Department of Physics

Thermal Physics

Problem Set #3, Solution

Date: 2013/03/29

1. If we apply the highly successful kinetic theory of gases to a metal, consider as a gas of electrons (in fact, back in 1900 Drude constructed the theory, hence the Drude theory of metals), and assume that the electron velocity distribution is given by the Maxwell-Boltzmann distribution, what would the most probable speed, average speed, and rms speed for electrons at room temperature? Compare those to H2.

The Drude theory was replaced by the Sommerfeld theory of metals, in which the

Maxwell-Boltzmann distribution gave way to the Fermi-Dirac distribution, a consequence of the Pauli exclusion principle. Further improvements will be discussed in solid state physics. Now question yourself what you expect the true speeds to be qualitatively. Discuss among your teammates your arguments and then look for solutions in advanced books. A good reference book is Solid State Physics by Ashcroft

& Mermin, Chapter 1-3. Write down your questions and ndings.

Solution: From the Maxwell-Boltzmann distribution f (v) = 4πv 2

m

2πkT

3/2

exp −

mv 2

,

2kT

(1)

The most probable speed, the average speed, and the rms speed are vmax = v = vrms =

2kT m 8kT πm 3kT

,

m

(2)

(3)

(4)

respectively.

The difference between hydrogen molecules and electrons is their mass. The mass of a proton is about 2000 times larger than an electron. A hydrogen molecule has

√

two protons, which means its m is about 60 times larger than an electron. So all corresponding velocities are about 60 times smaller.

1

2. Zhao & Luo, Problem 2-17 (page 128)

Solution: From the Maxwell-Boltzman distribution, we have fp (px , py , pz )dpx dpy dpz =

1

2πmkT

3/2

exp −

p2 + p2 + p2 x y z dpx dpy dpz

2mkT

3/2

p2 + p2 + p2 z y x dpp2 4π

2mkT

3/2

√

1

=

d 2m 2m 4π exp −

2πmkT

kT

√

2

(kT )−3/2 exp− /kT

=

d. π =

1