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.

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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:^[1]

${\displaystyle n_{\rm {Q}}=\left({\frac {MkT}{2\pi \hbar ^{2}}}\right)^{3/2}}$

where:

M is the mass of the particles in the system

k is the Boltzmann constant

T is the temperature as measured in kelvins

${\displaystyle \hbar }$ 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]

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