Please print this sheet, sign the statement at the bottom, and use it as cover page when you submit your homework. Remember to always include a complete explanation of your reasoning, and to show all calculations.

• 1: Variation of pressure with height [adapted from Kennett, Problem 6.2]
  We can calculate the variation of pressure in an isothermal atmosphere with a constant gravitational field g by equating the chemical potential at heights 0 and h. Use this argument to show that the pressure at height h is \[p(h) = p(0)\,{\rm e}^{-mgh/k T}.\] This result is not applicable to the Earth's atmosphere as it is not isothermal.

• 2: 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: Particle number fluctuations
  Find the relative fluctuation \(\sigma_N/\bar N\) in the number of particles for an ideal gas of particles of mass m in a grand canonical equilibrium state at temperature T and with chemical potential \(\mu\). Estimate the relative fluctuation numerically for a gas of helium atoms inside a container at room temperature and atmospheric pressure if the mean number of particles corresponds to 1 mole of gas.

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, and I did not use anyone else's worked out solution to any of these problems when writing mine.

Signed:   _______________________________________