BOILING water to make tea is much easier than freezing it to make ice. For thousands of years people have generated high temperatures – at least high enough to melt metals – but have only relatively recently managed to produce temperatures significantly below those of their surroundings. To explain why this is so, we must look carefully at the concept of temperature and how our ability to heat or cool things is limited by the laws of thermodynamics.
Common experience suggests that things get hotter as we add more heat, so a simple definition of temperature might be “the degree of hotness”. But it is not clear from this how to compare the temperatures of different objects; for example, the heat required to raise the temperature of water by 40 °C, enough for a bath say, would raise the temperature of a block of copper of the same mass by about 400 °C.
Sharing energy
Random exchanges
THERE are even situations – changes of states between solid and liquid phases or liquid and gas phases – where extra heat does not raise temperature. For example, ice absorbs heat as it melts to water but it stays at the same temperature – one reason why ice cubes are better at cooling drinks than cold water is. There is more to temperature than just heat content – it also depends on how that heat is distributed within a body.
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When a body is heated, the energy of the particles in that body increases. This energy may break bonds, allowing particles a greater freedom of movement and perhaps a change of state, as in melting ice to water, for example. It may also excite internal motions in the molecules, such as vibrations along bonds or rotations about some molecular axis. Or it may simply intensify the particles’ random motion by increasing their kinetic energy.
As the particles move randomly they interact with one another, continually exchanging energy. At any instant there will be a distribution of energy such that some particles have far more than average and others have far less. Higher energy particles are responsible for processes like evaporation and starting chemical reactions, and as the temperature rises so do the numbers of these high energy particles and the rates of these processes.
Bodies at high temperatures have relatively large numbers of high-energy molecules (see Diagram). But it is not just the quantity of energy supplied that matters. When water boils, the energy supplied to the liquid state is used to break bonds and separate particles rather than intensify their random movement. This energy, known as latent heat, does nothing to increase temperature. Similarly, if heat is supplied to a material whose molecules are capable of several internal motions, then the heat energy spreads more thinly and the rise in temperature is smaller than for a simple material in which added energy goes entirely to increasing molecular kinetic energies.
Now imagine placing a hot body in contact with a cold one. The hot body will have proportionally more molecules with high kinetic energy than the cold one and, since energy is randomly transferred across the boundary between the two bodies, the overwhelming likelihood is that energy will spread so that the proportion in the hot body falls while that in the cold body rises. (Although if the bodies are made of different materials, then the change in their temperatures as a result of transferring the same net amount of energy in either direction will not necessarily be the same.) Eventually they will reach a common temperature when the proportion of higher to lower energy molecules in both bodies is equal and simply that dictated by chance. In this condition, energy is still exchanged but the distribution, and hence the temperature, remains constant, apart from statistical fluctuations.
The essential idea that allows us to define temperature scales is that of thermal equilibrium – the fact that there is no net heat exchange when bodies at the same temperature are placed in contact with each other. This principle was formalised around 1930 into the Zeroth Law of Thermodynamics, which states that if bodies A and B are separately in thermal equilibrium with a third body C, then they are also in equilibrium with each other. All three share a common property, temperature. The Zeroth Law was the last law of thermodynamics to be formulated, a fact that underlines the depth and subtlety of this otherwise apparently familiar concept.
If temperature is linked to the distribution of energy among atoms or molecules, then it should come as no surprise that there is a lower limit to temperature as the heat energy of a body is removed. This absolute zero of temperature was first inferred by the French physicist Guillaume Amontons in 1702. At room temperature and above, it was found that the pressure of a fixed volume of any gas is directly proportional to its temperature – it fits the Ideal Gas Equation. If we extrapolate this back, we discover that an ideal gas would exert no pressure at about -273 °C. The classical interpretation is that a gas consists of particles in random motion and that its temperature is related to the average kinetic energy of these particles. Absolute zero is the temperature at which thermal motion would cease and the particles could therefore exert no pressure on the walls of the container. There is no upper ceiling for temperature since we can go on heating an object indefinitely.
