Niven's constant

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Quantum effects become appreciable when the particle concentration is greater than or equal to the quantum concentration, which is defined as:

$n_{\mathrm{Q}}={\left(\frac{MkT}{2\pi {\hslash }^2}\right)}^{3/2}$

where:

Template:Mvar is the mass of the particles in the system

Template:Mvar is the Boltzmann constant

Template:Mvar is the temperature as measured in kelvins

$\hslash$ is the reduced Planck constant

As the quantum concentration depends on temperature, high temperatures will put most systems in the classical limit unless they have a very high density e.g. a White dwarf.

For an ideal gas the Sackur–Tetrode equation can be written in terms of the quantum concentration as

$S(T,V,N)=N{k}_{\mathrm{B}}[\frac{5}{2}+\ln \left(\frac{n_{\mathrm{Q}}}{n}\right)]$

References

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