2015.08.31.
Thermodynamics.
Kinetic theory of gases
Department of Biophysics, Medical School
University of Pecs
Kinetic Theory of Gases
macroscopic properties (P, T)
ideal gas law
derived from experimental
observations (empirically)
microscopic behavior
of gas molecules
kinetic theory of gases
each molecule is a
physical body
that moves continually
in random directions
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2015.08.31.
Derivation of Kinetic Energy equation
Consider a gas molecule colliding elastically with the right
wall of the container and rebounding from it.
• In a given container which has a volume of V ,
there are N gas molecules. Each molecule has a
mass of m. And pressure of this system is equal
to P.
• Molecular number per unit volume:
N
n
V
dV  vx  dt  dA
Momentum  n  dV  m  vx
 n  vx  dt  dA  m  vx
 n  m  vx 2  dt  dA
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2015.08.31.
• According to momentum equation
m1  v1  m2  v2  F  t
• Because of elastic collision, after collision only
velocity’s direction changes to its opposite
direction. Time elapsed during collision is two
times dt since the molecule has to travel forward
and backward.
momentum  (momentum)  F  2  dt
F  n  m  vx 2  dA
• The pressure exerted on the wall
Force n  m  vx 2  dA
P

 n  m  vx 2
Area
dA
• But these gas particles can move in three
directions, they have three degree of freedom.
Hence, we have
Px  Py  Pz  P
1
P   n  m  v2
3
1
P V   n V  m  v 2
3
1
  N  m  v2
3
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2015.08.31.
Kinetic energy of ideal gas
• Boltzmann constant
P V  n  R  T 
N
N
 R T
NA
R
T  N  k T
NA
k: Boltzmann constant 1.38 × 10−23 J/K
1
P V   N  m  v 2  N  k  T
3
• Kinetic Energy:
1
3
KEavg   m  v 2   k  T
2
2
It shows the average translational kinetic energy of an
ideal gas.
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Kinetic energy of ideal gas
• This equation could be understood as total
kinetic energy in x, y and z directions
1
1

Kinetic Energy   m  v 2  3    k  T 
2
2

• Hence, in each direction, the kinetic energy is
equal to.
1
 k T
2
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• The distribution of speeds for nitrogen gas
molecules at three different temperatures. A
higher temperature will result in a faster
molecular speed.
• The distribution of speeds of three different
gases at the same temperature
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Distribution function
• Population distribution in Canada
Percentage(%)  f ( Age)d( Age)
Distribution function
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Maxwell-Boltzmann Distribution
• There are totally N molecules in one container. In
kinetic energy range between E and E+dE, the
number of molecule is dNE.
dN E
 f ( E )  dE
N
f(v) is called distribution function.
• Maxwell-Boltzmann Distribution function
f (E)  A  e

E
k T
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2015.08.31.
Most probable rate and Mathematical
average rate
• The most probable rate could be written for one
molecule or one mole of molecules:
vm 
vm 
2  k T
m
2  k T  N A

m  NA
2  R T
M
• Mathematical average rate
Ni  vi
v1  N1  v2  N 2  v3  N3  ... 
8  k T
i
vav 


N
N
 m
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Root-mean-square rate
• Root-mean-square rate
1
  vi 2 dN v  2
  3 k T
u i


N
m


This rate is rate used in the expression of average
kinetic energy
1
3
KEavg   m  u 2   k  T
2
2
• Ratio among these three rates:
2  k T
8  k T
3 k T
:
:
m
 m
m
 1:1.128 :1.224
vm : vav : u 
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