A rigid vessel of volume holds molecules of an ideal gas

PHYSICS · Subjective · 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 rigid vessel of volume VV holds N=500N = 500 molecules of an ideal gas. An imaginary plane divides the vessel into a left region of volume 0.400V0.400V and a right region. Each molecule is, independently of the others, in the left region with probability p=0.400p = 0.400 and in the right with probability q=0.600q = 0.600, so the number nn in the left region has the binomial distribution P(n)=(Nn)pnqN−nP(n) = \binom{N}{n}p^{n}q^{N-n}. In the text the mean and variance were obtained by adding NN single trials. Here obtain them by a second route, summing the distribution itself. (a) Prove the identity n(Nn)=N(N−1n−1)n\binom{N}{n} = N\binom{N-1}{n-1} and use it, with the binomial theorem, to show ⟨n⟩=Np\langle n\rangle = Np. (b) Prove n(n−1)(Nn)=N(N−1)(N−2n−2)n(n-1)\binom{N}{n} = N(N-1)\binom{N-2}{n-2} and hence show ⟨n(n−1)⟩=N(N−1)p2\langle n(n-1)\rangle = N(N-1)p^{2}. (c) Deduce that σn2=Npq\sigma_n^{2} = Npq. (d) Evaluate ⟨n⟩\langle n\rangle and σn\sigma_n for the numbers given.

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