1). Suppose a room is divided into the front one fifth, the middle two fifths, and the back two fifths. What is the ratio of the probabilities of finding a 20%:40%:40% split in the particles (the most likely distribution), as opposed to a 19.9%:40.2%:39.9% split? Assume that all the particles are identical.

2). For a magnetic system the work done by the system is u(sub not)HdM, and the Gibbs free energy is given by G = U - ST + u(subb not)*HM. Show that

a) dG = -SdT + u(sub not)HdM

b) (dS/dM)T = u (sub not) (dH/dT)M

c) If M = M(H/T) then magnetic contribution to the heat capacity at constant magnetization vanishes.

3). Consider the case of a gas in the atmosphere. Assume that the temperature is a constant. Based on the Maxwell Boltzmann distribution, at sea level the atmosphere contains 78.1% N2, 21% O2, 0.9% Argon, and 0.036 CO2. What are the ratios at the the top of Everest? (Molecular mass of N2 = 28, Molecular mass of O2 = 32, Atomic mass of Argon = 40, Atomic mass of CO2 = 22. Height of Everest = 8,848 m)