Page 51 - Physical chemistry eng

28 CHAPTER 2 Heat, Work, Internal Energy, Enthalpy, and the First Law of Thermodynamics

The preceding remarks about the difference between C and C have been qualita-
P
V
tive in nature. However, the following quantitative relationship, which will be proved in
Chapter 3, holds for an ideal gas:
C - C = nR or C P,m - C V,m = R (2.11)
V
P

2.6 State Functions and Path Functions

An alternate statement of the first law is that ¢U is independent of the path connecting
the initial and final states, and depends only on the initial and final states. We make this
statement plausible for the kinetic energy, and the argument can be extended to the
other forms of energy listed in Section 2.1. Consider a single molecule in the system.
Imagine that the molecule of mass m initially has the speed v . We now change its
1
speed incrementally following the sequence v : v : v : v 4 . The change in the
2
3
1
kinetic energy along this sequence is given by
1 2 1 2 1 2 1 2
¢E kinetic = ¢ m(v ) - m(v ) ≤ + ¢ m(v ) - m(v ) ≤
2
1
3
2
2 2 2 2
1 1
2
+ ¢ m(v ) - m(v ) ≤
2
4
3
2 2
1 1
2
= ¢ m(v ) - m(v ) ≤ (2.12)
2
4
1
2 2
Even though v and v can take on any arbitrary values, they still do not influence the
3
2
result. We conclude that the change in the kinetic energy depends only on the initial and
final speed and that it is independent of the path between these values. Our conclusion
remains the same if we increase the number of speed increments in the interval between
v and v to an arbitrarily large number. Because this conclusion holds for all molecules
1
2
in the system, it also holds for ¢U .
This example supports the assertion that ¢U depends only on the final and initial
states and not on the path connecting these states. Any function that satisfies this condi-
tion is called a state function, because it depends only on the state of the system and
not the path taken to reach the state. This property can be expressed in a mathematical
form. Any state function, for example U, must satisfy the equation
f
¢U = dU = U - U i (2.13)
f
3
i
where i and f denote the initial and final states. This equation states that in order for ¢U
to depend only on the initial and final states characterized here by i and f, the value of
the integral must be independent of the path. If this is the case, U can be expressed as
an infinitesimal quantity, dU, that when integrated, depends only on the initial and final
states. The quantity dU is called an exact differential. We defer a discussion of exact
differentials to Chapter 3.
It is useful to define a cyclic integral, denoted by the symbol , as applying to a cyclic
A
path such that the initial and final states are identical. For U or any other state function,
dU = U - U = 0 (2.14)
f
f
C
because the initial and final states are the same in a cyclic process.
We next show that q and w are not state functions. The state of a single-phase sys-
tem at fixed composition is characterized by any two of the three variables P, T, and V.
The same is true of U. Therefore, for a system of fixed mass, U can be written in any of
and
the three forms U(V,T), U(P,T), or U(P,V). Imagine that a gas characterized by V 1
is confined in a piston and cylinder system that is isolated from the surroundings.
T 1
There is a thermal reservoir in the surroundings at a temperature T 6 T 1 . We do com-
3
to
pression work on the system starting from an initial state in which the volume is V 1