A1: Isothermal atmosphere

Too many infinities if you let gamma go to 1 exactly, so try making it go to 1.001 and see what happens. When you're sure of that try gamma = 1 + epsilson.

A2: Ice skating

For the pressure p, you need an energy and a volume. A natural energy is the heat of vaporization for one molecule. Which should suggest a natural volume...

A3: Fog

Moist air cools at night and the water condenses into little droplets (fog). How much water condenses? How far can a light ray travel before hitting a droplet? Use your knowledge of mean free paths from ideal gases (or use dimensions).

B1: Vapor pressure

The Boltzmann factor gives the probability of finding a molecule in different energy states. Here the states are liquid (nice, low energy) or gas (much higher energy because all the bonds are gone).

As T goes to infinity, the vapor pressure equation predicts that the pressure asymptotes to p0. That result seems strange. But think about the world at infinite temperature. Then the energy difference between liquid and gas becomes miniscule, and the molecule no longer cares whether it is in the liquid or the gas. Near such huge temperatures, it cares almost equally little.

D1: Carnot cycle

The notes for 6 Feb have everything except the diagrams. But two points to watch for: no heat flow in the adiabatic legs (2 out of the 4 steps); and the net work done must be the net heat flowing in (conservation of energy, or the first law).