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The quantum concentration $n Q$ is the particle concentration (i.e. the number of particles per unit volume) of a system where the interparticle distance is equal to the thermal de Broglie wavelength.

Quantum effects become appreciable when the particle concentration is greater than or equal to the quantum concentration, which is defined as:^[1]

n_{\rm {Q}}=\left({\frac {Mk_{B}T}{2\pi \hbar ^{2}}}\right)^{3/2}

where:

$M$ is the mass of the particles in the system
$k_{B}$ is the Boltzmann constant
$T$ is the temperature as measured in kelvins
$\hbar$ is the reduced Planck constant

The quantum concentration for room temperature protons is about 1/cubic-Angstrom.

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^[1]

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

References

^ ^a ^b Charles Kittel; Herbert Kroemer (1980). Thermal Physics (2 ed.). W. H. Freeman. pp. 73–77. ISBN 978-0716710882.

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[Kittel-1] Charles Kittel; Herbert Kroemer (1980). Thermal Physics (2 ed.). W. H. Freeman. pp. 73–77. ISBN 978-0716710882.

[1]

Quantum concentration

References