Chemistry Reference
In-Depth Information
In the thermodynamics, there is an important notion, namely,
l
, the number
of the degrees of freedom of the molecule. The energy of the molecule is
E
∼
l
p
2
2
m
.
Assuming that the value of
l
depends on the momentum, we replace this relation
by the following one [6]:
E
σ
=
c
σ
|
p
|
2+
σ
,
c
σ
= const
.
(12)
We can say that both the average momentum and the number of the degrees
of freedom depend on the temperature, and hence the number of the degrees
of freedom depends on the momentum. We assume that this dependence is
the simplest one, namely, it is a power dependence (the parameter
σ
of in the
exponent characterizes the molecule). Here the potential
Ω
takes the form
Z
Ω
γ
=
π
1+
γ
V T
2+
γ
m
2+
γ
Γ(2 +
γ
)
∞
−(
t
−
T
)
dt,
t
1+
γ
e
(13)
0
where
3
2 +
σ
−
1 =
1
−
σ
2 +
σ
.
It is readily verified that, in this case, relation (11) will remain valid for an ideal
gas.
γ
=
Thus, we replace the parameter of the integer degrees of freedom by some
continuous parameter that characterizes the given molecule. In principle, this
parameter
γ
is of the same physical origin as the number of the degrees of
freedom. But since it is continuous, it takes into account more details of the
spectrum of the molecule.
Now we can rigorously formulate the axiom of thermodynamics corre-
sponding to the approximate conservation of the gas density (this corresponds
to the physicists' statement in equilibrium thermodynamics: “the density is
homogeneous in a vessel”; physicists consider equilibrium thermodynamics as
a separate discipline, and then, separately, fluctuation theory).
Main mathematical axiom of thermodynamics.
Consider a vessel of volume
V
containing
N >
10
19
identical molecules
corresponding to the parameter
γ
=
γ
0
.
Consider a small convex volume of
