Past exam of the physics B course of the University of Cambridge / 2013 / Paper 2 / Section A / A4

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- Solution A4

Solution

Use the rotating reference frame equilibrium model, in which the centrifugal potential energy of a particle of mass

m_{i}

is

U_{i} (r) = - m_{i} ω^{2} r^{2} /2

. The Boltzmann distribution then gives

\frac{n _{i} ( r )}{n _{i} ( a )} = exp [\frac{m _{i} ω ^{2} ( r ^{2} - a ^{2} )}{2 k _{B} T}] .

Define the concentration ratio as

c = n_{1} / n_{2}

. At the axis,

c_{axis} = c_{wall} exp [\frac{( m _{2} - m _{1} ) ω ^{2} a ^{2}}{2 k _{B} T}] .

Thus the axis favours the lighter species. The expression assumes a dilute classical mixture at one temperature, equilibrium in the rotating reference frame, and sampling sufficiently close to

r = 0

.

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