Page 85 - Physical Chemistry

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2
2
Chapter 2 state 2. The values of the integrals dq and dw depend on the path from 1 to 2.
L 1
L 1
The First Law of Thermodynamics In general, if b is not a state function, then db depends on the path. Differentials
2
L 1
of a state function, for example, dU, are called exact differentials in mathematics; the
differentials dq and dw are inexact. Some texts use a special symbol to denote inexact
differentials and write dq and dw (or Dq and Dw) instead of dq and dw.
From (2.82), it follows that, if the value of the line integral 2 db depends on the
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path from state 1 to state 2, then b cannot be a state function.
2
Conversely, if db has the same value for every possible path from state 1 to
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state 2, b is a state function whose value for any state of the system can be defined as
follows. We pick a reference state r and assign it some value of b, which we denote by
b . The b value of an arbitrary state 2 is then defined by
r
r
b b 2 db (2.83)
2
r
Since, by hypothesis, the integral in (2.83) is independent of the path, the value of b 2
depends only on state 2; b b (T , P ), and b is thus a state function.
2
2
2
2
If A is any state function, A must be zero for any cyclic process. To indicate a cyclic
process, one adds a circle to the line-integral symbol. If b is a state function, then (2.82)
gives db 0 for any cyclic process. For example, dU 0. But note that dq q
and dw w, where the heat q and work w are not necessarily zero for a cyclic process.
We now show that, if
db 0
2
for every cyclic process, then the value of db is independent of the path and hence
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Figure 2.13 b is a state function. Figure 2.13 shows three processes connecting states 1 and 2.
Processes I and II constitute a cycle. Hence the equation db 0 gives
Three processes connecting states
db db 0 (2.84)
1 and 2. 1 2
2 1
I II
Likewise, processes I and III constitute a cycle, and
1 db 2 db 0 (2.85)

2 1
I III
Subtraction of (2.85) from (2.84) gives
2 db 2 db (2.86)

1 1
II III
Since processes II and III are arbitrary processes connecting states 1 and 2, Eq. (2.86)
2
shows that the line integral db has the same value for every process between states
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1 and 2. Therefore b must be a state function.

Summary
2
If b is a state function, then db equals b b and is independent of the path from
1
L 1
2
state 1 to state 2. If b is a state function, then db 0.
2
If the value of db is independent of the path from 1 to 2, then b is a state func-
L 1
tion. If db 0 for every cyclic process, then b is a state function.