On the physical chemistry of ice cream and margaritas

When I was an undergrad at Stanford, senior biology majors were recruited to be TAs in the freshman biology class. I signed on for this and my first teaching experience at the undergraduate level involved attending lectures and leading discussion sections. In one lecture, the prof talked to the class about why salt is added to the ice in an old fashioned hand-cranked ice cream machine. He said, correctly, that the salt allowed the brine surrounding the freezing canister to get colder than 0°C, but that the mechanism was via the enthalpy of change solution of the salt dissolving in the water. I told my recitation section that the prof was wrong. The brine gets colder than 0°C due to freezing point depression, and that it was thanks to the ice starting at well below 0°C.

My questioning of his authority got back to the prof, who decided to add some material to a subsequent lecture to correct what he believed was his smart-aleck TA leading some of his freshmen astray. He pointed out that the enthalpy of solution for NaCl is +3.3 kJ/mol (or ~0.8 kcal/mol; we used kcal back then). I forget how much NaCl he thought was reasonable, but he came up with a back of the envelope calculation that disagreed with this

For each 58.44 grams (2.06 ounces) of salt that dissolves, 0.717 kilocalories (3 kilojoules) of heat is absorbed, meaning that dissolving salt causes the solution to become colder. The change is so slight you are unlikely to notice it in everyday life.

Fortunately for my prof, this was long before smart-ass students could use Google on their phones to find links to contradictory sources. And Wikipedia was far in the future. Saturated NaCl at 0°C is a 26% solution, which is ~4.45 M. So, starting with ice cold water with no ice you could drop the temperature to something on the order of 3 degrees. Which would eventually freeze the ice cream if you had massively excessive volume relative to ice cream, where you have to need a liquid to solid phase transition where the enthalpy of fusion on the order of 200 J/g (less than water, but still a lot relative to the heat of solution, according to this (pdf)).

So, in hindsight, I remain unconvinced by my old prof. I’ve often thought that one problem with intro bio textbooks is that they start with material that is taught more rigorously  in intro chem, by faculty who know the material better. I might have only been a freshman, but the elapsed time between when I had taken chemistry was decades shorter than it was for the ecologist teaching intro bio.

I was reminded of this experience earlier on Friday night, when I attended a very entertaining public lecture in the Physics Department, only peeking at the streaming video for the Stanford-S. Carolina Women’s basketball Final Four game on my phone a couple of times (Stanford lost, unfortunately… not enough enthalpy of shooting).

This was the event:

Now Harvard’s David Weitz is very different in background from a Stanford Ecologist/Intro Bio prof., and the elapsed time from my last physics/p. chem course is orders of magnitude longer than the last time he taught physics. Nevertheless, I think he made the same kind error as my old Intro Bio prof, and in fact I think his is worse in terms of the thermodynamics.

Toward the end of the lecture, he was using Peter Madden’s margarita preparation to illustrate temperature and phase transitions. In a shaker with ice and water, he asked the packed audience what they thought the temperatures were for the liquid and solid phases, i.e. the water and the ice. A young boy in the audience guessed that because they were in different phases, the ice was colder. He said something like: the ice is 31.999999 °F and the water is 32.000001 °F. Weitz said, no, they are both at 32 °F (or 0°C; there was a lot of shifting between C and F and I forget which). At that point, I leaned over to Debby and muttered that he was assuming that the system had reached thermal equilibrium, which was not knowable from the information provided.

OK, whatever… but then he had Madden pour out some tequila, which the measured as being at room temperature. They drained the water from the shaker, added the room temperature tequila, and shook it to mix. He then asked what people thought the temperature of the tequila would be. People guessed, they did the measurement, and lo and behold, it was significantly below 32°F/0°C.

What gave me a deja vu experience was his explanation of why the liquid phase was below the freezing point of pure water. We agree that the ethanol in the tequila is key. But unless I really misunderstood him both in real time and when I asked him about it afterward, he was arguing that the ethanol somehow allowed the liquid phase to lower the temperature of the solid phase! Which would require heat flow from the colder ice to the warmer tequila, looks to me like a flagrant violation of the Second Law of Thermodynamics.

This violation is only a problem if you think, as he insisted afterward, that the ice in the ice-water mix had reached thermal equilibrium and the solid phase starts uniformly at 0°C when the 20°C tequila is added. By contrast, if you agree with the kid in the audience that at least part the ice was colder than the final temperature of the liquid phase, there is no problem. The final temperature is just set by the freezing point depression from the ethanol and other solutes in the tequila.

The alternative hypothesis is that I’m misunderstanding what he said or missing something. This is plausible, because although it’s an appeal to authority argument, I think it’s a reasonable to think that a Harvard Physics Prof who specializes in phase transitions to have a higher probability of being right about this than me, a molecular biologist/annotation maven. Although I would estimate that the probability of me being right is still higher than the probability of getting a reservation at El Bulli before it closed.

But see the next post for an update!

Equation of the post:

ΔTF = KF · b · i,

  • ΔTF, the freezing-point depression, is TF (pure solvent) − TF (solution).
  • KF, the cryoscopic constant, which is dependent on the properties of the solvent
  • b is the molal concentration of the solute
  • i is the stoichiometry of the solute: the number of particles per molecule in solution. For ethanol this would be 1; for NaCl, it would be 2.

From Wikipedia.