Derivation of sackur-tetrode equation

[ralden]

how sackur-tetrode equation derive?, can it derive without the use of macrocanonical ensemble? only by classical thermodynamics? thank you.

 

[vanhees71]

The Sackur-Tetrode formula can only be derived properly as the classical limit of quantum statistics, which is Boltzmann statistics for both fermions and bosons, taking into account the indistinguishability of particles, which is a specific quantum-mechanical phenomenon. Classical statistics leads to the Gibbs paradoxon and a wrong (non-extensive) expression for the entropy which is solved by the Sackur-Tetrode formula.

 

[wbandersonjr]

You can derive the Sackur-Tetrode equation by solving for the entropy of an ideal gas using Stirling's approximation applied to the multiplicity formula. My thermodynamics text does not go through the whole derivation, but that is how it says to derive it.

 

[vanhees71]

Yes, that you can do, but you have to assume the indistinguishability of particles, which leads to an additional factor $1/N!$ compared to classical mechanics. This factor cannot justified without the indistinguishability argument that is generically quantum theoretical.

 

[PJC01]

The sackur-tetrode equation is a fundamental equation in thermodynamics that describes the entropy of an ideal gas. It was derived by the German physicist Carl Sackur and the Russian physicist George Tetrode in the early 20th century.

The derivation of the sackur-tetrode equation is based on the macrocanonical ensemble, which is a statistical ensemble that describes a system in equilibrium with a fixed number of particles, volume, and energy. This ensemble is used to derive the equation by considering the number of microstates available to an ideal gas at a given temperature, volume, and number of particles.

It is not possible to derive the sackur-tetrode equation without using the macrocanonical ensemble. This is because the equation is based on statistical mechanics, which is a branch of physics that uses statistical methods to describe the behavior of a large number of particles.

Classical thermodynamics, on the other hand, is a macroscopic approach that describes the behavior of a system in terms of its macroscopic properties such as temperature, pressure, and volume. While classical thermodynamics can provide some insights into the behavior of an ideal gas, it cannot derive the sackur-tetrode equation.

In summary, the sackur-tetrode equation can only be derived using the macrocanonical ensemble in statistical mechanics. Classical thermodynamics, which is a macroscopic approach, cannot derive this fundamental equation.