For eq 3b and c, respectively, using the definitions

In the absence of work, the isothermal equilibrium

of Gibbs free energy, enthalpy, and Helmholtz free

condition is *dA *= 0; furthermore, a spontaneous pro-

energy, differentiating and substituting eq 3b or 3c for

cess produces negative Helmholtz free energy. In other

words, a constant temperature system minimizes Helm-

holtz free energy.

(4a)

A similar derivation can be done for constant pres-

sure and temperature processes to show that the spon-

or

taneity condition is

(4b)

i

(11)

(5a)

Thus, at constant temperature and pressure and in the

or

absence of work, the equilibrium condition is *dG *= 0;

and, a spontaneous process produces negative Gibbs

(5b)

free energy.

i

(6a)

Substituting *TdS *≥ δ*Q *into the first law results in

or

(12)

(6b)

i

For an isolated system, *dU *= δ*W *= δ*Q *= 0; thus, eq

Equations 3 through 6 are known as the four funda-

12 applied to an isolated system is

mental equations of thermodynamics.

(13)

Since *dS *= (δ*Q*rev / *T *) , if a positive quantity of heat

Equilibrium thermodynamic relations are often used

passes from region *a *to *b *within an isolated system,

in soil freezing and frost heave models (analytical and

then

numerical); therefore, their derivations are now present-

ed. For a system undergoing cyclical changes in state,

1

1

the process is reversible if, at the end of each cycle, the

(14)

*T*b Ta

surroundings are restored to their original state. At every

stage of this process, the system departs from equilib-

and for a spontaneous process, *dS *> 0; therefore, *T*a >

rium only infinitesimally. Thus, the condition for revers-

ibility is an equilibrium condition (e.g., Castellan 1983):

in thermal equilibrium has the same temperature in all

(7a)

regions, and when it is not in equilibrium, heat flows

from regions of high temperature to low temperature.

and for irreversible (natural) processes

(7b)

For a constant-volume, constant-temperature system

divided into regions *a *and *b*, if region *a *expands revers-

The composite functions are used to describe equi-

ibly by *dV*a then region *b *contracts by *dV*b = *dV*a.

librium and spontaneous transformation conditions of

According to eq 6a, ( *A*/ *V*)T = *P*, or *dA *= *PdV*, and

systems under the constraints for which they were

defined. For example, at constant temperature, *d*(*TS*) =

δ*W *= 0 (for a constant volume), from eq 10 and the

second law, *dA *≤ 0 and, therefore, *P*a > *P*b. In other

first law and the substitution that δ*Q *= *TdS *= *d*(*TS*)

words, for a spontaneous expansion of region *a *into *b*,

results in

(8)

this type of analysis is helpful when less intuitive pro-

cesses are described as below.

(9)

For a constant-volume, constant-entropy system

consisting of two phases *a *and *b *with an interface, ψ,

or

(10)