No way down
Absolutely elusive
ABSOLUTE zero is used as the lower fixed point on the kelvin scale of temperature. The upper fixed point is the triple point of water – where ice, water and water vapour coexist at 0.01 °C. This is defined as 273.16 K. The degrees on the Kelvin scale are the same size as those on the Celsius scale. The conversion from Celsius to Kelvin is thus very simple:
Increasing the temperature involves adding energy, but it also leads to an increase in disorder, or entropy. Cooling involves the removal of heat and the reduction of entropy. The Second Law of Thermodynamics (see Inside Science, No. 75, “No Way Back!”) forbids any process whose net effect is to reduce the entropy of the Universe. So to cool something, which imposes order, we must create even more disorder elsewhere. Natural, spontaneous changes convert order to disorder all the time, and it is hard to reverse this. However, it is possible to harness the link between temperature and entropy to produce a cooling effect. If we increase the number of ways energy can be distributed, for example, by changing a gas into a liquid, then it will spread more thinly, the temperature will drop and heat can be absorbed from the surroundings. This is the principle that is used in most domestic fridges.
A refrigerator is designed to pump heat from a cold interior, at T1, and dump it in a warmer environment, at T2. One might suppose that this is possible without drawing on any other energy sources, since it doesn’t violate the First Law of Thermodynamics – that is, it conserves energy. However, the entropy change associated with a heat flow, Q, is Q/T, where T is the temperature in Kelvin of the object to which or from which the heat was given or taken. This means that the heat pump would violate the Second Law by causing a smaller increase in the entropy of the warmer environment than the decrease in entropy of the cooler interior of the refrigerator:
At the very least, we shall have to supply some extra energy so that more heat can be generated and dumped at the higher temperature exterior to pay for the order created in cooling the interior. This is the electrical work, W, drawn from the mains supply. The overall effect of refrigeration is to convert this work into heat. In other words refrigerators are actually heaters – if you leave a fridge running in a sealed room, the temperature of the room will rise. This is still true even if you leave the fridge door open.
In order not to violate the Second Law of Thermodynamics, the entropy generated by refrigeration must at least equal that destroyed. This gives a simple condition from which we can calculate the minimum quantity of work, W, needed to maintain any particular temperature below the ambient temperature:
The larger the ratio W/Q, the more work must be done per joule of energy extracted from the interior of the refrigerator. For example, to cool to a temperature that is 30 K below a room temperature of 300 K requires about 0.11 joules input per joule extracted. However, to cool to a temperature 150 K below room temperature needs one joule input per joule extracted. The work needed to remove heat increases more rapidly than the temperature difference that is to be produced and the efficiency of the process falls. Cooling to absolute zero would need an infinite input of energy per joule and is therefore impossible. This is true whatever temperature we cool from, since T2/0 is infinite for any T2. However, there is no theoretical limit to how closely we can approach absolute zero, although we can never attain it.
Another way of looking at the increasing difficulty of cooling as we go to lower and lower temperatures is to consider the ratio W/Q as we cool in steps. Consider cooling something from 16 K toward zero K by halving the temperature in each step:
16 K to 8 K W/Q = 1.00
8 K to 4 K W/Q = 1.00
4 K to 2 K W/Q = 1.00
2 K to 1 K W/Q = 1.00
1 K to 0.5 K W/Q = 1.00
There is clearly a law of diminishing returns here. Absolute zero cannot be reached in any finite number of such steps and so is unattainable. This is the Third Law of Thermodynamics, or at least one version of it. (see Illustration)
Getting colder
Cryogenics
THE DOMESTIC fridge would not be much use for liquefying air – oxygen liquefies at about 90 K and nitrogen at 77 K and the refrigerant would not change from liquid to gas at this temperature, no matter how low the pressure. However, gases such as oxygen and nitrogen were first liquefied using the same principles and a different refrigerant, by the French physicist Louis Cailletet and the Swiss scientist Raoul Pictet in 1877. Liquefaction of hydrogen at 20.2 K and helium at 4.2 K was trickier but was achieved by similar means. The British chemist James Dewar managed to liquefy hydrogen in 1898, and helium was liquefied by the Dutch physicist Heike Kamerlingh Onnes in 1908.
