For a gas of free electrons in d dimensions, compute the isothermal compressibil
ID: 3161652 • Letter: F
Question
For a gas of free electrons in d dimensions, compute the isothermal compressibility, kr(0) at zero temperature, in terms of the mean number of particles per unit volume, n = /V and the Fermi energy, epsilon_F. Give the answer to an overall numerical constant C_d. Estimate numerically the Fermi temperature of metallic copper by treating the electrons as a gas of free particles in 3-dimensions. The density of copper is 8920 kg. m^-3 its atomic weight is 63.5 and it may be assumed that there is one conduction electron per atom.Explanation / Answer
a) for free electron gas the electrons are considered as moving in a box of length l therefore energy eigen value using wave vector k for the electrons looks like
E= Hcut^2K^2/2m
Considering FD and BE statistics the Fermi energy may have the given expression
Ef= KbTf = Hcut^2/2m(3pie^2N/V)^2/3
but isothermal compressibility is based on Z and p where Z is the ratio of actual volume and ideal volume of gas (free electron)
from equation of state we know that Vg^-1 = P/ZnRT
using the same isothermal compressibility = -1/Vg (delVg/delP)T
now if we consider fermy temperature Tf then the compressibility factor becomes - 1/Vg(2V/2P)Tf
b)
Fermi temperature Tf= h^2/2piemK(N/1.504v)^2/3 = (6.626 *10^-34)^2/ 2*3.14* 8.92*10^-27*1.38*10^-23 (6.023*10^27/1.504*63.5)^2/3 = .074K
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.