PHYSICS 313
Optional supplementary problems

If you have missed one or more of the regular assignments you can hand in solutions to this problem set for marking, but do so before class Wednesday November 29.

1:Derivation of Equation of State. In a certain system the internal energy U is related to the entropy S, particle number N, and volume V through

$$U={\rm const} N \left(\frac{N}{V}\right)^d \exp\left[\frac{dS}{Nk}\right]$$

where d is a constant.

(a) Show that the system satisfies the ideal gas law

PV=NkT

independent of the value of the constant d.

(b) Find the coefficient $\gamma$ in the adiabatic  equation of state $PV^{\gamma} = const.$ and the molar specific heats C_P and C_V of the system.

2. If the ratio of oxygen to nitrogen is 1/4 at sea level, what would you expect the ratio to be at an altitude of 5000m. Assume an isothermal atmosphere and temperature T=0^oC.

3: A sealed cylindrical vessel is in contact with a heat bath at constant temperature T. A friction-less airtight piston of weight mg divides the container into two volumes V₁ and V₂=V-V₁. There are N₁ ideal gas atoms in the top partition and N₂ in the bottom partition. The area of the piston is A.

(a). Find the equilibrium height of the piston.

(b). The ideal gas is replaced by a real single component gas. At a certain temperature the bottom partition is found to contain a puddle of liquid coexisting with its vapor. Which of the following statements may be true at equilibrium:

(i). The top partition contains a liquid in coexistence with its vapor. (ii). The top partition contains only vapor. (iii). The top partition contains only liquid.

(c). Assume that the cylinder is turned around so that the piston moves horizontally and that N₁=N₂. Which of the three statements in part (b) may now be true?