Physical Chemistry Third Edition

4.1 Criteria for Spontaneous Processes and for Equilibrium: The Gibbs and Helmholtz Energies 153

If our system is simple,dw−P(transmitted)dV, and

dU−TdS+P(transmitted)dV≤0 (simple system,Tconstant) (4.1-6)

There are two important cases for isothermal closed simple systems. The first case

is that of a constant volume, so thatdV0:

dU−TdS≤0 (simple system,TandVconstant) (4.1-7)

TheHelmholtz energyis denoted byAand is defined by

AU−TS (definition) (4.1-8)

The Helmholtz energy has been known to physicists as the “free energy” and as the

“Helmholtz function.” It has been known to chemists as the “work function” and as

the “Helmholtz free energy.” The differential of the Helmholtz energy is

dAdU−TdS−SdT (4.1-9)

so that ifTis constant

dAdU−TdS (constantT) (4.1-10)

Equation (4.1-7) is the same as

dA≤0 (simple system,TandVconstant) (4.1-11)

The Helmholtz energy is named for
Hermann Ludwig von Helmholtz,
1821–1894, already mentioned
in Chapter 2 as the first person
to announce the first law of
thermodynamics.

The second important isothermal case is the case that the pressure of the system

is constant and equal toPextand toP(transmitted). We refer to this case simply as

“constant pressure.” In this case,

dU+PdV−TdS≤ 0

(simple system,

TandPconstant)

(4.1-12)

TheGibbs energyis defined by

GU+PV−TS (definition) (4.1-13)

The Gibbs energy is named for Josiah
Willard Gibbs, 1839 –1903, an American
physicist who made fundamental
contributions to thermodynamics and
statistical mechanics and who was the
first American scientist after Benjamin
Franklin to gain an international
scientific reputation.

The Gibbs energy is related to the enthalpy and the Helmholtz energy by the relations

GH−TSA+PV (4.1-14)

The Gibbs energy has been called the “free energy,” the “Gibbs free energy,” the “Gibbs

function,” and the “free enthalpy.” The symbolFhas been used in the past for both the

Helmholtz energy and the Gibbs energy. To avoid confusion, the symbolFshould not

be used for either of these functions and the term “free energy” should not be used.

The differential of the Gibbs energy is

dGdU+PdV+VdP−TdS−SdT (4.1-15)