Physical Chemistry , 1st ed.

The state functions U,H,A, and Gare the only independent energy quan-

tities that can be defined using p,V,T, and S. It is important to note that the

only type of work we are considering at this point is pressure-volume work. If

other forms of work are performed, then they must be included in the defini-

tion ofdU. (Usually, they appear as dwnon-pV. We will consider one type of

non-pVwork in a later chapter.)

Furthermore, it must be understood that the condition G 0 defines only

spontaneity, not speed. A reaction may be thermodynamically favorable but

might proceed at a snail’s pace. For example, the reaction

2H 2 (g) O 2 (g) →2H 2 O ()

has a very negative G. However, hydrogen gas and oxygen gas can coexist in

an isolated system for millions of years before all of the reactant gas has con-

verted into liquid water. At this point, we cannot address the speed of the re-

action. We can address only whether it can occur spontaneously.

The Helmholtz energy is named after the German physician and physicist

Hermann Ludwig Ferdinand von Helmholtz (Figure 4.1). He is known for the

first detailed, specific enunciation of the first law of thermodynamics in 1847.

The Gibbs free energy is named for Josiah Willard Gibbs, an American math-

ematical physicist (Figure 4.2). In the 1870s, Gibbs took the principles of ther-

modynamics and applied them mathematically to chemical reactions. In doing

so, Gibbs established that the thermodynamics of heat engines was also ap-

plicable to chemistry.

The usefulness of the Helmholtz energy,A, can be demonstrated by starting

with the first law:

dUdq dw

Since dS dq/T, we can rewrite the equation above as

dUT dSdw

IfdT0 (that is, for an isothermal change), this can be written as

d(UTS) dw

Since the quantity inside the parentheses is the definition ofA, we can substitute:

dAdw

which we integrate to get

Aw (4.10)

This says that the isothermal change in Helmholtz energy is less than or, for

reversible changes, equal to the work done by the system on the surroundings.

Since work done bythe system has a negative value, equation 4.10 means that

the Aof an isothermal process is the maximum amountof work a system can

do on the surroundings. The connection between work and the Helmholtz en-

ergy is the reason that Helmholtz energy is represented by A. It comes from the

German word Arbeit, meaning “work.”

A similar expression can be derived for the Gibbs free energy, but using a

slightly different understanding of work. So far, we have always discussed work

as pVwork, work performed by expanding gases against external pressures.

This is not the only kind of work. Suppose we define a sort of work that is non-

pVwork. We can write the first law of thermodynamics as

dUdq dwpV dwnon-pV

4.3 The Gibbs Free Energy and the Helmholtz Energy 93

Figure 4.1 Hermann Ludwig Ferdinand von

Helmholtz (1821–1894), German physicist and

physiologist. In addition to studying various as-

pects of physiology including sight and hearing,

Helmholtz made important contributions to the

study of energy. He was one of the first people to

clearly enunciate what became the first law of

thermodynamics.

© CORBIS/Bettmann

Figure 4.2 Josiah Willard Gibbs (1839–1903),

American physicist. Gibbs applied the mathemat-

ics of thermodynamics to chemical reactions in a

rigorous fashion, thereby extending the applica-

bility of thermodynamics from engines to chem-

istry. However, his work was so much over the

heads of his contemporaries that it took almost

20 years for his contributions to be recognized.

Photo by Gen. Stab. Llt. Anst, AIP Emilio Segrè Visual Archives