Chemical Potential Leads to More Goodies

Colligative properties, those that depend not on the identity of a solute but only on how much is dissolved, had been known experimentally for years before an underlying theory was established. We typically discuss four such properties of solutions (and are compared to the corresponding pure solvent): vapor pressure lowering, freezing point depression, boiling point elevation and osmotic pressure.

It should be apparent from Monday's lecture that vapor pressure lowering for ideal solutions is an obvious consequence of Raoult's Law, although all solutions experience the phenomenon (which is due to the fact that the chemical potential is lower for a solution than it is for the solvent: μ₁(soln)=μ*₁(l)+RTlnX₁). But all four of these phenomena are explained by setting the appropriate chemical potentials equal (where we designate 1 as the solvent and 2 as the solute):

vp lowering: μ₁(soln) = μ₁(g)
fp depression: μ₁(soln) = μ₁(s)
bp elevation: μ₁(soln) = μ₁(g)
osmotic pressure: μ₁(l) = μ₁(soln) + ΠV_m

Using some math elegance with a few key assumptions (like ∆_transH being relatively constant and the solution dilute), we are able to obtain the experimentally obtained formulae familiar to students of general chemistry:

bp elevation: ∆T_b=K_bm where [ebullioscopic constant] K_b=RM₁T_b²/∆_vapH
fp elevation: ∆T_f=K_fm where [cryoscopic constant] K_f=RM₁T_f²/∆_fusH
osmotic pressure: Π=MRT

Since we have thus far dealt only with molecular [nonionic] solutions, we needed to refresh some concepts from basic electrostatics [coulomb's law, electric permittivity, dielectric constants] before moving onto ionic solutions, which can be radically different (and never truly ideal).