Environmental Engineering Reference
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ferent vibrational and translational temperatures. However, the resonant character
of exchange of vibrational excitation takes place only for weakly excited molecules.
At moderate excitations, the resonant character is lost because of molecular anhar-
monicity. This leads to a particular type of distribution of molecular states that we
shall now analyze.
We consider a nonequilibrium gas consisting of diatomic molecules where the
translational temperature T differs from the vibrational temperature T ν .Theequi-
librium between vibrational states is maintained by resonant exchange of vibra-
tional excitations in collisions of molecules, as expressed by
v 1 )
v 2 ) ,
M (
v 1 )
C
M (
v 2 )
$
M (
C
M (
(1.71)
where the quantities in the parentheses are vibrational quantum numbers. Assum-
ing molecules to be harmonic oscillators, we obtain from this the condition
D v 1 C v 2 .
C v 2
v 1
The excitation energy of a vibrational level is
1/2) 2
E v D„ ω
( v C
1/2)
ω
x e ( v C
,
where
is the harmonic oscillator frequency and x e is the anharmonicity parame-
ter. The second term of this expression is related to the establishment of equilibri-
um in the case being considered, where translational and vibrational temperatures
are different. Specifically, the equilibrium condition leads to the relation
N ( v 1 ) N ( v 2 ) k v 1 , v 2
ω
! v 1 , v 2 D
N ( v 1 ) N ( v 2 ) k ( v 1 , v 2
! v 1 , v 2 ),
where N ( v ) is the number density of molecules in a given vibrational state and
k ( v 1 , v 2
! v 1 , v 2 ) is the rate constant for a given transition. Because these tran-
sitions are governed by the translational temperature, the equilibrium condition
gives
k v 1 , v 2
! v 1 , v 2 D
k v 1 , v 2
! v 1 , v 2 exp(
Δ
E / T ),
E ( v 1 )
E ( v 2 ) is the difference of the energies
where
Δ
E
D Δ
E ( v 1 )
C Δ
E ( v 2 )
Δ
Δ
1/2) 2 . From this, one finds the
for a given transition, where
Δ
E ( v )
D„ ω
x e ( v C
number density of excited molecules to be [67, 68]
N 0 exp
,
ω v
T ν
ω
x e v
(
v C
1)
N ( v )
D
C
(1.72)
T
where N 0 is the number density of molecules in the ground vibrational state. This
formula is often called the Treanor distribution.
Formula (1.72) gives a nonmonotonic population of vibrational levels as a func-
tion of the vibrational quantum number. Assuming the minimum of this function
to correspond to large vibrational numbers, we have for the position of the mini-
mum
1
2 x e
T
T ν
D
v min
1 ,
(1.73)
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