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ProblemSet6

# ProblemSet6 - ChE 101 2012 Problem Set 6 Read Schmidt...

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ChE 101 2012 Problem Set 6 Read Schmidt, Chapter 7 1. Derivation of the Langmuir isotherm: In this problem, we will use the tools of statistical mechanics to derive the Langmuir adsorption isotherm and understand how it changes with temperature. All of the theory we will need was developed in Ph 2B and can be found in Chapter 5 of Thermal Physics by Kittel if you need a refresher. We will model our heterogenous catalyst as a collection of N possible adsorption sites on the surface of a solid, onto which gas molecules can adsorb. Each site has binding energy and can accommodate at most one molecule. The adsorbed phase is in equilibrium with a gas surrounding the solid. (a) What three intensive properties of the two phases must be equal in order for equilibrium to be established? (b) If there are M adsorbed molecules, what is the energy E ( M ) of the system? What is the degeneracy Γ( M ) of the energy, treating molecules as indistinguishable? (c) In statistical mechanics, any system that is allowed to exchange both energy and particles with a larger reservoir (in our case the surrounding gas) is called a “grand canonical ensemble”. All of the thermodynamic properties of such an ensemble are encoded in its Gibbs sum Z , which is an extension of the partition function to systems with fluctuating particle numbers. Letting { s i } denote the set of all microstates of the system, the Gibbs sum for a system at temperature T and chemical potential μ can be expressed as follows: Z = X { s i } e β ( N i μ - E i ) Here, N i and

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