Mpemba

The Mpemba Effect


Mpemba was a secondary school student in Tanzania in 1963 who had the fortune of re-discovering some interesting physics during one of his cookery lessons.

Having boiled some milk for making ice cream, his class were told to let the mixture cool before putting it in the refrigerator. Mpemba, however was anxious of ensuring himself a space so put his mixture in straight away.

The other students waited and put their mixtures in later, after they'd cooled down. Having noted the time his ice cream entered the freezer compared with the rest of his class, Mpemba realised his mixture had frozen significantly faster than everyone else's.

He came away with the simple observation that "hot liquids freeze faster than cold liquids".

His science teacher told him this was impossible and he must have got mixed up. So why exactly is this impossible?

In a subsequent year at high school Mpemba was taught about Newton's law of cooling in science: the rate at which a body cools is proportional to the temperature difference between that body and its surroundings, given by...

dT(t)/dt = -k[T(t)-T_a]

...which leads to the solution...

T(t) = T_a + (T_0 - T_a)e^(-kt)

Any set of cooling curves plotted from such a function will never cross, no matter what the initial starting temperature. So a curve which starts at a higher temperature will never undercut a curve starting at a lower temperature and will therefore always take longer to cool. Varying the parameter k on the other hand could well cause graphs to cross. But this parameter is determined from some initial conditions, if both systems are not identical in such things as geometry or arrangement, with the exception of starting temperature, then it is hardly appropriate to compare cooling times for different initial temperatures.

However, Mpemba was undeterred by a theory which did not seem to support his observations: he asked a friend who sold ice cream in a nearby town who told him he routinely used hot mixtures because they froze more quickly.

Still persisting with this, in 1969 a visiting academic from University College in the capital called Dr Osborne came to visit Mpemba's school and he jumped at the opportunity to quiz him about this apparent violation of Newton's Law. Thankfully he didn't dismiss it outright, and upon returning to Dar es Salaam, he instructed a lab-assistant to carry out an experiment to see if hot water would freeze more quickly than cold water.

The lab-assistant reported the hot water had frozen first, but not to worry, "I'll keep on repeating the experiment until we get the right result." After several attempts it seemed Mpemba was right - hot water would freeze faster than cold water.

Publish or perish

Osborn and Mpemba published these results in a journal called Physics Education, coincidentally the same year that George Kell at the National Research Council of Canada in Ottawa reported the same phenomenon that year in the American Journal of Physics.

I said Mpemba re-discovered this; having mentioned this "Mpemba Effect" in one of their articles, the New Scientist was subsequently flooded with anecdotes from all over the world of only the hot water pipes freezing during a short cold snap, ice-rink operators preferring to use hot water and so on.

So this clearly wasn't unheard of.

Surely the validity of this effect can be deduced by carrying out experiments - however this has proved surprisingly difficult. The Mpemba effect is only observed under certain conditions - there are clearly many factors which could affect how quickly water cools such as the geometry of the container, the volume of water and the temperature of the refrigerator.

In 1977, Jearl Walker published results in the Scientific American whereby the time to freeze was measured against the initial temperature for a variety of containers. His results showed two things. Firstly where negative gradients occur, water at an initially higher temperature appeared to be freezing more quickly. Secondly, this is by no means a universal effect, since most of the curves showed very little (if any) in the way of negative gradients.

On the aspect of repeatability, Walker reported that whilst most of his results were repeatable, he sometimes observe large variations in his results and said "I have not been able to resolve the controversy".

So assuming both Newton and Mpemba are correct - how do we understand what is going on here? Can we somehow reconcile these two arguments?

Conduction

Imagine two containers of equal geometry and material, one containing hot water and another containing an equal amount of cold water. Both of these are placed on a shelf in a freezer. Now any frost that collects on the container is likely to be melted by the warmth of a container made of a good conductor.

This has the effect, later on, when the water inside has cooled somewhat, such that the frost outside refreezes, of creating very good thermal contact between, say, the cold freezer shelf and the vessel of water. Hence heat is drawn out of warmer water more quickly.

