Biomedical Engineering Reference
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our present discussion, but the important result is not. Boltzmann distribution for energies
is given by
E
m
i
exp
n
i
n
T
¼
k
T
B
(6.6)
P
j
exp
E
m
j
k
T
B
where n
i
is the number of molecules at equilibrium temperature T, with an energy E
mi
, n
T
is
the total number of molecules in the system, and k
B
is the Boltzmann constant. Because
velocity and speed are related to energy,
Eqn (6.6)
can be used to derive relationships
between temperature and the speed of molecules. The denominator in this equation is known
as the canonical partition function.
For the case of an “ideal gas” consisting of noninteracting atoms (hard spheres) in the
ground state, all energy is in the form of kinetic energy. That is,
v
x
þ
v
y
þ
v
z
E
m
¼
m
(6.7)
m
2
where m
m
is the mass of a molecule, and (v
x
, v
y
, v
z
) are the velocity components in the Carte-
sian coordinates (x, y, z). We can then rewrite
Eqn (6.6)
as
!
!
v
x
þ
v
y
þ
v
z
v
x
þ
v
y
þ
v
z
n
i
n
¼
1
^
¼
1
^
g
exp
m
g
exp
M
(6.8)
m
2
k
T
2
RT
T
B
where
g
is the partition function, corresponding to the denominator in
Eqn (6.6)
. M is the
molecular mass and R is the ideal gas constant. This distribution of n
i
/n
T
is proportional
to the probability density function P( ) for finding a molecule with these values of velocity
components. Thus,
^
!
M
v
x
þ
v
y
þ
v
z
v
z
Þ¼
c
P
ð
v
x
;
v
y
;
g
exp
(6.9)
^
2
RT
where c is the normalizing constant, which can be determined by recognizing that the prob-
ability of a molecule having any velocity must be 1. Therefore, the integral of
Eqn (6.9)
over all
v
x
, v
y
, and v
z
must be 1.
Z
N
Z
N
Z
N
P
ð
v
x
;
v
y
;
v
z
Þ
d
v
x
d
v
y
d
v
z
¼ 1
(6.10)
N
N
N
It can be shown that
M
2p
3=2
c
^
g
¼
(6.11)
RT
Substituting
Eqn (6.11)
into
Eqn (6.9)
, we obtain
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