as a measure of spontaneity is limited in application, since it applies to isolated
systems on which no work is done and which are adiabatic, so that both wand
qare zero. We recognize, however, that many processes occur with w 0
and/or q0. What we really want is a way to determine spontaneity for ex-
perimental conditions that are common in real life. These conditions are con-
stant pressure(because many processes occur when exposed to atmospheric
pressure, which is usually constant over the course of the experiment) and
constant temperature(which is the easiest state variable to control).
Internal energy and enthalpy can also be used to determine spontaneity un-
der appropriate conditions. Consider equation 4.1. Since w0 and q0, the
process is occurring at constant U, and we can label the infinitesimal change
in entropy dSwith these constant state variables:
(dS)U,V 0 (4.2)
where the subscripts U,Vindicate what variables are held constant. Let us de-
termine different spontaneity conditions for different conditions. The Clausius
theorem for a spontaneous change is:dq
TrevdSWe can rewrite this as
dq
TrevdS 0and since we know that dUdqrevpextdV,or dqrevdU pextdV,dU
TprevdV
dS 0The “equal to” part of the sign applies if the process is reversible. Multiplying
through by T, we get for a spontaneous change,
dU pdVTdS 0
If the process occurs under conditions of constant volume and constant en-
tropy, that is,dVand dSare zero, this equation becomes
(dU)V,S 0 (4.3)
as a spontaneity condition. Because this condition depends on volume and en-
tropy staying constant,Vand Sare called the natural variables of internal en-
ergy. The natural variablesof a state function are the variables for which
knowledge of how the state function behaves with respect to them allows one
to determine all thermodynamic properties of the system. (This will become
clearer with examples later on.)
Why did we not introduce equation 4.3 as a spontaneity condition earlier?
First, it depends on our definition of entropy, which we did not get to until the
previous chapter. Second—and more importantly—it requires a process that is
isentropic; that is, where dS0 infinitesimally and S0 for the overall
process. One can imagine how difficult it must be to perform a process on a
system and ensure that the order, on an atomic and molecular level, does not
change. (Contrast that with how easy it is to devise a process where dVis zero
or, equivalently,Vfor the entire process equals 0.) To put it bluntly, equation
4.3 is not a very useful spontaneity condition.
Since dHdU d(pV), we can substitute for dUin equation 4.3:
dHpdVVdp pdVTdS 090 CHAPTER 4 Free Energy and Chemical Potential