The cooler container on the other hand won't have the opportunity to melt any surrounding frost and will just sit on top of a layer of ice, which isn't the best conductor of heat - so takes longer to cool down.

This account seeks to 'explain-away' the Mpemba effect in terms of bad experimental technique: if you don't allow one container to gain better thermal contact - you won't observe the effect. Well, the effect of conduction can be dramatically reduced by using a vessel made of a better insulator, in fact Mpemba himself used wooden buckets and still observed the effect.

So assuming measures are taken to prevent conduction, convection seems the next likely candidate.

Convection

As the warmer water cools rapidly at the surface it will develop convection currents within the container since warmer water is, at most temperatures, less dense than cooler water - creating an uneven distribution of temperature with hot water nearer the surface.

So when the hot-water container reaches the temperature the cool water container started at, the hotter water is nearer to the surface, the so-called "hot-top". This assists quicker evaporation and hence faster cooling since there is greater evaporation from hot water than from cold.

This shows that the initially hot water cools faster, but of course it also has further to go. So whether it actually reaches 0oC first,is not immediately clear. In fact, to know which one finishes first would require theoretical modelling of the convection currents, which nobody has done.

To add to the confusion, there are "cold tops". Cooler water is not always more dense than hot water - below 4oC cold water is actually lighter than the surrounding warm water. This means that once the coolest part of the water get - below 4oC it rises to the top and soon freezes - creating a insulating plug slowing down further cooling. Convection currents in the warmer water might help to reduce this process.

Evaporation

The next phenomenon is evaporation. An evaporating substance will loose mass, which takes with it an associated latent heat of vaporisation. With less mass, the hot water has less heat to loose, and so it cools faster. Assuming this explanation, hot water freezes first, but only by virtue of the fact there's less of it to freeze.

George Kell actually conducted some calculations that showed that if the water cooled solely by evaporation with a uniform temperature, the warmer water would freeze before the cooler water.

This explanation is often citied by many as the explanation of the Mpemba effect - whilst it's very important other experiments show that it cannot be the sole mechanism that drives the Mpemba effect. Dr Osborne measured the mass lost due to evaporation in his original experiment and found it incomparably less than that predicted by Kell's article.

Super-cooling

Finally, the last effect to offer an explanation is super-cooling. Once water reaches its freezing point, water molecules attempt to adopt the lowest energy state, which is an ice crystal. However they cannot do this without first encountering some irregularity in their surroundings, a nucleation site, which forces them to arrange themselves in a certain way, allowing an ice crystal to develop.

But if the molecules do not encounter such an irregularity they continue to cool below zero whilst still remaining in the liquid phase for a while longer. This is super-cooling. So a liquid that undergoes super-cooling will take longer to freeze since it stays liquid despite having reached 0oC

There have been some claims that initially hot water doesn't super cool for very long - say only as far -2oC whereas initially cool water may remain super cooled as far -8oC. This is no more an explanation than a replacement problem - how can water remember what temperature is was at before it reached 0oC ?

One possible explanation is that a heated water has more of its dissolved gases expelled in the boiling process. This supposedly helps the flow of convection currents and thus assists in cooling.

But one would expect that with less dissolved gas to act as a nucleation point, the boiled water which starts off hotter would super cool for longer whilst the molecules searched for a comparatively rare nucleation point.

Supporters of the super cooling theory point to symmetric molecules like nitrogen and methane, which are non-polar solvents, the solubility of which don't necessarily vary linearly with temperature.

More recently in 2005 Monwhea Jeng published some work with the most probable conclusion there simply isn't a unique explanation, certainly not yet, as to why hot water sometimes cools more quickly than cold. So it's tempting to believe, since freezing requires sufficiently cool molecules to encounter nucleation sites that it could largely be a matter of probability. This might explain why the Mpemba effect can sometimes be hard to reproduce and doesn't always lead to consistent results.

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