Environmental Engineering Reference
In-Depth Information
The number densities of atoms
N
a
and molecules
N
m
are connected by the Saha
formula (1.52), so
N
a
/
N
m
D
/
T
), where
D
is the dissociation energy
and
F
(
T
) has a weak temperature dependence compared with the exponential de-
pendence. Because
N
a
D
F
(
T
)exp(
N
m
and
D
T
,wehave
@
N
m
/
@
T
D
(
D
/
T
2
)
N
m
,and
from (4.43) it follows that
D
T
2
D
D
m
N
m
,
(4.44)
i
where
D
m
is the diffusion coefficient for molecules in an atomic gas. Comparing
this expression with the thermal conductivity coefficient (4.40) due to translational
heat transport, we have
D
T
2
N
m
N
a
i
t
.
(4.45)
In the regime we have been treating, we have the
N
a
T
.There-
fore, the ratio (4.45) can be about unity for comparatively low concentrations of
molecules in the gas. Note that these results are valid if the dissociation equilibri-
um is established fast compared with molecular transport for the distances under
consideration.
N
m
and
D
4.2.6
Thermal Capacity of Molecules
Equation (4.41) for heat transfer allows us to obtain expressions for thermal capac-
ities. This equation refers to a gas contained within a fixed volume. If the process
proceeds in a small part of a large volume of gas, it is necessary to replace the ther-
mal capacity at constant volume
c
V
in (4.41) by the thermal capacity at constant
pressure
c
p
.Itisconvenienttorewrite(4.41)intheform
@
T
@
t
C
w
r
T
D
Δ
T
,
(4.46)
where
/(
c
V
N
) is the thermal diffusivity coefficient.
To find the connection between the thermal capacities
c
V
and
c
p
, let the tem-
perature of a element of gas containing
n
molecules vary by
dT
. Then the energy
variation of this gas element is
dE
D
D
nc
p
dT
. This energy change can also be stat-
ed as
dE
pdV
,where
p
is the gas pressure and
V
is the volume. The
equation of state (4.12),
p
D
nc
V
dT
C
D
Tn
/
V
, allows us to rewrite the second energy change
relation as
dE
D
nc
V
dT
C
ndT
. Comparing the two expressions for
dE
yields
c
p
D
c
V
C
1 .
(4.47)
The thermal capacity
c
V
permoleculecanbewrittenasthesumoftermsdueto
different degrees of freedom, that is,
c
V
D
c
tr
C
c
rot
C
c
vib
,
(4.48)
