Thermodynamics, Statistical Physics, and Quantum Mechanics

234 SOLUTIONS

The partition function is then

and the average number of particles

the same as (S.4.102.8).

4.103 Thermionic Emission (Boston)

a) We can consider the electron gas outside the metal to be in equilibrium
with the electrons inside the metal. Then the number of electrons hitting
the surface from the outside should be equal to the number of electrons
leaving the metal. Using the formula for chemical potential of a monatomic
ideal gas (see Problem 4.39), we can write

where for an electron gas. Rewriting (S.4.103.1), we have

The state of equilibrium requires that this chemical potential be equal to
the potential inside the metal, which we can take as i.e., the
energy is required to take an electron from the Fermi level inside the
metal into vacuum. So, the pressure of the electron gas is given by

On the other hand, the number of particles of the ideal gas striking the
surface per unit area per unit time is