1. Consider a heat engine that has three steps. One step is isothermal, one is adiabatic, and the third is isochoric. (a) There are two orders that the steps could be carried out in. Sketch a PV diagram for each order. (b) For each order, determine the efficiency of the cycle in terms of the minimum volume Vi in the cycle, the maximum volume V2 in the cycle, the temperature TH of the hot reservoir, the number of moles of ideal gas n in the system, and the molar heat capacity at constant volume Cv. (c) Calculate To for each cycle and compare the efficiency for each cycle to the Carnot Efficiency, ecarnot = 1 - Tc/TH. (d) For each cycle, is the efficiency closer to the Carnot Efficiency for a gas with Cy = =R or one with Cv = = R? 2. The internal energy of an ideal gas only depends on the number of moles present and the temperature. What third state variable does the internal energy of a real gas depend on and why? 3. Suppose you have a pot with very well insulated sides and lid, but the bottom is pure copper and 3.00 mm thick. You fill the pot of water to a depth of 12.0 cm with 80.0 .C water, place the lid on it and set it on a giant block of ice at 0 .C. Assume that the water all cools at the same rate, meaning when the water at the bottom is 78 "C all of the water in the pot is at 78 "C. (a) Determine the rate that the water is initially cooling. (b) Determine the time it takes the water to cool to 75 .C. (c) Determine the time it takes for the water to cool to 2 "C. Hint: Refer back to the homework problem of the lake freezing