JOST A MON

Immediately after we were taught Newton's Law of Cooling in high-school, the physics teacher asked us which vessel of water would freeze faster - one with water at 70°C or an identical vessel with the same mass of water at 20°. The consensus among my classmates was that the former vessel would freeze quicker, in accordance with Newton's law. I, however, objected. Newton's Law states that the rate of cooling of a body is proportional to the difference between its temperature and the ambient temperature. So, at 70°C, indeed, the hotter vessel's rate of cooling would exceed that of the colder vessel. As the hot water cooled rapidly, at one point it would catch up to the same temperature as the initially colder water - and from that time onwards, both of them should cool at the same rate and become ice simultaneously. Unless, of course, the point of coincidence was below freezing. The teacher shook his head sadly at me, but didn't explain if I was wrong.

As it happens, I'm still not sure if I was in error. Here's an alternative argument: the hotter water should first cool down to 20°, and then do whatever the 20° water does to freeze. So, the hotter water takes a little more time to freeze than the colder water.

As pointed out in this paper, which surveys the problem and discusses its ramifications on the thinking and doing of physics, the error in this last argument is that it implicitly assumes that temperature is the sole factor that determines the rate of freezing.

Physicists have long scoffed at this. It contradicts the laws of thermodynamics! It is not experimentally verifiable! The reports support discredited theories of heat! And so on. This raises the question of why physicists should dismiss the reports of laymen.In a related vein, Canadians have long known that they should not use hot water on their cars in winter:

In the 1960s, a Tanzanian high-school student named Erasto Mpemba asked his teacher to explain this cooling effect. He was laughed at, but by perseverance and the good offices of a more open-minded college professor, showed in a reproducible fashion that this effect, much disputed by theoretical physicists, did exist in reality.

So how to explain it? It turns out that this is a very hard problem to pin down experimentally. There have been contrarian reports on which factors are relevant to the effect and which are not. As pointed out above, the temperature of the water is not the sole determinant of the rate of cooling. Newton's law ignores such effects as evaporation, convection, and the fact that cold water is denser than hot water at 4° and contrarily below. The hot vessel loses more of its contents as steam than the cold vessel, and so the cooling masses are not the same, affecting their individual rates of cooling. Also, the fact that cold water is denser than hot water means that water that is uniformly at 30° would be quite different from one that had cooled to that (average) temperature from 70° - there would be a temperature differential through the vessel. And, the shape of the container affects the convective currents created in the water it holds, affecting the rate of cooling. More importantly, the shape of the refrigerator can affect the cooling - it is essential to consider the convection in the air surrounding the vessel! Lastly, the chemical composition of the water can skew results, particularly if there's a lot of dissolved air in it.

Clearly the Mpemba effect provides an opportunity for much experimental fun in a high-school physics lab. Not for me, however. I detested experimentation. Inherent laziness, I guess. Plus the equipment was badly calibrated and (horrors!) dusty. Had the teacher insisted that we should analyse the cooling law, I'd probably have hated doing it. Such a relief to have all this behind me now.

Reference:

1. Monwhea Jeng, Hot water can freeze faster than cold?!? Am. J. Phys. 74, 514 (2006).