The formula for the absolute entropy of a monatomic ideal gas is named after Otto Sackur and Hugo Tetrode who independently derived it in 1912 [1-3]. In classical thermodynamics the entropy of a monatomic ideal gas is
where E, V and N are the kinetic energy, the volume and the number of atoms, respectively. In classical physics the constant s0 is undetermined. The achievement of Sackur and Tetrode was to compute s0. At first sight this does not look very exciting, however, in order to compute s0 they had to work out the size of “elementary cells or domains” in phase space. Only with this phase space discretization it is possible to count the number of states in classical phase space. This is a prerequisite for the application of Boltzmann's absolute entropy formula
where W is the number of possibilities to realize a macroscopic system. Sackur and Tetrode determined the volume of phase space cells as where h is Planck's constant and n is the number of degrees of freedom. Until then, h was primarily associated with harmonic oscillators and photons. With the work of Sackur and Tetrode it became clear—more than ten years before the advent of quantum mechanics—that Planck's constant was ubiquitous in statistical physics.
Before we delve into its derivation, we abridge the history of the Sackur–Tetrode (ST) equation. While in his 1911 paper  Sackur develops the formula for the entropy S of a monatomic ideal gas as a function of the size of the elementary cell, in the subsequent paper  he postulates that its size is . At the same time, Tetrode gives an illuminating derivation of S , assuming that the size of the elementary cell is . Since the paper contains some numerical errors, he writes an erratum—see , where he corrects the numerics. He obtains now from a fit of z to data on the vapor pressure of mercury, and acknowledges the papers [1, 4] of Sackur. Finally, in  (received October 1912) Sackur presents a derivation slightly different from that of , obtains good agreement () with the data on the vapor pressure of mercury, and comments on the paper by Tetrode where he observes that Tetrode's result, which is the correct one, is larger by than his. It is amusing to note that in  Sackur had obtained the correct result, though this previous derivation is by no means clearer. For further details we refer the reader to .
The starting point of Tetrode's reasoning is the entropy formula (2) which should, according to Nernst's heat theorem , give the correct value of the entropy without any additive constant. He considers a system with n degrees of freedom and phase space coordinates , and associates W with the number of configurations of phase space points. In order to have a finite entropy, Tetrode states that it is necessary to discretize phase space and introduces “elementary domains” of volume
where z is a dimensionless number. This is a purely dimensional argument. Tetrode furthermore argues that, in a system of N molecules or atoms which are “equal and interchangeable” [gleichartig und vertauschbar], configurations being related solely by exchange of particles should not be counted as different. Therefore, denoting by the number of configurations in phase space for degrees of freedom, the entropy for such a system is . In this way the Gibbs paradox is avoided and S is obtained as an extensive quantity, though Tetrode does not mention Gibbs in this context. In the next step, Tetrode computes the phase space volume of a monatomic gas consisting of N atoms with mass m in the spatial volume V, and maximal total energy E. That volume is times the volume of the 3N-dimensional sphere in momentum space defined by
Then, according to the arguments above, the entropy is given by
Leaving out the details, Tetrode finally arrives at equation (1) with
This modern looking derivation is, 100 years later, one of the standard methods in textbooks. Of course, nowadays it is clear that quantum mechanics fixes the size of the elementary domain to , i.e. ; the latter result was obtained by Tetrode through a fit of z to the data on the vapor pressure of mercury.
In sketching Sackur's derivation, we have opted for the version in , where more details are found than in . In this paper he first derives Planck's law of radiation by considering a system of radiators, before he moves on to the ideal monatomic gas. In both cases Sackur defines a time interval τ in which the system is monitored and an energy interval for the discretization of energy. For the gas the time τ is assumed to be so small that, during this time, collisions between atoms can be neglected. Therefore, in this time interval, each of the kinetic energies associated with the three directions in space, , , , of every atom, can be assumed to lie in a well-defined energy interval of length . In this way, Sackur defines a three-dimensional energy space divided into cubes of volume . Numbering the cubes, Sackur makes the Ansatz that the number of atoms in the i-th cube is given by
arguing that this corresponds to the lowest order in an expansion in , with one factor for each axis. In this Ansatz, is the energy of an atom in the i-th cube. Furthermore, the total number of atoms and the total energy are
respectively. He goes on by distributing the N atoms into r energy cubes, in exactly the same way as in the case of harmonic oscillators and photons. The number of possibilities for putting N1 atoms into cube 1, N2 atoms into cube 2, etc. is given by
Notice that Sackur computes W for a given decomposition of N into the numbers , which clearly implies that he assumes distinguishable atoms; for indistinguishable atoms, a fixed decomposition would simply correspond to a single state and thus . According to equation (2), the entropy is obtained as
for large numbers . The most probable distribution is given by the maximum of S under the conditions (8), which leads to the Boltzmann distribution, i.e.
