148
DOC.
12
THERMODYNAMIC DEDUCTION
(2)
8Ss
+
8Sg
+ 8Sr
=
0,
where
Ss
is
the
entropy
of the
radiation,
Sg
that of the
gas
mixture,
and
Sr
that of the
reservoir. We
consider the
following
virtual
displacement:
One
gram-molecule
of
AB
decomposes
with the
absorption
of radiation
energy
e
in the interval dv which
is
in
the
vicinity
of
the
frequency v.
Thus,
we
have,
in the first
place:
[6]
(a)
8S"
=
-
e
g
ft9
S
if
T's
denotes the
temperature
of the
radiation
of
frequency
v
that
corresponds
to
the
[7] density
p'
=
a
For the
entropy
of
the
gas
mixture
we
have the familiar
equation
Sg
=
E
nx(ax +
R
log V
~
R
log
nx),
where
nx
=
number of
gram-molecules
of the
gas
of
kind
X
ox
=
dux/T
(ux
=
energy
of
the
gas
of kind
A per gram-molecule)
Eg
=
Enxux
is
hence the total
energy
of the
gas
mixture
V
=
volume.
R
is
the universal
gas
constant.
This
results in
(b)
8Sg =
Y,
8ng(ax'
-
tf'ogV),
where
bnx
is the
change
in the
number
of
gram-molecules
in the virtual
displacement
(6n,
=
-1,
bn2
=
6«3
=
+1)
a'x =
ox
-
R
rj'x
=
volume
concentration
of
the
gas
of kind
A
in
case (1a)
It should be noted that
owing
to
(1a), one
also has
(b') Y,SniloSvi
=
EK'opIn
-
log«
=
EHlog1i
+
log-£-.
P
Finally,
let
Es
denote the
energy
of the
radiation,
and
Eg
that of the mixture.
By
virtue of the
equivalence
principle,
(SEs
+
SEg)
will be
equal
to
the heat communi-
cated
to
the
reservoir,
so
that
.
.
£^i5
T
or,
if
SEs
and
SEg
are replaced by
their
values,