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Kinetic theory of gases
A glass of water (again)
A glass of water can have potential energy
(because I lift it from the table)
It can have kinetic energy (because I drop it)
These are “bulk properties”
Look at the water in detail: disordered motion
(“thermal motion”) of the molecules. Energy
associated with it
Internal energy
Internal energy: due to disordered motion of
molecules
Glass of water on microscopic scale:
kinetic energy (molecules in motion)
potential energy (attraction between molecules)
Temperature
Temperature measures translational kinetic energy
(so T1 = T2 does not imply U1 = U2!)
State variables
state variable: precisely measurable physical
property which characterizes the state of a
system, independently of how the system was
brought to that state
Examples: p, V, T, U
Any property that is a combination of state
variables is a state variable itself
Empirical gas laws
V  N when p,T constant (Avogadro)
p  T when N,V constant (Charles/Gay-Lussac)
p  1/V when N,T constant (Boyle)
Ideal gas law:
N
pV  NkT or p  kT
V
k is the same for all gases: 1.381  10-23 J K-1
Avogadro’s number
NA= the number of atoms in 12 g of 12C.
Value: 6.0221023 mol-1
A mole of molecular species has NA molecules
Rewrite:
N / NA
n
p
 (kN A )  T  RT
V
V
R = 8.3145 J mol-1 K-1
Kinetic theory of gases
Ideal gas:
neglect intermolecular attractions
all collisions perfectly elastic
dilute gas, volume occupied is negligible
Pressure due to collisions with wall
Newton’s Second Law:
P
F
t
Kinetic theory of gases II
Force due to collisions with wall
 momentum transferr ed  number of 



per collision per molecule  molecules 


round trip time 
Works because total momentum is conserved
in molecular collisions
Kinetic theory III
collision 1:
t=0
v
vx
vy
L
collision 2:
t = 2L/vx
Kinetic Theory IV
So:
F
2mvx   N
2L
vx

2
Nmvx
L

2
Nmvx A
V
Not all molecules have same vx: use
vx2  v 2y  vz2  13  v 2
F 1 mv 2 N 2 1 2 N
 3  2 mv 
Substitute: p   3 
A
V
V
Kinetic Theory V
Compare with empirical ideal gas law:
p
2  1 mv 2
3 2
N N
  kT 
V V
1 mv 2
2
 32 kT
For ideal monatomic gases this translational
kinetic energy is the only form of energy:
U  32 kT
Kinetic Theory – Summary
Using Newtonian mechanics we have
established:
the relationship between p, N/V, T;
the universality of the gas constant;
the relationship between temperature and K.E.
the internal energy of a monatomic ideal gas
Question time!
Consider a fixed volume of gas. When N or T
is doubled the pressure doubles since pV=NkT
T is doubled: what happens to the rate at which a
molecule hits a wall?
(a)  1 (b) 2 (c) 2
N is doubled: what happens to the rate at which a
molecule hits a wall?
(a)  1 (b) 2 (c) 2
Question 2
Container A contains 1 l of helium at 10 °C,
container B contains 1 l of argon at 10 °C.
a) A and B have the same internal energy
b) A has more internal energy than B
c) A has less internal energy than B
Question 3
Container A contains 1 l of helium at 10 °C,
container B contains 1 l of argon at 10 °C.
a) The argon and helium atoms have the same
average velocity
b) The argon atoms are on average faster than the
helium atoms
c) The argon atoms are on average slower than the
helium atoms
Question 4
Container A contains 1 l of helium at 10 °C,
container B contains 1 l of helium at 20 °C.
a) The average speeds are the same
b) The average speed in A is only a little higher
c) The average speed in A is about 2 higher
d) The average speed in A is about twice as high
Van der Waals gases
Two phenomena that we have neglected so far
can easily be included
molecules are not point particles
molecules attract each other
Volume occupied: replace V by V-Nb
b is about 4 times the spatial volume occupied by
a molecule (b depends on the distance at which
they “feel” each other)
Attractive forces
Molecules near the wall are only attracted by
other molecules from the other side
The gas is less dense near the wall
p  kT  reduced density 
N 
aN 
p  kT   1 

V  VkT 
We won’t derive this, but: the average velocity
is the same throughout the gas
Van der Waals equation
This leads to an improved formula
2

aN
p
V  Nb   NkT
2 

V


Not as easy to use but agrees better with
experiment at high densities, near phase
transitions, etc.
Van der Waals gas & ideal gas I
Consider two equal amounts of gas at identical
temperature. One can be treated as an ideal gas,
the other is a Van der Waals gas.
a) The internal energies are the same
b) The Van der Waals gas has more internal energy
c) The Van der Waals gas has less internal energy
d) We can’t be sure
Van der Waals gas & ideal gas II
Consider two equal amounts of gas at identical
temperature. One can be treated as an ideal
gas, the other is a Van der Waals gas. The
specific heat at constant volume is
a) The same for both gases
b) Higher for the Van der Waals gas
c) Lower for the Van der Waals gas
d) We can’t be sure
Van der Waals gas & ideal gas III
An ideal gas and a Van der Waals gas at the
same temperature expand isothermally by the
same amount. The work done is
a) The same for both gases
b) Higher for the Van der Waals gas
c) Lower for the Van der Waals gas
d) We can’t be sure
Van der Waals gas & ideal gas IV
An ideal gas and a Van der Waals gas at the
same temperature expand isothermally by the
same amount. The heat added is
a) The same for both gases
b) Higher for the Van der Waals gas
c) Lower for the Van der Waals gas
d) We can’t be sure
PS225 – Thermal Physics topics
The atomic hypothesis
Heat and heat transfer
Kinetic theory
The Boltzmann factor
The First Law of Thermodynamics
Specific Heat
Entropy
Heat engines
Phase transitions