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Take any solution, then start freezing it at sufficiently low temperature. Soon a small amount of ice forms. I'd like to know how the composition of dissolved salts and gases in the bit of ice differs from that in the water that's still liquid.

Assume that the solution is far from saturation and whatever other extreme conditions there can be. I don't have any particular solution in mind - it's the general principles that I'm interested in.

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It sounds like you're talking about freezing-point depression. If the solid component of a system is pure (oft assumed so; crystals are good at excluding impurities), the equilibrium is shifted because the chemical potential of the solution drops.

For a system with two pure species, the chemical change from say, water ($\star$ denoting pure) to ice is in equilibrium when their chemical potentials (denoted $\mu$) are equal:

$$\mu_{\ce{H_2O}(\text{s})} = \mu^\star_{\ce{H_2O}(\text{l})}$$

Note: Chemical potentials are always dependent, to varying degrees, on both temperature and pressure, so I'm not going to bother writing $\mu(T, P)$ everywhere

On the following graph that's at $T_0$; the usual freezing point of water where the "pure water" and "ice" curves intersect. If water exists at a temperature lower than $T_0$, its higher chemical potential translates to a higher free energy, so it's thermodynamically driven towards freezing.

Ice, pure water, and solution chemical potential graph

(image source: a website that is currently down; retrieved via the Web Archive)

If something non-volatile (e.g. salt) is dissolved in the water, its chemical potential changes to $\mu_i^\star+RT\ln \chi_i$, where $\mu_i^\star$ is the chemical potential of the pure species $i$, $R$ is the gas constant, and $\chi_i$ is the molar fraction of species $i$.

Originally, the fraction was $1.0$ (it was pure), and because $\ln1 = 0$, that term was ignored. But now, because some of the solution species is something else, $\chi_i$ drops, thus $\ln\chi_i$ becomes negative, so the term lowers the overall chemical potential. Because the chemical potential is lower, its free energy is lower, so it is thermodynamically more stable at lower temperatures. At some $T$, we will find that:

$$\mu_{\ce{H_2O}(\text{s})} = \mu_{\text{solution}} = \mu^\star_{\ce{H_2O}(\text{l})} + RT\ln\chi_i$$

And there, the two species will be in thermodynamic equilibrium.

For more, check this site, or Wikipedia articles on chemical potential and freezing point depression (a basic Physical Chemistry text book, or chapter in a Gen. Chem. one might be better; the WP articles are not that great to get started with).

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  • $\begingroup$ Thanks, that was informative, but I'm not sure it answers my question. In my problem the graph also has a curve named "salty ice". Or rather a spectrum of curves for ice with different amounts of salt in it. Where do those go? $\endgroup$ – Karolis Juodelė Sep 6 '13 at 18:00
  • $\begingroup$ @KarolisJuodelė For your problem you'd just need to find the intersection of curves at the appropriate temperature. If impurities are being trapped in the ice (at a constant rate) it's still the same problem, you just don't make the assumption the ice is pure, so its chemical potential curve will be at some other value (your 'spectrum of curves'). If you find the equilibrium temperature and solution composition you can determine the chemical potential of the ice (at that temperature) which should allow you to determine its composition. $\endgroup$ – Nick T Sep 6 '13 at 19:47
  • $\begingroup$ That said, in practical systems the concentration of entrapped solute probably increases as the temperature drops...a much more difficult problem to solve theoretically. Analytically it's easy, chip off some of the frozen bit and measure. $\endgroup$ – Nick T Sep 6 '13 at 19:49
  • $\begingroup$ So the equilibrium I'd need to solve for $T$ is $\mu_{water}(T) + RT\ln \chi_{water_1} = \mu_{ice}(T) + RT \ln \chi_{ice_1}$ with a constraint $\chi_{water_1}\chi_{salt+water_1} + \chi_{ice_1}\chi_{salt+ice_1} = \chi_{water_0}$ (intended to be conservation of mass) where $_1$ indicates state in equilibrium and $_0$ indicates state before any freezing? Does the chemical potential of "salty ice" have some other expression? And do $\mu(T)$ have algebraic approximations? And, most importantly, how come there is no term describing the salt in the equations? $\endgroup$ – Karolis Juodelė Sep 7 '13 at 6:46
  • $\begingroup$ $\chi$'s sum to 1...multiplying them is throwing me a bit. There is no salt in my simpler equations because it never changes state. In any case, these are vaguely the "general principles", and you're getting beyond what I know how to theoretically solve. $\endgroup$ – Nick T Sep 7 '13 at 7:26

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