A chamber holds argon (molar mass

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles

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

Problem

A chamber holds argon (molar mass 39.9 g mol−139.9\,\mathrm{g\,mol^{-1}}, collision cross-section σ=3.6×10−19 m2\sigma = 3.6\times10^{-19}\,\mathrm{m^2}), treated as an ideal gas at the reduced pressure 1.0×104 Pa1.0\times10^{4}\,\mathrm{Pa} and temperature 300 K300\,\mathrm{K}. The gas is still, so no convection occurs. Model an atom as a one-dimensional random walker taking 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 and successive steps independent and of random sign. 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)) How long does the walk take to reach a root-mean-square displacement of 1.0 cm1.0\,\mathrm{cm} along one axis, and how many steps is that? ((b)) An imaginary region of volume 1.0×10−18 m31.0\times10^{-18}\,\mathrm{m^3} is drawn in the gas and one quarter of that volume is singled out, so each of the NN atoms in the region is independently inside the quarter with probability p=14p = \tfrac14. Using the Gaussian limit of the binomial, find the probability of the mean count ⟨n⟩\langle n\rangle, and the probability of a count 10001000 atoms above the mean.

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