Hi there. I'm a little lost as to why 31 is B (energy added to system in which process --> isotherm only). I thought that energy is added to the system when the gas is compressed, which means positive work on the gas or a movement to the left. May someone help explain this one to me.

Kailey, In path 3 to 1, that is an isothermal process - where the internal energy stays the same (depends on temperature), but heat is added to the gas so it can do work on the surroundings.  The heat adds energy to the gas so it can perform positive work. Please check out slide 137 in the presentation.  John

Hi John. Thanks for your explanation. I can agree that going from 3 to 1 (in that direction), that energy would be added to the system and work is positive. I also referenced slide 137 and saw that Q=-W when moving to the right (A to B; opposite direction of 3 to 1). In problem 31, it states that the gas starts at A and then moves to one of three possible processes. For this reason, I thought the answer had to be A (the isobaric process) since we can't go from 3 to 1 and only A to 3 or A to 1.

Sorry, my answer was not a good one - there is no 3 to 1 as you pointed out in your reply - all processes start at A. From A to 3, an isothermal process where delta U = 0, we have Q = -W' (W' is the work done by the external force). 

The gas is expanding at a constant temperature and a decreasing pressure, performing work (W) on the environment. The energy to perform that work comes from heat energy added.

Or use the equation: W' is negative, as the external force is opposite to the expansion of the gas, so Q = -W', a positive Q - heat energy added to the gas.

For A to 1, the gas is being compressed at a constant pressure. The work done on the gas is positive. Since the internal energy is decreasing (the temp is decreasing), heat must be released by the gas to the environment to account for the energy added by the external work. delta U = Q + W'.

John