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• 1: Particle Number Fluctuations in a Grand Canonical State: The probability that a system in the grand canonical ensemble has exactly N particles is given by

p(N) = z^N Z_N(V, T) / Z_g(z, V, T) .

Verify this statement and show that in the case of a classical, ideal gas the distribution of particles among the members of a grand canonical ensemble is identically a Poisson distribution. Calculate the root-mean-square fluctuation ΔN for this system, and estimate the fractional fluctuation ΔN/N in the number of particles for 1 mole of helium atoms in equilibrium inside a container at STP (T = 300 K, p = \(10^5\)kPa).

• 2: Atmospheric Pressure: Kennett, Problem 6.2. Follow the approach Kennett suggests and remember to justify all your steps, including the use of the grand canonical ensemble if you use it.

• 3: Ultra-Relativistic Ideal Gas: Kennett, Problem 6.3.
   

Aside from discussions I may have had with other students in the class about how to solve these problems, the solutions to this homework set I am submitting are entirely my own. While I was writing these solutions I did not look at anyone else's solutions, and I was not told by anyone what to write.

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