(53)

particles in the frost heave process by applying the gen-

eralized Clapeyron equation (GCE) to the thermody-

since Π, the osmotic pressure, is equal to RTc .

~

namic equilibrium between ice and water in soil. In

If the symbol *P*w is used to describe the total soil

order to clarify that the GCE is based on sound thermo-

water potential, then

dynamics, one of Miller's students, J.P.G. Loch (1978)

published a detailed derivation of the GCE. That deri-

(54)

vation is now summarized.

Using the Gibbs-Duhem equation (eq 31) for soil

Note that *P *is the pressure of the water excluding os-

motic effects and

ical potential as the Gibbs free energy per unit mass of

a substance so that ηi refers to mass and not moles of a

substance, *i*:*

(55)

ηwdw = *SdT *+ *VdP * ηsds

Equation 55 is integrated after making the substitu-

(48)

tion that at equilibrium,

where the subscripts *w *and *s *refer to water and salt,

respectively. The equation for the chemical potential

(56a)

for salt in soil water (a form of eq 25) is

to obtain

s =

s (*T*, *P*) +

∆*T *

ln *x*s

(49)

*M*s

w = *H *ln1 +

+ *V pw*

where so(*T*, *P*) is the chemical potential of the pure

or for small ∆T,

salt at the same temperature and pressure as the sys-

tem, *M*s is the molecular weight of the salt, and *x*s is the

w ≈ -

+ *V P*w

mole fraction of the salt. Making the approximation that

(56b)

*M * η

*M*s ηw

where H is the enthalpy per unit mass of the solution.

In eq 56a, *T*o is the freezing point of pure water and

∆*T *is the freezing point depression (K).

we obtain

Assuming that ice contains no solutes, the chemical

*RT * *M*w *RT * ηs

s = s (*T*, *P*) +

+

ln

ln

.

potential for pore ice is

*M*s *M*s *M*s ηw

*A*

(50)

i = -

+ *V*i P + *V*i Ψiw r .

(57)

*V*

Differentiating eq 50 with respect to ηs gives *d*s =

(*RT*/*M*s)(1/ηs)*d*ηs, or

Setting the chemical potentials of pore ice and water

equal (at equilibrium) results in

η

ηsds = *RTd* s .

(51)

*M*s

*H*

*A*

- ∆*T *+ *V P*w = - i

+ *V*i P + *V*i Ψiw r . (58)

*V*

*T*o

Substituting eq 51 into eq 48 gives

Using the following definitions for the pressure of ice,

ηs

(52)

*M*sηwV

*A*

where *S *and *V *are entropy and volume of solutions per

(59)

*V*

gram of water, respectively. Since the expression in the

parentheses of eq 52 is equal to the concentration of

(60)

~

the solute in solution, c , eq 52 can be rewritten as

and substituting eq 59 and 60 into eq 58 results in the

equation of chemical equilibrium between pore ice and

water, or the generalized Clapeyron equation:

* Loch (1978) apparently redefined the chemical potential in this

way in order to arrive at the generalized Clapeyron equation (eq 61)

∆*T *

in a form that is convenient to work with because the specific vol-

.

(61)

ume of a substance is equal to the inverse of its density.

*T*o