A bell jar contains methane, (molar mass )

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Kinetic theory and transport in gases · Statistical foundations and the equilibrium ensembles

Thermal Physics, Ch Probability, random walks and the statistics of large numbers

Problem

A bell jar contains methane, CH4\mathrm{CH_4} (molar mass 16.0 g mol−116.0\,\mathrm{g\,mol^{-1}}), treated as an ideal gas at pressure 1.00×105 Pa1.00\times10^{5}\,\mathrm{Pa} and temperature 300 K300\,\mathrm{K}. The collision cross-section of a methane molecule is σ=5.0×10−19 m2\sigma = 5.0\times10^{-19}\,\mathrm{m^2}. The gas is still: there are no draughts or convection currents. Model a molecule as a one-dimensional random walker that takes a step of fixed length equal to the mean free path λ\lambda every collision time τ=λ/vˉ\tau = \lambda/\bar v, with vˉ\bar v the mean speed of the Maxwell distribution; successive steps are independent and equally likely to be in either direction. Use kB=1.381×10−23 J K−1k_B = 1.381\times10^{-23}\,\mathrm{J\,K^{-1}} and NA=6.022×1023 mol−1N_A = 6.022\times10^{23}\,\mathrm{mol^{-1}}. ((a)) Find the mean free path λ\lambda and the collision time τ\tau. ((b)) Find the diffusion coefficient D=λ2/2τD = \lambda^2/2\tau of the walk and the root-mean-square displacement along one axis after 60 s60\,\mathrm{s}. ((c)) An imaginary cubical cell of volume 1.0×10−18 m31.0\times10^{-18}\,\mathrm{m^3} is drawn inside the gas, and a plane divides it into two equal halves. Each of the NN molecules in the cell is independently equally likely to be in either half. Using the Gaussian limit of the binomial, find the probability that at one instant the left half holds exactly the mean number of molecules.

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