If the solution is dilute, then ln*x *= ln(1 *x*2) = *x*2, where

P

374, 220 atm

- *x*2 ≈ - 2

(44)

where *n *refers to the number of moles. Thus, substitut-

ing eq 44 into eq 43 yields

1 atm

Π=

(45)

but V ≈ *nV *, so

Π = *cRT*

(46)

~

611 Pa

where *c *is the solute concentration (mol m3). The con-

~

cept of osmotic pressure is useful in formulating expres-

sions for total soil water pressure because soil water

0 0.01

100

contains solutes. Furthermore, the influence of soil par-

T ( C)

ticle surfaces on the chemical potential of the soil water

can be "approximated" as solutes.

the pressure and temperature conditions for phases in

equilibrium.

Fundamental thermodynamic principles have been

reviewed in the above sections. The relations and con-

cepts that are particularly useful in studying the physi-

Osmotic pressure, Π, is the pressure required to

cal processes associated with freezing soil are

maintain equilibrium between a solution and the pure

1. Thermodynamic equilibrium requires a balance

solvent across a semi-permeable membrane through

of thermal, mechanical, and chemical forces. Thermal

which the solvent, but not the solute, can diffuse. The

equilibrium is reached when temperatures are equal,

osmotic pressure is easily determined using the chem-

ical potential requirement of equilibrium. For the sol-

ance of mechanical forces, and chemical equilibrium

vent on both sides of the membrane,

is reached when the chemical potentials of all compo-

nents of the system are equal.

(*T*, *P *+ Π, *x*) = o (*T*, *P*).

(39)

2. The general equation for mechanical equilibrium

between two phases--i.e., the interface is curved, rather

From eq 25 we know that (*T*, *P *+ Π, *x*) = o(*T*, *P *+ Π)

than planar--is (*P*a *P*b)*dV *= Ψ*d*Ar (eq 17).

+ *RT*ln*x*. Substituting this into eq 39 results in

This equation applies to interfaces between all phases

of a substance (solid/liquid, vapor/liquid, and solid/

o (*T*, *P *+ Π) + *RT*ln*x *= o (*T*, *P*).

(40)

vapor interfaces) and is the physical reason for capil-

lary rise of liquids in porous materials.

Using eq 34, d o = *V dP *and integrating from *P *to

3. The GibbsDuhem equation,

∑ ηi d i = -*SdT *+ *VdP*

∫

o (*T*, *P *+ Π) - o (*T*, *P*) =

(41)

i

P

(eq 31), is useful when applied to water in freezing soils.

4. The concept of osmotic pressure is useful in for-

Substituting eq 41 into 40 yields

mulating expressions for total soil water pressure. This

is because soil water contains solutes; in addition, the

∫

(42)

influence of soil particle surfaces can be "approximated"

as solutes. Expressions for the osmotic potential of a

P

dilute solution are

For an incompressible solvent in an ideal solution the

- *RT *ln*x*

molar volume remains constant, and

Π=

- *RT *ln*x*

Π=

(43)

.

(eq 45) and Π = *cRT *(eq 46).

~