One molecule of mass moves freely in a box held at temperature by contact with a…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles · Statistical mechanics of ideal classical systems

Thermal Physics, Ch The Boltzmann factor, the canonical ensemble and the partition function

Problem

One molecule of mass

m

moves freely in a box held at temperature

T

by contact with a reservoir; its energy is purely kinetic,

E = \tfrac12 mv^{2}

. The counting of translational states gives, for a large box, a number of states with speed between

v

and

v+\mathrm{d}v

proportional to

4\pi v^{2}\,\mathrm{d}v

, with the same constant of proportionality at every speed. You may use

\int_0^{\infty}e^{-u^{2}}\,\mathrm{d}u = \frac{\sqrt{\pi}}{2},\qquad \int_0^{\infty}u^{2}e^{-u^{2}}\,\mathrm{d}u = \frac{\sqrt{\pi}}{4}.

((a)) Derive the normalised speed distribution

f(v)

, such that

f(v)\,\mathrm{d}v

is the probability that the molecule's speed lies between

v

and

v+\mathrm{d}v

. State which factor comes from the Boltzmann weight and which from the count of states. ((b)) By differentiating

f(v)

, show that the most probable speed is

v_p = \sqrt{2k_BT/m}

. ((c)) Evaluate

v_p

for an argon atom,

m = 6.63\times10^{-26}\,\mathrm{kg}

, at

T = 350\,\mathrm{K}

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