1. An ideal gas of N identical particles, each of mass m, is confined to a one-d
ID: 947580 • Letter: 1
Question
1. An ideal gas of N identical particles, each of mass m, is confined to a one-dimensional line of length L. The gas is in equilibrium at temperature T, which is large enough for the gas to be in the classical regime. Find the force f produced by the gas on the container confining it to the interval L: force f produced by the gas on the container confining it to the interval L: OF where F is the free energy of the gas 2. Write down the partition function Z and calculate the free energy F of a gas of photons in a cavity of volume V in equilibrium at temperature T directly from the partition function 2Explanation / Answer
Show that the partition function of a photon gas is given by
Z = Yn [1 exp((-hn)/ )]ˆ1
The partition function for a single mode with frequency n is determined by looking at the energies: E = n¯hn. The partition function of a single photon mode is then
Z1 = (,n=0)[ exp(n¯hn/ )] = 1/ 1 exp((-hn)/ )
The free energy is found directly from Z equation as
F = X n ln[1 exp(¯hn/ )] (106)
Transform the sum into an integral; integrate by parts to find
F = (ˆ2 x V x ˆ4 /45 x hˆ3 x cˆ3)
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