Four short counting exercises.

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

Thermal Physics, Ch Microstates, multiplicity and the statistical entropy

Problem

Four short counting exercises. ((a)) A row of 99 distinguishable two-state units, each fixed in place and showing up or down, all arrangements having the same energy. How many microstates has the macrostate with exactly 55 units up? ((b)) An Einstein solid of 77 identical oscillators at fixed sites shares q=3q = 3 indistinguishable units of energy hνh\nu. How many microstates has this macrostate? ((c)) A nitrogen molecule (N2\mathrm{N_2}, molar mass 28.0 g mol−128.0\,\mathrm{g\,mol^{-1}}) has mean translational kinetic energy 32kBT\tfrac32 k_BT at a temperature of 0 °C. Find its root-mean-square momentum mvrmsm v_{\rm rms}. ((d)) Treating that molecule as a free point particle in a cubical box of volume V=0.50 LV = 0.50\,\mathrm{L} with rigid walls, so that its de Broglie standing waves give allowed momentum components pi=nih/2ap_i = n_ih/2a with ni=1,2,3,…n_i = 1,2,3,\ldots, how many single-particle states have momentum magnitude less than the value found in (c)?

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