Chemistry Reference
In-Depth Information
Table 4.3
Ideal Gas as an Entropy Spring,
First and second laws of thermodynamics applied to compression of a gas:
dU
5
dq
1
dw
(i)
where U
5
internal energy function, dq
5
heat absorbed by substance, and dw
5
work
done on substance by its surroundings.
dU 5TdS2PdV
(ii)
Equation (ii)
yields
P5Tð@S=@VÞ
T
2 ð@U=@VÞ
T
52ð@A=@VÞ
T
(iii)
where S
5
entropy and A
5
Helmholtz free energy
U
TS.
From (iii), the pressure consists of two terms:
entropy contribution: T(
@
S/
@
V )
T
(called kinetic pressure)
internal energy contribution:
V )
T
(called internal pressure)
To evaluate the terms in
Eq. (iii)
experimentally, substitute
entropy contribution: T
2
(
@
U/
@
ð@
S
=@
V
Þ
T
5
T
ð@
P
=@
T
Þ
v
Thus,
P
5
T
ð@
P
=@
T
Þ
v
2 ð@
U
=@
V
Þ
T
(iiia)
internal energy contribution to the total pressure:
P
2
T
ð@
P
=@
T
Þ
v
(iv)
Definition of ideal gas is gas which obeys the equation of state:
PV
nRT
and for which the internal energy U is a function of temperature only, i.e.,
ð@
5
U
=@
V
Þ
T
5 ð@
U
=@
P
Þ
T
5
ð
Þ
0
ideal gas
(vi)
It follows that
P
5
T
ð@
P
=@
T
Þ
v
ð
Þ
ideal gas
(vii)
and the pressure is due only to the entropy contribution.
(A reversible process is the thermodynamic analog of frictionless motion in
mechanics. When a process has been conducted reversibly, we can, by performing
the inverse process in reverse, set the system back in precisely its initial state,
with zero net expenditure of work in the overall process. The system and its sur-
roundings are once again exactly as they were at the beginning. A reversible pro-
cess is an idealization which constitutes a limit that may be approached but not
attained in real processes.)
For a reversible process,
Eqs. (4-6) and (4-7)
yield
dU 5TdS1dw
(4-8)
We define the Helmholtz free energy
A
as
A U 2TS
(4-9)
