Solid-Liquid Solutions

We now consider solutions of sparingly soluble solids in liquids. In these solutions there is always a clearly identifiable solute and solvent. This is different from the liquid-liquid solutions we considered previously in that the solute is no longer miscible in all proportions in the solvent. By convention we denote the solvent as component 1 and the solute as component 2. The activities of these solutions are defined with respect to the appropriate standard state.

a₁ = P₁/P₁* (Raoult's law standard state for the solvent)

a_2x = P₂/k_H,x (Henry's law standard state for the solute)

The x indicates that the Henry's law constant used here is derived for a mole fraction scale (as opposed to the molality scale we derive below). The vapor pressure of the solute may be exceedingly small and yet the ratio P₂/k_H,2 will still be finite.

For dilute solutions of a solid in a liquid molality is a more convenient measure of concentration than mole fraction. The molality, m is the number of moles of solute per 1000 grams of solvent.

m = n₂/1000 g. solvent

where n₂ is the number of moles of solute (subscript 2). The units of molality are moles/kg. A solution containing 2.00 moles of KCl in 1.00 kg of water is 2.00 molal, or that it is a 2.00 mol/kg KCl(aq) solution. The relation between the mole fraction of solute (x₂) and molality (m) is

where M₁ is the molar mass (g/mol) of the solvent.

We define the solute activity in terms of molality

a_2m ® m as m ® 0

where subscript m indicates that a_2m is based on a molality scale. We can express Henry's law in terms of the molality rather than the mole fraction by P₂ = k_H,mm. In terms of k_H,m, the activity of the solute is defined by

a_2m = P₂/k_H,m

Similarly the Henry's law constant can be defined on a molarity scale where the molarity, c is the number of moles of solute per 1000 mL of solution. In each case, the activity coefficient is defined by dividing the activity by the appropriate concentration.

g_m = a_2m/m for molality

g_c = a_2c/c for molarity

Standard State	Activity Coefficient	Condition
Raoult's Law	a1 = P1/P1*	a1 ® x1 as x1 ® 0
Mole fraction	g1 = a1/x1	P1 ® P1^*x1 as x1 ® 0
Henry's Law	a2x = P2/kH,x	a2x ® x2 as x2 ® 0
Mole fraction	g2x = a2x/x2	P2 ® kH,xx2 as x2 ® 0
Henry's Law	a2m = P2/kH,m	a2m ® m as m ® 0
Molality	g2m = a2m/m	P2 ® kH,mm as m ® 0
Henry's Law	a2c = P2/kH,c	a2c ® c as c ® 0
Molarity	g2c = a2c/c	P2 ® kH,cc as c ® 0

When the vapor of a solute so low that it is not measurable the Gibbs-Duhem equation provides a means of determining the activity. We consider a solution of sucrose in water. The activity can be expressed in terms of the osmotic coefficient, f. The definition of the osmotic coefficient is:

ln a₁ = -mf/55.5 mole/kg

The deviation of f from 1 represents a deviation of the solution from ideality. Experimentally this is determined from a deviation of the measured vapor pressure from Raoult's law behavior as a function of the sucrose concentration. Significant deviations are observed when the mole fraction of water is less than 0.97. To derive this expression we used

ln a₁ = ln x₁ = ln(1 - x₂) » -x₂ » - m/55.5 mol/kg

The activity coefficient of sucrose can be calculated from

n₁d(ln a₁) + n₂d(ln a₂) = 0

In terms of molality, m, n₁ = 55.5 mol and n₂ = m, so the Gibbs-Duhem equation becomes

55.5 mol d(ln a₁) + m d(ln a₂) = 0

From above

d(ln a₁) = -d(mf)/55.5 mole/kg

and a_2m = g_2mm

so we have

- d(mf) + md(ln g_2mm) = 0

- fdm - mdf + m{d(ln g_2m) + d(ln m)} = 0

By integrating this equation we can find the activity cofficient of the solute from the vapor pressure data of the solvent.

This illustrates a general procedure for starting with vapor pressure data for a solvent to obtain activities and activity coefficients for both solvent and solute.