The Absolute Zero of temperature is thought of as an unattainable goal. Whatever the design and power of your refrigerator, it will never be able to reach this limit. However when I began to look at the arguments for this, I found them unconvincing. See what you think...

The science of thermodynamics started with the question of the efficiency of steam engines. It was realised that to extract useful work it was necessary to have a hot source (burning coal) and a cold sink (the surroundings), and that useful energy could be extracted in the process of transferring heat from the hot source to the cold sink. However, such idealised heat engines can also be thought of as working in reverse, in which case there is an input of useful work and the result is to transfer heat from a cold place to a hotter one - this is how a refrigerator works.

If T_H is the temperature of the environment and T_C that of the cold object then a perfect refrigerator will use (T_H/T_C-1) units of energy for each unit of energy extracted from the cold object. So if we consider matter with a heat capacity of C Joules/Kelvin then the work required to decrease the temperature of the object down to T₀, with the environment at T_H is,

∫

TH   T0

(

TH T

-1

)

CdT

Hence if C is constant, then this tends to infinity as T tends to 0, implying that it would take an infinite amount of energy to cool a sample down to absolute zero - an unattainable limit!

The above ideas got me thinking. If you used energy to cool an object down then in theory you could get this energy back by using the object as the cold sink for a heat engine. This might be used as a way of storing energy. Indeed liquid nitrogen has been used to power an engine, but it own produces about 5% of the energy of the same weight of gasoline - although it has its uses in safety critical situations. If you do the sums for liquid helium it doesn't work out much better. But if there's the possibility to store an infinite amount of energy in a finite amount of matter then you would think that there would be plenty of useful applications - for instance getting into orbit.

The trouble is that the above calculations assume a constant heat capacity over temperature, and heat capacity is certainly not constant when you decrease the temperature - it drops to zero. If you look at the theoretical energy required to cool something to absolute zero you then get a finite answer. Indeed this enables us to define absolute zero as the temperature at which everything has zero entropy - the Third Law of Thermodynamics. The third law is often said to mean that absolute zero is unattainable, but I don't see how to deduce this. True, it might seem that you could use an object at absolute zero as a cold sink for a heat engine, and thus generate useful energy without increasing the temperature of the cold sink, but in fact the increase in temperature isn't 0, it's 0/0, implying we need a new way of looking at it.

In the case of a normal solid, the temperature is related to the atoms vibrating with respect to each other. However, quantum theory tells us that vibration of the atoms is quantised. Cooling in these circumstances is a case of removing the quanta of energy from the system, and reaching absolute zero means removing the last quantum of energy. This doesn't seem that simple, at least using technology we can think of today, but I'm not sure that it's theoretically impossible. Of course low temperature quantum systems lead to things such as Bose-Einstein condensates, where a collection of atoms are cooled so much that they all adopt the same quantum state, and it's harder to understand the concept of temperature to such a state.

So if the amount of energy input to reach absolute zero is finite, what other problems might there be. Well the usual argument is that a cooling process like adiabatic demagnetisation proceeds in cycles. Firstly the sample is magnetised and then cooled by demagnetisation. When you look at this you find that it would require an infinite number of cycles to reach absolute zero. This is a reasonable argument, but to my mind it has too much of Zeno's paradox. If Achilles gives the tortoise a 100 metre head start, but goes at 10 times the speed then first Achilles has to cover the 100 metres - but the tortoise has covered an extra 10 metres. Achilles covers this, and the tortoise is 1 metre ahead. It seems that Achilles will never overtake the tortoise as there are an infinite number of such episodes. However, we know that in fact he does so after 111 1/9 metres. Although it isn't as straightforward, it's possible to imagine that there may be some process which can reduce the temperature to absolute zero without going through an infinite number of cycles

Do I think that it might be possible to reach absolute zero? Well, on balance I think the belief that it's impossible is probably right. However, I would like to see more a more convincing argument to be certain.