QUESTION:

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 pdV on their surroundings, which means that the total energy \(E_{\text {tot }}\) of the electrons will change by an amount \(d E_{\text {tot }}=-p d V\). Hence \(p=-d E_{\text {tot }} / d V\).

(a) Show that the pressure of the electrons at absolute zero is

                             \(p=\frac{3^{2 / 3} \pi^{4 / 3} \hbar^{2}}{5 m}\left(\frac{N}{V}\right)^{5 / 3}\)

(b) Evaluate this pressure for copper, which has a free-electron concentration of \(8.45 \times 10^{28} \mathrm{~m}^{-3}\). 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?

Equation Transcription:

Text Transcription:

dV

pdV

E_tot

dE_tot=-pdV

p=-dE_tot/dV

p=3^2/3 pi^4/3 h^2 over 5m(N overV)^5/3

8.45x10^28 m^-3