Page 96 - Modern physical chemistry

86 Entropy and the Second Law

I
States that cannot be I States from which state s
reached from s with I cannot be reached with
q=O I q=O
I
•
• s I
o
I
I Removal of heat needed FIGURE 5.1 Correlation of states
I of a uniform system with points
I on aline.

We are thus led to write

dS = c dqrev' [5.1 ]
1
Since S is a function of state, expression [1 is an integrating factor for dqrev'

5.2 Integrability of the Reversible Heat
Let us now look at a general form for dqrev' and determine how it is integrable.
From section 4.2, the work done on a system in a reversible process has the form

n-l
[5.2]
dWrev = LF j dxj
j=l
in which Fj is the jth generalized force and Xi is the jth generalized coordinate. Let the
nth coordinate be the temperature.
The internal energy depends on all the coordinates; so

[5.3]

and
n 8E
dE=L-dx j . [5.4]
j=18xj

With formula (4.12), we have

[5.5]

where
8E
Xj=--Fj . [5.6]
8xj
The independent variables may be considered as Cartesian coordinates in an n-dimen-
sional Euclidean space. At the origin these variables would all equal zero. A radius vector
r drawn from the origin to the point (Xl' X 2, ••• , xJ then defines this point. Coefficient ~
may be considered the jth component of vector function R. See figure 5.2.
At a given entropy S, differential dqrev equals zero and
[5.7]