Several stages may be involved in cooling to very low temperatures. Liquid nitrogen or liquid helium are used as preliminary coolants. We can then get down to about 0.7 K by evaporating liquid helium at reduced pressure – if the vapour is continually pumped away then sufficiently energetic molecules will be able to escape from the liquid surface and the remaining liquid will cool. If the less massive helium-3 isotope is used instead of helium-4, then temperatures slightly below 0.3 K can be reached using this evaporation cryostat.
Getting very cold
Demagnetisation
THE KINETIC energy of molecules at very low temperatures is extremely small and when we attempt to cool them still further we need to exploit comparably sensitive phenomena in order to extract this heat. One phenomenon used is magnetism. A paramagnetic material contains individual atomic magnetic dipoles which are randomly oriented until the body is placed in a magnetic field. This random disordered arrangement has high entropy, and it persists in some paramagnetic materials even at very low temperatures.
The demagnetisation of a paramagnetic material at constant pressure -adiabatic demagnetisation – uses the transition between this disordered state and the order imposed by an applied magnetic field in a similar way to the change of state in a fridge’s vapour cycle (see “Chilling Work”). The temperature of samples which have been cooled in liquid helium can be further reduced to well below 0.1 K.
A second technique exploits the quantum mechanical properties of molecules. The dilution refrigerator, proposed by the German physicist Heinz London in 1951, uses a mixture of helium-3 and helium-4 isotopes in a cycle similar to that of the domestic fridge. At close to absolute zero, thermal energies are extremely small and quantum mechanical behaviour has to be taken into account. The mixture splits into two phases, one of which is light and the other rich in helium-3. The former acts rather like helium-3 gas while the latter is more like helium-3 liquid. By pumping helium-3 out of the “gas” phase, it will “evaporate” from the liquid phase and extract heat from it. This is analogous to the evaporation of a refrigerant in the vapour cycle. Very low temperature refrigerators, which exploit this phenomenon to produce temperatures as low as 0.002 K, were develop by the 1960s.
Paramagnetic demagnetisation eventually fails because the process depends on the disordered state of the magnetic dipoles, and this only exists because of thermal agitation. At temperatures that are low enough, separate dipoles interact and “crystallise” into an ordered state, even without an applied magnetic field. But the tiny magnetic dipoles associated with the “spin” of atomic nuclei interact much more weakly and their disordered state remains at even lower temperatures. So nuclear demagnetisation is used to reach these temperatures. The technique is very similar to paramagnetic demagnetisation -a magnetic field is applied to order the dipoles, and the temperature rises; this heat is removed and the temperature falls back to that of the prechilling stage, 0.002 K; the field is removed and the dipoles become disordered, so that the temperature drops and heat is absorbed from the sample. Nuclear cooling can reach temperatures as low as 0.00001 K.
Nernst’s theorem
Unattainable goal
FOR A long time, it was thought that entropy changes were all important and that it was meaningless to talk in terms of an absolute scale of entropy. (An absolute scale of entropy would be measured from the fixed point of an absolute zero of entropy, just as the Kelvin scale is measured from an absolute zero of temperature.) However, in 1906 the German physicist Walther Nernst postulated that the entropy of all objects would be zero at a temperature of absolute zero. But there is no process that remains disordered right down to a temperature of absolute zero, so an absolute zero of entropy cannot be achieved. This is another way of expressing the Third Law of Thermodynamics, which says that a temperature of absolute zero is also ultimately unattainable.
Even if there were a process that remained disordered right down to a temperature of absolute zero, successive cooling cycles would result in ever smaller reductions in temperature. Absolute zero cannot be attained by any finite number of steps. In practice, cooling is limited because at very low temperatures the ordered state is frozen in.
Weird science
Near absolute zero
ONE might well wonder why such efforts are made in taking ever shorter steps toward the ultimately impossible goal of achieving absolute zero. The reason is quite simple – strange things happen when the randomising influence of thermal vibrations is removed.
In 1911, Kamerlingh Onnes was investigating how the electrical resistance of metals varied at low temperatures. Close to room temperature, the electrical resistance increases with temperature – the electrons which carry electrical charge are scattered by thermal vibrations in the lattice of the material. With mercury, he discovered that the resistance steadily dropped with the temperature until 4.2 K, where it suddenly and dramatically fell to a value too low to measure and now believed to be zero. This superconductivity is not common to all metals. The best room-temperature conductors – copper, silver and gold – never become superconducting.
