How do colligative properties affect freezing point?

If you examine the following graph:

It shows the change in the chemical potential of a solution versus a change in the temperature, at a constant pressure. (All compounds want to minimize chemical potential, similar to minimizing energy.)

There is an equation which describes colligative properties:

#\mathbf(mu_j = mu_j^"*" + RTlna_j)#

where #mu_j# is the chemical potential of a solvent #j# that already contains some amount of solute, #mu_j^"*"# means pure solution (i.e. just the solvent without added solute), and #a_j# means activity of solvent #j# in the solution.

The activity is defined as:

#a_j = x_jgamma_j#

where #x_j# is mole fraction of compound #j# and #gamma_j# is the activity coefficient of compound #j#. Thus, you can infer that a lower mole fraction of compound #j# gives a lower activity and vice versa.

Since mole fractions are always #<= 100%#, the activity can never be higher than #100%#, and thus #a_j <= 1#.

Because of this, and because of the fact that #ln(a_j)# is negative when #0 < a_j < 1# (#ln1 = 0#), if you add any solute at all to the solution, #a_j# will go down, and thus #mu_j < mu_j^"*"#.

What this says, then, is that the chemical potential decreases, and so on the graph above, if you are examining a liquid's freezing point, move the straight line that corresponds to the liquid downwards a certain amount.

You would see that the freezing point is shifted left, and the boiling point is shifted right.

Therefore, colligative properties cause freezing point depression (and boiling point elevation).