66
FOUNDATIONS OF THERMODYNAMICS
or, noting
that
according to
§6
the
quantities
2h1E1
-
c1,
2h2E2
-
c2,...
are
identical with
the
entropies
S1,S2...
of
the partial
systems
up
to
a
universal
constant,
(8)
S1
+
S'2
+
...
S1 +
S2
+
... ,
i.e.,
the
sum
of
the entropies
of
the partial
systems
of
an
isolated
system
after
some
arbitrary
process
is
equal to
or
larger
than the
sum
of
the
entropies
of the
partial
systems
before the
process.
§9.
Derivation
of
the second
law
Let
there
be
an
isolated total
system
whose
partial
systems
shall
be
called
W,
M,
and
E1,E2....
Let the
system
W,
which
we
shall call heat
reservoir,
have
an
energy
that
is infinitely
large
compared
with the
system
M
(engine).
Similarly,
the
energy
of
the
systems
E1,E2...,
which
interact
adiabatically
with
each
other, shall
be
infinitely
large
compared
with that
of
M.
We
assume
that all the partial
systems
M,
W, E1,
E2... are
in
a
station-
ary
state.
Suppose
that the
engine
M
passes
through
a
cyclic
process
during
which
it
changes
the distributions
of states of
the
systems
E1,E2...
infinitely
slowly through
adiabatic
influence, i.e.,
performs work, and
receives the
amount
of heat
Q
from
the
system
W.
The
reciprocal adiabatic influence
of
the
systems
E1,E2...
at
the
end
of
the
process
will
then differ
from
that
before the
process.
We
say
that the
engine
M
has
converted the
amount
of
heat
Q
into
work.
We
now
calculate the increase in
entropy
of
the individual partial
systems
during
the
process
considered.
According
to
the results
of
§6
the
entropy
increase
of
the heat reservoir
W
equals
-Q/T
if
T
denotes the
absolute
temperature.
The entropy
of
M
is
the
same
before
and
after
the
process
because the
system
M
has
undergone
a
cyclic
process.
The
systems
E1,E2...
do
not
change
their
entropies
during
the
process
at
all because
these
systems
only experience
an
adiabatic influence that
is
infinitely
slow.
Hence
the
entropy
increase
S'
-
S
of
the total
system
has
the value