It is often assumed that quantum theory only applies to small systems, such as atoms and fundamental particles. But quantum theory can describe systems of macroscopic size and does so much better than classical mechanics. Indeed, some of the problems in classical physics that instigated the discovery of quantum theory were to do with the classical theory of solids, gasses and other macroscopic objects. One such problem is that the classical theory of solids violated the third law of thermodynamics.
The third law of thermodynamics was first proposed by the German physicist and chemist Walther Nernst in 1906. Nernst was motivated to explore the behaviour of matter at extremely low temperatures because of his interest in the properties of matter in the vicinity of absolute zero.
Nernst argued that when the temperature of a substance approaches zero Kelvin, all of its constituent particles will be in their lowest possible energy states so that the system is at its most ordered. The entropy of the substance will therefore be at its minimum value at zero Kelvin, so Nernst proposed that, by convention, a system's entropy should be equal to zero at zero Kelvin – this is now known as Nerst’s formulation of the third law of thermodynamics.
Nerst realised that his formulation of the third law provided a reference point for calculating the absolute entropy of substances at higher temperatures. So to test his hypothesis, Nernst conducted experiments on the heat capacities of substances at low temperatures, which allowed him to calculate their entropies. He found that the entropy of the substances he investigated approaches zero as the temperature approaches absolute zero, supporting the third law of thermodynamics and helping establish it as a fundamental principle of thermodynamics.
Nerst’s experimental results contradicted classical physics, which predicted that entropy could not approach zero at zero Kelvin. The reason for this is that a system’s heat capacity is a constant with respect to temperature – heat capacity is defined as the amount of energy required to increase the temperature of a system by one Kelvin while the gas or solid is held at constant volume:
Here dQ and dT are infinitesimal changes in a system's heat and temperature, respectively.
In classical physics, the heat capacity of a system of N particles can be derived by using the equipartition theorem, which states that for a system in thermal equilibrium, each degree of freedom of the system contributes an equal amount of energy TkB/2 to its total energy, where kB is the Boltzmann constant, and T is temperature. For a system with N particles and f degrees of freedom, the equipartition theorem tells us that the total energy is given by U = fNTkB/2. By taking the derivative of this expression with respect to temperature, we get:
Hence, in classical physics, the heat capacity of a system is constant with respect to temperature. From the heat capacity, one can compute the system’s entropy by using the definition of entropy, namely
Using the above equations, one can integrate the expression for entropy using the fact that the heat capacity is constant to obtain
so when simplified
Evidently, when T approaches zero, S(T) becomes arbitrarily large, regardless of the value of S(T₀), demonstrating that classical physics conflicts with the third law of thermodynamics.
The problem of heat capacity was solved by Albert Einstein, who used quantum theory to study solids. Einstein treated the atoms in a solid as quantum harmonic oscillators. In such a substance, now known as an Einstein solid, the heat capacity is temperature dependent, and in the low-temperature limit, the heat capacity can be shown to be proportional to
where hbar is Planck's reduced constant, and omega is the characteristic frequency of the Einstein solid. For this heat capacity, the entropy vanishes when T goes to zero, so Einstein’s model of a solid is in accordance with the third law. To demonstrate explicitly that the third law is satisfied, one can derive the expression for entropy using Einstein's heat capacity:
so taking the limit in which T goes to zero, one finds
as required by the third law.
Although Einstein's model was perfectly in agreement with the third law of thermodynamics, it was not entirely in agreement with experimental data on heat capacity. Einstein’s model was later refined by Debye, who argued that a solid also contains lattice vibrations, which should be considered as a kind of emergent particle—so-called quasiparticles—each of which contributes some energy to the system. Consequently, at low temperatures, the heat capacity in Debye's model has temperature dependence
Similarly, all other properties of solids – such as their rigidity, conductivity, and – are explained by quantum theory; the same is true for liquids, gasses, and other states of matter. Hence, even at large scales, quantum theory is needed to explain the behaviour of matter.
This post is based on one from my old blog; the original can be found here.