Until 1986, the highest known temperature for a material to become superconducting – its transition temperature – was 25 K. Since then, high-temperature superconductors have been discovered with transition temperatures above 100 K. Low-temperature superconductivity is thought to be due to a quantum coupling between conduction electrons which enables them to travel through the lattice as if it were not there. Above the transition temperature, the interaction which binds the electrons together is disrupted by thermal energy and the metal loses its superconductivity.
Another strange effect of temperatures close to absolute zero is superfluidity. At atmospheric pressure, helium liquefies at less than 4 K but it never becomes a solid. Classical physics has no explanation for this; liquid helium is an example of a quantum liquid.
The Uncertainty Principle (see Inside Science, No. 25, “Quantum Rules, OK!”) forbids particles to stop in one place, since then they would have a definite position and momentum. As we approach 0 K, therefore, the particles cannot have zero kinetic energy, and are left with a zero point energy. Helium’s low mass means that it has a zero point energy comparable to the energy required to dislodge it from the lattice that it would crystallise into as a result of interatomic interactions. An atom of helium is unable to maintain a fixed position in such a structure. If we cool liquid helium to below 2.2 K, its viscosity suddenly drops to zero and the atoms can flow without frictional opposition – it has become a superfluid. Superfluids have the uncanny ability to spontaneously siphon themselves out of any open container. Once again, low temperatures have revealed subtle but dramatic new physics.
1: Chilling work
WORK has to be done to cool things, but how is it to be expended? All refrigerators (see Diagram), from domestic fridges and freezers to the gas liquefiers and cryostats used in low-temperature physics, use the same four thermodynamic steps.
In the first step, work (W) is done to put the substance which will absorb the heat – the refrigerant – into a more ordered state. There are then fewer ways in which the substance’s heat can spread and so its temperature rises. In a fridge, this is achieved by compressing and liquefying a suitable fluid using a pump.
In the second step, the warmer, more ordered refrigerant is cooled to the temperature of its surroundings. In a fridge, the warmer liquid cools by losing heat (Q2) to the kitchen from the long tubes at the back.
In the third step, the refrigerant is placed in thermal contact with the object to be cooled and the “force” which imposed order on it is removed. In a fridge, this is done by passing the pressurised liquid through a valve so that it expands rapidly.
In the final step, the refrigerant resumes its previous disordered state but with less energy than before, so its temperature drops and it absorbs heat (Q1) from is new environment. In a fridge, the expanding liquid evaporators inside tubes within the ice box. In the process, the liquid absorbs heat from the interior of the fridge so that it can break the bonds between its constituent molecules, change state and expand into a gas.
Work is then done by the refrigerator to return the refrigerant to its initial ordered state. In a domestic fridge, the cold refrigerant is returned to the pump and the process – called the vapour cycle – is repeated.
2: Lowest of the low
LOW temperatures are essential in the sensitive detectors used to observe low-energy phenomena. Refrigerators with exceptionally high cooling powers are used at CERN, the European particle physics laboratory, to produce very cold targets for experiments. And large aluminium cylinders weighing several tonnes have been cooled to below 0.1 K to reduce thermal noise when looking for the tiny resonance vibrations induced by passing gravitational waves. Similarly, microwave detectors surveying the cosmic background radiation must be cooled to very low temperatures. Warmth from the detector could mask tiny fluctuations in the 2.7 K radiation from space which might otherwise give us clues to the distribution of matter in the early Universe and hence to the origin of the galaxies.
The quest for ever lower temperatures has lead to advances in electronics. The development of an electronic microrefrigerator – announced earlier this year by John Martinis – which cools to temperatures between 1 K and 0.1 K might significantly improve the performance of some electronic devices by reducing thermal noise. This novel refrigerator is a superconducting quantum device which allows only the most energetic electrons to tunnel away. Thus, it reduces the temperature in the same way that evaporation cools a liquid, but on a quantum scale.
And among the applications for superconductivity are ultra-sensitive magnetic field detectors that are capable of detecting the very tiny fields produced by human brain activity, quantum electronic switches and the extremely strong magnets used in particle accelerators. If room-temperature superconductors were developed or discovered, then electrical power could be transmitted via the electricity grid without loss.
