Thermodynamics developed from the study of heat engines, which convert heat into useful work over the course of a thermodynamic cycle. For instance, a car's internal combustion engine converts the heat released by igniting gasoline to do the "useful work" of getting a car moving. Thermodynamics places strong constraints on how this can be done. We will explore an example of such a constraint in this problem. So far, we've seen the Second Law stated as "the average entropy of the universe cannot decrease." But there are other, equivalent, ways of stating it, like "heat cannot spontaneously flow from a cold object to a hotter object." This problem is to prove a corollary of the latter statement: "it is impossible to design a thermodynamic cycle that can extract useful work from heat absorbed from a single thermal reservoir." A quick reminder: a "thermal reservoir" is a source of energy so large that it always remains at the same temperature.
Imagine a cylinder filled with 1 mole of monatomic ideal gas. We will specify a "state" by the variables (P, V,T). For example: state i is specified by (P., Vi, T;).
A) The cylinder starts in state 0. We hold it at constant volume, and put it in contact with a thermal reservoir at temperature Th > To. It exchanges heat with the reservoir until it comes to an equilbirium in state 1. In terms of only Po, Vo, To, and Th, what is state 1 (P1, V1,Tı)?
B) Now, remove the cylinder from the thermal reservoir, and allow it to adiabatically (Q = 0) expand until its pressure equilibrates back to Po in state 2. For this part, you will need the adiabatic relation for a monatomic ideal gas T! = constant. (2.1) Р What is T2, in terms of only To and Th?
C) Rank To, Th, and T2, from highest to lowest.
D) From state 1 to 2, how much work is done on the cylinder?
E) To turn this into an engine, we need to create a cycle by returning the gas to state 0. Imagine we achieve this with a constant-pressure recompression of the cylinder back to V. Show whether heat exchanged during this process positive, negative, or 0.
F) If the cylinder can only exchange energy with the single reservoir at Th, prove that the last step is impossible.