This procedure superficially resembles the derivation of the canonical ensemble, however, its spirit is completely different. We know that the ST equation is only valid for a dilute gas, and Tetrode's derivation implicitly assumes that the occupation numbers, i.e. the numbers of particles occupying the energy levels of single-particle states, are very small; otherwise the expression for the number of distinguishable configurations in phase space would be much more complicated than and effects of spin and statistics would have to be taken into account. However, Sackur in his derivation assumes the opposite, namely occupation numbers .
The aforementioned considerations yield the entropy
In this formula there are three unknowns: , α and β. At this point, referring to Sommerfeld , Sackur states that the smallest action that can take place in nature is given by Planck's constant h, therefore,
This replacement he had already successfully made in the derivation of Planck's law of radiation in the same paper. The other two parameters are determined by equation (8), in which summation is to be replaced by integration. That latter step requires several manipulations. First, Sackur sets , where the are momentum components and the are average Cartesian components of the distance covered by an atom during the time τ. Then Sackur connects the product of the three average distances with the volume V of the gas by equating it with the volume per atom: . It is hard to understand why this equation should hold, but in this way the summation in N and E of equation (8) is replaced by the integration
which allows to obtain the quantities α and β as functions of E, V, and N. Finally, inserting these α and β into equation (12) almost yields the ST equation, with 3/2 instead of the correct 5/2 in s0 of equation (6).
How could Sackur and Tetrode subject the absolute entropy of a monatomic ideal gas to experimental scrutiny? Consider the latent heat of a monatomic substance for the phase transition from the liquid to the gaseous phase. In terms of the absolute molar entropies, the latent heat is given by
where denotes the vapor pressure and the indices v and l refer to vapor and liquid, respectively. If the vapor behaves in good approximation like a monatomic ideal gas, one can substitute for the ST equation in which is replaced by the molar gas constant R, , and . For the liquid phase, neglecting in good approximation its pressure dependence, the absolute entropy can be expressed as an integral over the heat capacity:
Note that the integration begins at and includes, therefore, the solid and liquid phases, and the latent heat of melting. After insertion of and into equation (15), one obtains an expression for the vapor pressure:
This equation can be directly tested against experimental data and serves thus as a check for the ST equation. In 1912 the most comprehensive set of data was available on mercury. The test comprises two tricky issues. One is the latent heat L whose temperature dependence was not sufficiently known. Thus, using Kirchhoff's equation, Sackur and Tetrode took refuge to the approximation
where L1 is the measured latent heat at a reference temperature T1 and is assumed to be constant. The other issue is the integration over the solid phase in integral (16). It was known that Einstein's model of the heat capacity of a solid  gives the wrong behaviour at low temperatures. Therefore, both Sackur and Tetrode used a model by Nernst , which is a modified Einstein model and gives a better description at low temperatures. It is interesting to note that the Debye model  also came out in 1912, after Tetrode's paper but before Sackur's paper ref. .
It is amusing to use presently available thermodynamic data on mercury and repeat the test of Sackur and Tetrode. We follow Tetrode's approach of replacing h by in equation (17), plug in for h the mean value as recommended by CODATA, and fit z to the data on the vapor pressure of mercury taken from the CRC Handbook of Physics and Chemistry, in the temperature range from the melting point to 200° C. In order to achieve an accuracy as good as possible, we also take into account the slight temperature dependence of the heat capacity of liquid mercury. As further input we use the values of the enthalpy of formation and the standard molar entropy from the CODATA Key Values for Thermodynamics; the latter value saves us from the awkward task of determining the integral (16) at a reference temperature. The result of our computation is . For details we refer the reader to . We conclude that with present data the test of Sackur and Tetrode works perfectly well and Planck's constant can be determined with an accuracy better than one percent. This is not a bad result, considering the notorious difficulty of thermodynamic measurements.
The ST equation is one of the very first confirmations of Planck's quantum hypothesis and represents a fundamental step towards modern physics. Nevertheless, nowadays its history is an almost forgotten episode of early quantum theory. One may speculate why this is so. One point is that the development of the ST equation was a pioneering work which soon became common knowledge. But at the same time, the protagonists ceased to play a role in the academic world. Sackur (1880–1914), who was actually a physical chemist, tragically died in an explosion in the laboratory of Fritz Haber, only two years after the ST equation. On the other hand, Tetrode (1895–1931) was a wunderkind who published his first research paper, namely the paper on the ST equation, at the age of 17. He never finished his studies and never held any academic position. He rather lived in seclusion, though he did publish five more papers, some of which were appreciated by the community, and kept a few contacts with eminent contemporary physicists via correspondence before he prematurely died of tuberculosis.