Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 2 / A7 / a

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- Solution a

Solution

Let

Λ = h / 2 πm k_{B} T

be the thermal de Broglie wavelength. The two free momentum components of one adsorbed atom give the phase-space integral

q_{trans}^{(2)} = \frac{1}{h ^{2}} \int_{σ} d^{2} r \int_{R^{2}} d p_{x} d p_{y} e^{- β (p_{x}^{2} + p_{y}^{2}) / (2 m)} = \frac{σ}{h ^{2}} (2 πm k_{B} T) .

Each momentum integral is a Gaussian integral, so

q_{trans}^{(2)} = \frac{σ}{Λ ^{2}}, Z_{N}^{(2)} = \frac{1}{N !} (\frac{σ}{Λ ^{2}})^{N} .

The second expression is the classical ideal-gas partition function for mobile, indistinguishable surface atoms. Its validity requires a dilute nondegenerate gas,

N Λ^{2} / σ ≪ 1

, and neglects interactions and excited motion normal to the surface. The

N!

appears here because particles move between positions, unlike the fully occupied fixed-site model of A6.

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