D O C . 4 2 7 Q U A N T U M T H E O R Y O F I D E A L G A S 4 2 1
§4. Adiabatic Compression
Let the gas be enclosed in a parallelepiped vessel of the side lengths l1, l2, l3. The
velocity distribution is isotropic but otherwise arbitrary. Let the collisions with the
wall be elastic. Then the distribution of the state does not change over time. Let it
be given by
, (5)
where ρ is an arbitrarily given function of L.
If we displace the walls adiabatically infinitely slowly in such a way that
, (6)
then the distribution remains isotropic, hence of the form (5). How does the distri-
bution change in the process?
If signifies the absolute value of p1 of a molecule, one easily obtains by ap-
plying the laws governing an elastic collision
. (7)
Analogous equations apply for and . From this one obtains, taking (7)
into account,
. (8)
From (4) furthermore follows
,
or according to (8)
, (9)
therefore also
. (10)
In all these formulas, Δ indicates the change that the value being looked at un-
dergoes through the adiabatic change in volume.
Now, under an adiabatic change in volume, the number dN of molecules consid-
ered in (5) undergoes no change. Therefore,
is valid; or because of (10),
. (11)
dn
V
h3
---- -ρdΦ =
Δl1
l1
------- -
Δl2
l2
------- -
Δl3
l3
------- -
1ΔV
3
--------- -
V
= = =
[p. 21]
p1
Δ p1 p1
Δl1
l
------- - –=
Δ p2 Δ p3
ΔL
1
m
--- -( p1 δ p1 • • + + )
2
3
-
ΔV
V
–--L------- = =
ΔdΦ 2π(2m)3
/2§
L1 /2ΔdL
1
2
--L - 1 2ΔLdL¹ /– +
©
·
=
ΔdΦ dΦ-------
ΔV
V
–=
Δ( VdΦ) 0 =
0 Δdn Δ(VρdΦ) = =
Δρ 0 =