Colligative Properties

Colligative ("collective" or "linked together") properties of a solution refer to the following changes that occur when solute is added to a pure solvent:

·        Vapor pressure lowering

·        Boiling point elevation

·        Freezing point depression

·        Osmotic pressure elevation

 

The simplest systems that exhibit colligative properties consist of:

(1)   a pure solvent phase, which may be vapor, lquid, or solid

(2)   a solution phase

(3)   an interface between the two phases that is not crossed by the solute

 

So, if pure solvent phase is vapor phase, increased flow of solvent into solution lowers the vapor pressure. Higher T is then needed to restore the vapor pressure, resulting in increased boiling point.

 

Increased flow of solvent from solid phase into solution results in melting of the solid. Lower T is required to re-freeze the solid, i.e., the freezing point is lowered.

 

If the pure solvent phase is liquid (as in osmotic pressure experiments) one actually observes flow of the solvent into the solution phase.

Thermodynamic Explanation

Let's examine the expression for the chemical potential of solvent in an ideal solution:

Since x £ 1, ln x < 0, and the chemical potential of solvent in a solution is always lower than that of the pure solvent.

 

Now let's revisit the m vs. T plot that was used to explain phase diagrams. It is important to recall:

·        Substance i flows towards phase with lowest mi

·        If phases a and b are in equilibrium, then

In all cases of the colligative properties, the solute acts by lowering the m of the solvent in solution, thus causing solvent to flow into the solution phase.

The lower m curve of the solvent shifts the melting and boiling points as shown above.

Freezing Point Depression

Start with equilibrium condition (pure solid solvent in equilibrium with solvent in solution)

In the last step, we recognize that mi  for a pure substance is just Gm for that substance, and the difference between m in the liquid and solid states is just the free energy of fusion.

 We can transform the above equation by taking /T and applying the Gibbs-Helmholtz equation

as follows:

If we integrate this starting from the reference point of pure solvent (x1 = 1, Tf = Tf*), we obtain

This is sometimes called the ideal solubility equation.

 

Digression: Does the ideal solubility equation look familiar? Compare it to the other equilibrium expressions we have encountered:

         (Clausius-Clapeyron equation for sublimation or vaporization)

       (van't'Hoff equation for chemical reaction equilibrium)

 

All of these are derived from the Gibbs-Helmholtz equation, which expresses the relationship between G and H and accounts for the two separate influences of DH and TDS on chemical equilibrium.

 

The freezing point effect is typically expressed in terms of the solute mole fraction x2 = 1- x1:

A graph of Tf vs. x2 appears as follows:

The standard freezing point depression expression we learned in freshman chemistry was a linear relationship between Tf and x2. This relationship represents the behavior of the ideal solubility equation in the limit x2 ® 0. We therefore make some simplifying approximations in this limit:

1)       [first term of Taylor series for ln (1-x2) ]

2)         since n2 << n1.  Substituting and  into this expression gives x2 » m2M1 .

3)        Since the difference between Tf* and Tf is small for small values of x2, then Tf Tf* » (Tf*)2 and

 

Substituting these approximations into the above expression and solving for DTf gives

The expression in parentheses is a constant that depends upon solvent properties only. It is called the freezing point depression constant for the solvent, Kf.

Osmotic Pressure


If the solution is separated from the pure solvent phase by a membrane that is permeable to the solvent but not the solute, solvent will still tend to flow into the solution. The result in this case is the development of osmotic pressure P in the solution, which builds until the chemical potential of the solvent is the same for both the solution and the pure liquid solvent.

 


The equilibrium condition is then

Integrating the first derivative relation

 

Thus, equating the two results,

where the second step applies the approximation of taking the first term of the Taylor series expansion for the function ln(1-x2) as was done for the freezing point depression derivation, in the limit of small x2. Also in this limit,

With these substitution, we have the result

which is very similar to the ideal gas equation of state. Since c2 = n2/V, we may also write

Van’t’Hoff factor

The osmotic pressure expression may also be written

where i is the van’t’Hoff factor that represents the number of moles of particles produced in the solution by each mole of solute. Thus, if NaCl dissociates completely in water, it would have i=2 (actually i is a little less than 2, indicating that there is some slight association between Na+ and Cl- ions in solution, to the point where Na+Cl- pairs behave as a single particle.

Nonideality of Osmotic Pressure

If the approximations used to derive the simple equation above are not valid (typically as one goes away from the limit of extreme dilution) the equation must be modified. The modification used is exactly analogous to the Virial Equation used to describe deviation from ideal gas behavior.

In practice, osmotic pressure determinations are made by studying P as a function of added mass of solute. Thus, we write the osmotic pressure in terms of r2, the mass of solute per unit volume (substituting r2/M2 for c2):

The most accurate method is to plot P/r vs. r:

 and extrapolate the curve to r2=0. The extrapolated intercept then gives the molecular mass, M2.