Concise Physical Chemistry

c08 JWBS043-Rogers September 13, 2010 11:25 Printer Name: Yet to Come

112 A STATISTICAL APPROACH TO THERMODYNAMICS

8.3 GIBBS FREE ENERGY AND THE PARTITION FUNCTION

Taking logarithms of the previous equation, we obtain

lnKeq=−

ε 0

RT

+ln

QB

QA

whereε 0 =εB 0 −εA 0. Multiplying by−RT, we get

−RTlnKeq=ε 0 −RTln

QB

QA

but−RTlnKeq=G◦,so

G◦=ε 0 −RTln

QB

QA

Comparing this equation with the definition of the Gibbs chemical potential

G◦≡H◦−TS◦, we see that the classical enthalpy changeH◦for a re-

action is largely controlled by the energy separating the ground statesε 0 , and the

entropy changeTS◦is largely controlled by the ratio of the multiplicity of levels

available in each stateRTln (QB/QA). In very many cases, the enthalpy change is

dominant in chemical reactions, which explains why some very good nineteenth-

century scientists thought thatH◦was the only factor controlling equilibrium and

reaction spontaneity.

Given that reactions and physical processes exist that are not solely controlled or

dominated by the enthalpy change, it is the ratio of partition functions that interests

us. The partition function gives us the distribution of molecules over the energy levels

within a state. We may think of these asmicrostateswithin a state. If the enthalpy

balance is close, the reaction tends to go in the direction with the most microstates.

Such a reaction or process is maximizing the number of choices a particle (molecule)

has in which to reside within a state. A system providing a maximum number of

choices provides the maximum disorder, just as the number of places youmayfind

that piece of paper you are looking for is determined by the disorder of the desktop.

A disordered desktop has more microstates than an orderly one.

Looking again at the equationG◦=ε 0 −RTln (QB/QA), we can see another

reason why thermodynamicists prior to Gibbs could have missed the importance of

entropy in equilibrium. The second term includesT, so it may be small at ordinary

temperatures. The ratio of microstatesQB/QAmay be negligible at lower tempera-

tures, but it becomes important at higher temperatures and it may favor or disfavor

the product side of the equilibrium according to which is larger,QAorQB.Asis

true in classical thermodynamics, there are four possibilities for the two terms in

G◦=ε 0 −RTln (QB/QA): They may be FF, FD, DF, or DD, where F denotes

a term that favors the product side of the equilibrium and D denotes a term that

disfavors the product side.