature
Low kinetic energy
Particles move slowly
High temperature
High kinetic energy
Particles move fast
Transfer of kinetic energy
Thermal equilibrium: temperature is the same everywhere
MSU Physics 231 Fall 2015
17
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
MSU Physics 231 Fall 2015
18
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
MSU Physics 231 Fall 2015
19
Potential
Energy
Rmin
0
-Emin
R
Ideal gas: we are neglecting the potential energy between
The atoms/molecules
Potential
Energy
Kinetic energy
0
R
MSU Physics 231 Fall 2015
20
Properties of gases
V = volume
P = pressure
T = temperature in K (Kelvin)
n = number of moles
Example balloon
MSU Physics 231 Fall 2015
21
Molecular mass
m  mass of one atom (or molecule)
N A  6.02 10 23
Avagodro's numbers
M molar  m N A
molecular (molar) mass
for example M molar (carbon)  12.0 g  0.0120 kg
0.012
- 26
m (carbon) 

2.00

10
kg
23
6.02 10
MSU Physics 231 Fall 2015
22
Name
Number of electrons
Z
X
A
molar
mass
in grams
MSU Physics 231 Fall 2015
23
Weight of 1 mol of atoms
1 mol of atoms weighs A grams
Examples:
1 mol of Hydrogen weighs
1 mol of Carbon weighs
1 mol of Oxygen weighs
1 mol of Zinc weighs
(A is the molar mass)
1.0 g
12.0 g
16.0 g
65.4 g
What about molecules?
H2O 1 mol of water molecules:
2 x 1.0 g (due to Hydrogen)
1 x 16.0 g (due to Oxygen)
Total: 18.0 g
MSU Physics 231 Fall 2015
24
Number of atoms and moles
N  total number of atoms (or molecules)
N
n
 number of moles or
NA
N  n NA
so one mole contains N A atoms (or molecules)
M  m N  total mass of all atoms (or molecules)
M  n M molar
MSU Physics 231 Fall 2015
25
Example
A cube of Silicon (molar mass 28.1 g) is 250 g.
A) How much Silicon atoms are in the cube?
Total number of moles
n = M / Mmolar = 250/28.1 = 8.90
N = n NA = (8.9) (6.02x1023) = 5.4x1024 atoms
B) What would be the mass for the same number of
gold atoms (molar mass 197 g)
M = n Mmolar = (8.90) (197 g) = 1750 g
MSU Physics 231 Fall 2015
26
Question
1) 1 mol of CO2 has a larger mass than 1 mol of CH2
2) 1 mol of CO2 contains more molecules than 1 mol of CH2
a) 1) true
2) true
b) 1) true
2) false
c) 1) false
2) true
d) 1) false
2) false
MSU Physics 231 Fall 2015
27
Properties of gases
V = volume
P = pressure
T = temperature in K (Kelvin)
n = number of moles
Example balloon
MSU Physics 231 Fall 2015
28
Boyle’s Law (fixed n and T)
½P0 2V0
P0 V0
2P0 ½V0
At constant temperature: P ~ 1/V
implies that PV = constant
MSU Physics 231 Fall 2015
29
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
MSU Physics 231 Fall 2015
30
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
MSU Physics 231 Fall 2015
31
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
MSU Physics 231 Fall 2015
32
Boyle & Charles & Gay-Lussac
IDEAL GAS LAW
Does not depend on what type or
atom or molecule
PV  nRT
n = number of moles
R = universal gas constant 8.31 J/mol·K
If the number of moles is fixed
PV
 constant
T
or
P1V1 P2V2

T1
T2
MSU Physics 231 Fall 2015
33
Example
An ideal gas occupies a volume of 1.0 cm3 at 200 C at 1 atm.
A) How many atoms are 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
B) If the pressure is reduced to 1.0x10-11 Pa, while the
temperature drops to 00C, how many atoms remained
in the volume?
T = 00C = 273K , P = 1.0x10-11 Pa, V = 1x10-6 m3
n=4.4x10-21 mol
N=2.6x103 particles (almost vacuum)
MSU Physics 231 Fall 2015
34
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
= 1.013 x 105 + (1.0x103)(9.8)(950)
= 94.2 x 105 Pa
P1V1 P2V2

T1
T2
V2 
P1 T2
V1  (93.0)(1.046) V1
P2 T1
Vsurface=146 cm3
Expanded by a factor of 97
MSU Physics 231 Fall 2015
35
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) 0.4
b) 1
c) 2.5
d) 5
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 P1
a factor of 10.
MSU Physics 231 Fall 2015
36
“Standard temperature and pressure”
(STP)
P  1 atm  1.013  10 5 Pa
T  0 o C  273.15 o K
MSU Physics 231 Fall 2015
37
Moles
N
n  number of moles 
NA
N  total number of objects
N A  6.02  10 23  Avagadro' s number
MSU Physics 231 Fall 2015
38
macroscopic to microscopic
PV  n R T
macroscopic quantities
PV  N k BT
N = number of atoms or molecules (microscopic)
R
8.31
 23
kB 


1
.
38

10
(J/K)
23
N A 6.02 10
(Boltzman's constant)
n R  N kB
MSU Physics 231 Fall 2015
39
Quiz
Given P1 = 1 atm
V1 = 2 m3
T1 = 100 K
N1 = NA
P2 = 2 atm
V2 = 10 m3
N2 = 10 NA
T2 = ? K
A)
B)
C)
D)
E)
200
500
2000
5000
100
MSU Physics 231 Fall 2015
40
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
MSU Physics 231 Fall 2015
41
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
MSU Physics 231 Fall 2015
42
Movie of gas in two dimensions
MSU Physics 231 Fall 2015
43
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 = 10-12 Pa
d = 108 m
MSU Physics 231 Fall 2015
44
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
vrms  v 2
MSU Physics 231 Fall 2015
45
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
1 2
K   mv 
2

average translation kinetic energy
average squared velocity
mass of the object (atom or molecule)
MSU Physics 231 Fall 2015
46
MSU Physics 231 Fall 2015
47
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
2N K
PV 
3
combine with PV  N k BT
to get
3
K  k BT
2
How the average kinetic energy of one
atom is related to temperature
MSU Physics 231 Fall 2015
48
root-mean-square (rms) velocity for
one atom or molecule
combine
vrms
3
K  k BT
2
with
1

K   mv 2 
2

3k BT
 v 

m
2
MSU Physics 231 Fall 2015
3RT
M molar
49
Example
What is the rms speed of air at 1 atm and room temperature (293 K)?
Assume it consist of molecular Nitrogen only (N2)?
vrms
3k BT
 v 

m
2
3RT
M molar
R = 8.31 J/mol K
T = 293 K
Mmolar = (2 x 14)x10-3 kg/mol
vrms = 511 m/s = 1140 mph !
MSU Physics 231 Fall 2015
50
Total thermal energy
Eth 
d
d
d
d
N K  N k B T  nRT  PV
3
2
2
2
(since Nk B  nR )
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)
MSU Physics 231 Fall 2015
51
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?
MSU Physics 231 Fall 2015
52
Diffusion
MSU Physics 231 Fall 2015
53