Environmental Engineering Reference
In-Depth Information
where the thermal capacities
c
tr
,
c
rot
,and
c
vib
correspond to translational, rotation-
al, and vibrational degrees of freedom. Because the mean kinetic energy of particles
in thermal equilibrium is 3
T
/2, we have
c
tr
3/2. The classical energy of the rota-
tional degree of freedom is
T
/2 and coincides with the translational energy. Hence,
for a diatomic molecule with two rotational degrees of freedom we have
c
rot
D
D
1
in the classical limit
B
T
(
B
is the rotational constant). Polyatomic molecules
have three rotational degrees of freedom instead of two as for a diatomic molecule.
Hence, the rotational thermal capacity of polyatomic molecules is 3/2 if the thermal
energy is much greater than the rotational excitation energy. This condition is satis-
fied at room temperature for rotational degrees of freedom of most molecules, but
can be violated for the vibrational degrees of freedom. Therefore, we determine the
vibrational thermal capacity by any available relation between the excitation energy
for the first vibrational level and the thermal energy of the molecule.
Using the Planck formula (1.56) for the number of excitations for the harmon-
ic oscillator, we have the following expression for the average oscillator excitation
energy above the minimum of the potential energy
D
„
2
C
„
ω
ε
,
vib
exp(
„
ω
/
T
)
1
where
is the frequency of the oscillator. We find that the vibrational thermal
capacity corresponding to a given vibration is
ω
„
T
2
D
@
exp(
„
ω
/
T
)
vib
c
vib
T
D
.
(4.49)
1
/
T
)
2
@
exp(
„
ω
This expression yields
c
vib
D
1 in the classical limit
„
ω
T
. For the opposite
limit,
T
, the vibrational thermal capacity is exponentially small.
The above analysis yields
„
ω
3
2
C
n
rot
2
C
c
V
D
c
vib
(4.50)
for the thermal capacity of a molecule, where
n
rot
is the number of rotational de-
grees of freedom of the molecules, and the vibrational thermal capacity is a sum of
individual vibrations, for which (4.49) can be used. Note that the analysis uses the
assumption of thermodynamic equilibrium between different degrees of freedom.
4.2.7
Momentum Transport and Gas Viscosity
Transport of momentum takes place in a moving gas such that the mean velocity of
the gas particles varies in the direction perpendicular to the mean velocity. Particle
transport then leads to an exchange of particle momenta between gas elements
with different average velocities. This creates a frictional force that slows those
gas particles with higher velocity and accelerates those having lower velocities. We
can estimate the value of the viscosity coefficient by analogy with the procedures
