Page 94 - Essentials of physical chemistry

56 Essentials of Physical Chemistry

Conservation of energy is a general principle and U is a state variable, which allows us to quantify
the amount of an energy change in a given process. This brings up the question as to how the
process is carried out. The conservation of energy means that it does not matter how the process is
carried out; all that matters is the difference between the end and the beginning of the process. We
say that U is a state variable and the change in U is independent of the path. However, we must be
careful regarding dq and dw since we know by common sense that surely dw is dependent on the
path in terms of what we call ‘‘efficiency.’’ A general definition of efficiency used in this text is

work done
:
energy change used
Efficiency
The concept of efficiency can be subjective since it usually implies ‘‘useful work’’ while work done
against resistive friction is still work but not ‘‘useful’’ and it generates heat as well. However, the
first law says there is a nonsubjective state variable called the internal energy, U, which obeys the
conservation principle even if dq and dw vary according to the process used.
Again the notion of efficiency involves the amount of heat energy dq, which might be just
sufficient for a given process or wasteful. Thus, dU is an ‘‘exact differential’’ and is independent of
path while dq and dw are not exact differentials. Some texts write d =q and d=w with a slash ‘‘=’’
through the ‘‘d’’ and the ‘‘w’’ but we only need the student to realize that dU is path independent
while dq and dw are not. This is important later because we will discover and use other state
variables that have the property of path independence nicely explained by Denbigh [4]. Suppose a
process involves two steps from (I) ! (II) ! (III), we can calculate the energy change for
(I) ! (III) as:

ð II ð III
dU ¼ (U II U I ) þ (U III U II ) ¼ (U III U I ):
dU þ
I II
Now suppose (III) is really just a return to (I), then we obtain a special condition for which we have
a special symbol indicating integration over a cyclic path with the result of zero,

þ
dU ¼ 0:

Other cyclic integrals might not be zero but a state variable will sum to zero; that is the main
characteristic of a state variable. The circle on the integral sign means that the process is carried out
over a cyclic process. We can offer a simple insight in that state variables are universal variables and
characteristics of the universe while q and w are subjective variables, which depend on how a
process is carried out.
A simple example is the idea of gravitational potential energy. If Jack goes up a hill and later
tumbles down, we can say two things from common sense. First, if he ended up where he started
there is no net change in gravitational potential energy. Second, we cannot tell how much Jack
wandered side to side or how much energy he may have expended in other forms, which may not
have been recovered in his descent since as far as gravity goes, the only change is between the
‘‘after-minus-before’’ height change. A third observation is that for this process the cycle of
gravitational energy satisfies the gravitational cyclic process whose ‘‘path integral’’ (the integrated
path) is zero. The path Jack took is typical of a thermodynamic process where we can evaluate the
beginning and end of the process but have no information on what happened along the path. If we

qU qU
dP:
qT qP
write the differential of U in terms of T and P we obtain dU ¼ dT þ
P T