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Homework #5 Stat. Mech. CHAP E4120x Prof. Ben O’Shaughnessy

1

Homework #7

A bulk gas is in contact with a “sticky” surface which has A sites available for molecular adsorption. Each gas molecule lowers its energy by an amount ε if stuck to one of the A surface sites. (Note ε > 0 so the net energy change is – ε .) Each site can only contain at most 1 molecule and please DO NOT assume that the surface molecular density is small. (You cannot use the diluteness assumption.) The chemical potential of the gas is µ and the temperature of this system is T.

1) You may assume the chemical potential and temperature of the surface- adsorbed phase equals the bulk chemical potential µ and T. What does this rely on? (Hint : Think about adsorption-desorption timescales.)

2) Calculate the grand potential Ω(T,A,µ) for the surface. Show that kTe /Ω− can

be expressed as a summation over N ( ∑ =0N

LL ) and show that this sum is

dominated by a particular value of N. 3) Express the surface density of molecules (the fraction of sites occupied by

molecules) as a function of ε, µ and T.

Bulk at µ and T

Surface

A sites