Consider a system of \(N\) free electrons within a volume \(V\). Even at absolute
zero, such a system exerts a pressure \(p\) on its surroundings due to the
motion of the electrons. To calculate this pressure, imagine that the volume
increases by a small amount \(dV\). The electrons will do an amount of work \(p\)
\(dV\) on their surroundings, which means that the total energy \(E_{tot}\) of the
electrons will change by an amount \(dE_{tot} = -p dV\). Hence \(p =
-dE_{tot}/dV\). (a) Show that the pressure of the electrons at absolute zero is
\(p = \frac{3^{2/3}\pi^{4/3}\hbar^{2}}{5m} \lgroup \frac{N}{V}\ \rgroup^{5/3}\)
(b) Evaluate this pressure for copper, which has a freeelectron concentration
of \(8.45 \times 10^{28} m^{-3}\). Express your result in pascals and in
atmospheres. (c) The pressure you found in part (b) is extremely high. Why,
then, don't the electrons in a piece of copper simply explode out of the
metal?
