Why isn't 0°F the lowest possible temperature for a salt/ice/water mixture?

	Vocabulary
	Celsius temperature scale equilibrium eutectic mixture Fahrenheit temperature scale freezing point depression melting point phase diagram solute

1. Fahrenheit was a victim of Hofstadter's Law, which states:

It always takes longer than you think it's going to take, even when you take into account Hofstadter's Law. [2]

Fahrenheit originally chose an ice/salt/water mixture to use as a calibration point because it provided a steady, fairly reproducible temperature. To see why, consider the phase map for salt water, which shows which phases are present at different salt compositions and temperatures.

Phase map for salt water. Drawn from a diagram by R. E. Dickerson (Note 3)

The lowest temperature possible for liquid salt solution is -21.1°C. At that temperature, the salt begins to crystallize out of solution (as NaCl·2 H₂O), along with the ice, until the solution completely freezes. The frozen solution is a mixture of separate NaCl·2H₂O crystals and ice crystals, not a homogeneous mixture of salt and water. This heterogeneous mixture is called a eutectic mixture. The point at which a eutectic mixture is in equilibrium with the solution makes a wonderful calibration point for a temperature scale. This is because a mixture of salt and ice and saltwater is stable at one temperature and at one temperature only. If you see all three substances present in a mixture, and their amounts are not changing over time, you must be at -21.1°C.

By mixing salt, ice, and water so that at least 23.3% of the mixture was salt, Fahrenheit should have reached the eutectic point- in theory. But in practice:

• Salt dissolves very slowly in cold water.
• Salt dissolves very slowly in solutions that already contain high concentrations of salt.
• The salt dissolves more slowly if it has not been finely ground, and if the mixture is not stirred.

2. Fahrenheit didn't actually use NaCl solution. Fahrenheit calibrated 0°F as "the limit of the most intense cold obtained artificially in a mixture of water, of ice, and of sal-ammoniac or even of sea-salt" [1]. Sal-ammoniac is ammonium chloride, and sea-salt is not just NaCl. The eutectic points for ammonium chloride and sea salt mixtures are not at exactly -21.12°C (the eutectic point for NaCl and water), and they are not exactly equal to each other, either.

3. Fahrenheit abandoned the original 0°F calibration point. He chose new calibration points twice. He first adjusted the scale to make 32 the ice melting point and 96 the temperature of "the blood of a healthy man" [1]. It's easy to mark scale divisions that are multiples of two with simple tools, and the adjustment produced a scale with 64 divisions between the ice melting point and blood temperature.

He eventually replaced the upper calibration point at blood temperature with the boiling point of water. He discovered that the boiling point of water could be reliably used as a calibration point if atmospheric pressure was fixed, a fact that earlier temperature scale architects had overlooked. The boiling point at normal atmospheric pressure was set to 212°F, exactly 180 degrees between the freezing point of water and the boiling point of water, perhaps because 180 degrees is the number of degrees between opposite poles.

References and Notes

1. D. G. Fahrenheit,Phil. Trans. (London) 33, 1, 1724. Carmen Giunta provided the excerpts quoted at http://webserver.lemoyne.edu/faculty/giunta/fahrenheit.html.
2. From Douglas R. Hofstadter's excellent book Gödel, Escher, Bach: An Eternal Golden Braid, an anthem to self-reference in music, math, art, and stuff like that.
3. The saltwater phase diagram is based on Figure 6-59, p. 376 of R. E. Dickerson's Molecular Thermodynamics (Pasadena, California), 1969.
4. For a thermodynamic explanation of freezing point depression, see How can freezing point depression be explained in terms of free energies?

Author: Fred Senese senese@antoine.frostburg.edu