Physical Chemistry , 1st ed.

For clarity, we are dropping the “ext” label on the pressure variable. The two
pdVterms cancel to give us

dHVdpTdS 0

for a spontaneous change. If this change were to occur under conditions of
constant pressure and constant entropy, then dpand dSboth equal 0, so the
spontaneity condition becomes

(dH)p,S 0 (4.4)

Again, this is not a useful spontaneity condition unless we can keep the process
isentropic. Because pand Smust be constant in order for the enthalpy change
to act as a spontaneity condition,pand Sare the natural variables for enthalpy.
Equation 4.4 does suggest why many spontaneous changes are exothermic,
however. Many processes occur against a constant pressure: that of the atmo-
sphere. Constant pressure is half of the requirement for enthalpy changes to
dictate spontaneity. However, it is not sufficient, because for many processes
the entropy change is not zero.
Notice a certain trend. Equation 4.1, the spontaneity condition for entropy,
states that the entropy change is positive for spontaneous processes. That is,
entropy increases.On the other hand, the spontaneity conditions for both in-
ternal energy and enthalpy, both measures of the energy of a system, require
that the change is less than zero: the energy of the system decreasesin sponta-
neous changes. Changes toward increased entropy and decreased energy are
generally spontaneous if the proper conditions are met. However, we still lack a
specific spontaneity test for constant pressure and temperature, our most use-
ful experimental conditions.

Example 4.1

State whether or not the following processes can be labeled spontaneous un-

der the following conditions.

a.A process in which His positive at constant Vand p

b.An isobaric process in which Uis negative and Sis 0

c.An adiabatic process in which Sis positive and the volume does not

change

d.An isobaric, isentropic process in which His negative

Solution

a.Spontaneity requires that Hbe negative if pressure and entropy are con-

stant. Since we do not know the constraints on pand S, there is no require-

ment that this process mustbe spontaneous.

b.An isobaric process has p0. We are also given a negative Uand

S0. Unfortunately, the negative Uspontaneity condition requires an

isochoric (that is,V0) condition. Therefore, we cannot say that this

process mustbe spontaneous.

c.An adiabatic process implies q0, and with volume not changing we have

V0; therefore w0 and thus U0. The constant Uand Vallow us to

apply the strict entropy spontaneity test: ifS0, the process is spontaneous.

Since we are given that Sis positive, this process must be spontaneous.

d.Isobaric and isentropic imply pS0. These are the proper variables

for using the enthalpy spontaneity test, which requires that Hbe less than

zero. This is in fact the case, so this process must be spontaneous.

4.2 Spontaneity Conditions 91