Three 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

Three short counting exercises. ((a)) A row of 88 distinguishable two-state units ("spins"), each fixed in place and showing either up or down. All arrangements have the same energy. How many microstates belong to the macrostate with exactly 33 units up? ((b)) An Einstein solid consists of 44 identical oscillators at fixed sites, sharing q=3q = 3 indistinguishable units of energy hνh\nu; any oscillator may hold any whole number of units. How many microstates does this macrostate have? ((c)) A single argon atom is treated as a free point particle in a cubical box of volume V=1.00 L=1.00×10−3 m3V = 1.00\,\mathrm{L} = 1.00\times10^{-3}\,\mathrm{m^3} with rigid walls, so that its de Broglie standing waves give allowed momentum components pi=nih/2ap_i = n_i h/2a with ni=1,2,3,…n_i = 1, 2, 3, \ldots along each edge of length aa. How many single-particle states have momentum magnitude less than p=1.00×10−23 kg m s−1p = 1.00\times10^{-23}\,\mathrm{kg\,m\,s^{-1}}?

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