# 17.7: Partition Function of Indistinguishable Components

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## Breakdown of Boltzmann statistics

From quantum mechanics, it can be shown that the above reasoning is only valid *if* the number of available states is much larger than the number of particles. For that to be the case the following relationship should hold:

\[ \dfrac{N}{V} \left( \dfrac{h^2}{8mk_BT} \right)^{3/2} << 1\]

If this inequality is not fulfilled a different type of statistical distribution needs to be applied. Table 17.1 in the book gives an idea when this happens. The electron gas inside a metal is a clear example of breakdown. Of the physical gases only the lightest two start to deviate a little at low temperatures.