We have to show that at absolute zero of temperature the average energy of the conduction electrons in a metal is equal to , where is the Fermi energy.

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For a gas of electrons the average energy is defined by the relation

where is the number of electrons per unit volume having energy between and . At absolute zero of temperature the probability of occupation is given by the function

As the number density of electrons having energy between and is

,

and ,

we have

The number of electrons per unit volume at absolute zero of temperature is given by

therefore,

.