1.

Assuming the average energy of electrons to be 3/5 of the Fermi energy estimate the pressure of the electron gas in a metal. Do the calculations for aluminium.

Answer»


Solution :Find the total energy of the electron GAS `W=3/5n epsi_(F)V`, where V is the VOLUME of the metal. Imagine the gas compressed by a small amount dV. This will require work against the forces of pressure equal to `DeltaW=-PdV`, where P is the pressure of the electron gas. This work is equal to the change in the energy of the electron gas `DeltaW=dW`. To find the differential of energy let US express the electron gas concentration in terms of its volume : `n-N//V`, where N is the total number of electrons. We have
`W=3/5Nh^(2)/(2m)(3/(8pi))^(2//3)N^(2//3)V^(-2//3)`
Differentiating, we obtain
`dW=-2/3xx3/5h^(2)/(2m)(3/(8pi))^(2//3)N^(5//3)V^(-5//3)dV=-(2H^(2))/(5m)(3/(8pi))^(2//3)n^(5//3)dV=-2/5n epsi_(F)dV`
The pressure of the electron gas is `P=2/5n epsi_(F)`


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