Quantum_ Einstein, Bohr and the Great Debate About the Nature of Reality - Manjit Kumar [27]
Lord Rayleigh had originally proposed this other formula in June 1900, but Planck had taken little, if any, notice of it. At the time he did not believe in the existence of atoms and therefore disapproved of Rayleigh's use of the equipartition theorem. Atoms are free to move in only three ways: up and down, back and forth, and side to side. Called a 'degree of freedom', each is an independent way in which an atom can receive and store energy. In addition to these three kinds of 'translational' motion, a molecule made up of two or more atoms has three types of rotational motion about the imaginary axes joining the atoms, giving a total of six degrees of freedom. According to the equipartition theorem, the energy of a gas should be distributed equally among its molecules and then divided equally among the different ways in which a molecule can move.
Rayleigh employed the equipartition theorem to divide up the energy of blackbody radiation among the different wavelengths of radiation present inside a cavity. It had been a flawless application of the physics of Newton, Maxwell and Boltzmann. Aside from a numerical error that was later corrected by James Jeans, there was a problem with what became known as the Rayleigh-Jeans law. It predicted a build-up of an infinite amount of energy in the ultraviolet region of the spectrum. It was a breakdown of classical physics that many years later, in 1911, was dubbed 'the ultraviolet catastrophe'. Thankfully it did not actually happen, for a universe bathed in a sea of ultraviolet radiation would have made human life impossible.
Einstein had derived the Rayleigh-Jeans law on his own and knew that the distribution of blackbody radiation that it forecast contradicted the experimental data and led to the absurdity of an infinite energy in the ultraviolet. Given that the Rayleigh-Jeans law tallied with the behaviour of blackbody radiation only at long wavelengths (very low frequencies), Einstein's point of departure was Wilhelm Wien's earlier blackbody radiation law. It was the only safe choice, even though Wien's law managed to replicate the behaviour of blackbody radiation only at short wavelengths (high frequencies) and failed at longer wavelengths (lower frequencies) of the infrared. Yet it had certain advantages that appealed to Einstein. He had no doubts about the soundness of its derivation, and it perfectly described at least a portion of the blackbody spectrum to which he would restrict his argument.
Einstein devised a simple but ingenious plan. A gas is just a collection of particles, and in thermodynamic equilibrium it is the properties of these particles that determine, for example, the pressure exerted by the gas at a given temperature. If there were similarities between the properties of blackbody radiation and the properties of a gas, then he could argue that electromagnetic radiation is itself particle-like. Einstein began his analysis with an imaginary blackbody that was empty. But unlike Planck, he filled it with gas particles and electrons. The atoms in the walls of the blackbody, however, contained other electrons. As the blackbody is heated, they oscillate with a broad range of frequencies resulting in the emission and absorption of radiation. Soon the interior of the blackbody is teeming with speeding gas particles and electrons, and the radiation emitted by the oscillating electrons. After a while, thermal equilibrium is reached when the cavity and everything inside it is at the same temperature T.
The first law of thermodynamics, that energy is conserved, can be translated to connect the entropy of a system to its energy, temperature and volume. It was now that Einstein used this law, Wien's law and Boltzmann's ideas to analyse how the entropy of blackbody radiation depended on the volume it occupied 'without establishing any model for the emission