Homework 5 In Statistical Mechanics

Homework 5 Stat Mech

The assignment involves analyzing the adsorption of gas molecules on a sticky surface with A available sites. Each molecule lowers its energy by an amount ε when adsorbed, and each site can hold at most one molecule. The surface's chemical potential and temperature are assumed equal to those of the bulk gas, with consideration of the timescale dependence of adsorption-desorption equilibria. The task is to calculate the grand potential, show how the ratio kT/e^−Ω can be written as a sum over N, identify which N dominates this sum, and derive an expression for the surface molecular coverage as a function of ε, μ, and T.

Paper For Above instruction

Understanding adsorption phenomena at a molecular level is fundamental to statistical mechanics and thermodynamics. In this context, the problem describes a system where a bulk gas interacts with a surface containing A discrete adsorption sites. Each site can be either vacant or occupied, with the energy of an occupied site lowered by ε relative to the free state of a molecule in the bulk. The assumption that the chemical potential (μ) and temperature (T) of the surface phase are equal to the bulk's corresponds to the condition of rapid equilibration relative to the adsorption-desorption timescale, which ensures that both phases are in thermal and chemical equilibrium. This key assumption implies that the exchange of molecules between the bulk and surface is frequent enough to maintain such an equilibrium, justifying the use of a common μ and T in their thermodynamic descriptions.

Calculating the grand potential Ω (T, A, μ) for the surface involves the grand canonical ensemble. Since each surface site has two states—occupied or unoccupied—the partition function for a single site sums over these possibilities: for an unoccupied site, the contribution is 1; for an occupied site, it contributes exp(β(μ + ε)), where β = 1/(k_BT). Therefore, the grand partition function for the entire surface with A sites is the product of the individual site partition functions: [1 + exp(β(μ + ε))]^A. From this, the grand potential is Ω = -k_BT A ln[1 + exp(β(μ + ε))].

Expressing the ratio k_BT / Ω as a sum over N involves expanding the logarithm term via a combinatorial or binomial expansion, which reveals that the sum is dominated by a certain value of N, corresponding to the most probable number of occupied sites. The dominant N is given approximately by the expectation value of the number of occupied sites, which is A * exp(β(μ + ε)) / [1 + exp(β(μ + ε))]. This form resembles a Fermi-Dirac distribution, reflecting the exclusion principle that each site can hold at most one molecule.

Finally, the surface density of molecules—the fraction of occupied sites—can be expressed as θ = N / A, where N is the number of molecules adsorbed. This fraction derives from the probability that a given site is occupied, which is given by the Fermi-Dirac-like function:

θ = 1 / [1 + exp(-β(μμ + ε))]

or equivalently,

θ = exp(β(μ + ε)) / [1 + exp(β(μ + ε))]

This expression incorporates the effects of molecular energy lowering (ε), the chemical potential (μ), and temperature (T), and describes the surface coverage explicitly, capturing the full non-dilute behavior without assuming low surface density.

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