The Sackur–Tetrode equation is an expression for the entropy of a monatomic ideal gas.

It is named for Hugo Martin Tetrode (1895–1931) and Otto Sackur (1880–1914), who developed it independently as a solution of Boltzmann's gas statistics and entropy equations, at about the same time in 1912.

Formula

The Sackur–Tetrode equation expresses the entropy <math>S</math> of a monatomic ideal gas in terms of its thermodynamic state—specifically, its volume <math>V</math>, internal energy <math>U</math>, and the number of particles <math>N</math>:

:S<sub>0</sub>/R = for p<sup><s>o</s></sup> = 101.325&nbsp;kPa.

Information-theoretic interpretation

In addition to the thermodynamic perspective of entropy, the tools of information theory can be used to provide an information perspective of entropy. In particular, it is possible to derive the Sackur–Tetrode equation in information-theoretic terms. The overall entropy is represented as the sum of four individual entropies, i.e., four distinct sources of missing information. These are positional uncertainty, momenta uncertainty, the quantum mechanical uncertainty principle, and the indistinguishability of the particles. Summing the four pieces, the Sackur–Tetrode equation is then given as

<math display="block">

\begin{align}

\frac{S}{k_{\rm B} N} & = [\ln V] + \left[\frac 32 \ln\left(2\pi m e k_{\rm B} T\right)\right] + [ -3\ln h] + \left[-\frac{\ln N!}{N}\right] \\

& \approx \ln \left[\frac{V}{N} \left(\frac{2\pi m k_{\rm B} T}{h^2}\right)^{\frac 32}\right] + \frac 52

\end{align}

</math>

The derivation uses Stirling's approximation, <math>\ln N! \approx N \ln N - N</math>. Strictly speaking, the use of dimensioned arguments to the logarithms is incorrect, however their use is a "shortcut" made for simplicity. If each logarithmic argument were divided by an unspecified standard value expressed in terms of an unspecified standard mass, length and time, these standard values would cancel in the final result, yielding the same conclusion. The individual entropy terms will not be absolute, but will rather depend upon the standards chosen, and will differ with different standards by an additive constant.

References

Further reading

  • .
  • . (This derives a Sackur–Tetrode equation in a different way, also based on information.)
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