Please print this sheet, sign the statement at the bottom, use it as cover page when you submit your homework, and write your solutions on separate sheets of paper, leaving some blank space between problems. Remember to always include a complete explanation in full sentences of your reasoning, and to show all calculations.

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

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

and 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: Particle Adsorption on a Surface
  A surface with \(N_0\) adsorption centers has N (\(\le N_0\)) gas molecules adsorbed on it. Show that the chemical potential of the adsorbed molecules is given by \[ \mu = kT \ln {N\over(N_0-N)\, a(T)}\;,\]
  where \(a(T)\) is the partition function of a single adsorbed molecule. Solve the problem using the grand partition function of the system. (Neglect the interactions among the adsorbed molecules.)
   
• 3: Variation of Atmospheric Pressure with Height
  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.

You are allowed, in fact encouraged, to discuss these problems and how to solve them with other students in the class, but you are not allowed to copy or use in any way anyone else's written solution to any of these problems. This includes solutions that may be provided to you by a student who took the course previously, or solutions you may find online or in print. This policy will hold for all assignments in this class.

Aside from oral discussions I may have had with other students in the class, the solutions to this homework set I am submitting are entirely my own.

Name:   __________________________________   Signed:   ___________________________________