Chapter 20, Problem 20.57P

DATA You are conducting experiments to study prototype heat engines. In one test, 4.00 mol of argon gas are taken around the cycle shown in Fig. P20.57. The pressure is low enough for the gas to be treated as ideal. You measure the gas temperature in states a, b, c, and d and find T_a = 250.0 K, T_b = 300.0 K, T_c = 380.0 K, and T_d = 316.7 K. (a) Calculate the efficiency e of the cycle. (b) Disappointed by the cycle’s low efficiency, you consider doubling the number of moles of gas while keeping the pressure and volume the same. What would e be then? (c) You remember that the efficiency of a Carnot cycle increases if the temperature of the hot reservoir is increased. So, you return to using 4.00 mol of gas but double the volume in states c and d while keeping the pressures the same. The resulting temperatures in these states are T_c = 760.0 K and T_d = 633.4 K. T_a and T_b remain the same as in part (a). Calculate e for this cycle with the new T_c and T_d values. (d) Encouraged by the increase in efficiency, you raise T_c and T_d still further. But e doesn’t increase very much; it seems to be approaching a limiting value. If T_a = 250.0 K and T_b = 300.0 K and you keep volumes V_a and V_b the same as in part (a), then T_c/T_d = T_b/T_a and T_c = 1.20T_d. Derive an expression for e as a function of T_d for this cycle. What value does e approach as T_d becomes very large?

Figure P20.57