Question

4. The pressure-volume diagram below shows a special reversible cycle called the Carnot cycle A mole of an ideal gas starts off in state 1 in contact with a large thermal reservoir at temperature Th. The gas then undergoes an isothermal expansion from Vi to V2. Upon reaching state 2, the gas container is removed from contact with the thermal reservoir and covered with thermal insulation. Next the gas is allowed to expand adiabatically from V2to Vs. Because the expansion is adiabatic, in doing work the gas cools to temperature Te. Once state 3 is reached, the thermal insulation is removed, and the gas is put in contact with another thermal reservoir at the temperature Te. The gas is then compressed isothermally at Te until state 4 is obtained. From state 4, the container is removed from the thermal reservoir, and the gas is returned to its original state (state 1) via an adiabatic compression, thereby heating the gas back up to Th ck up to state a) Write down an expression for the total work done during the Carnot cycle in terms of the variables defined above. Refer to the figure below. You must perform the necessary integrals explicitl b) Write down an expression for the total heat that flows during the Carnot cycle. ladiabatic ou

Answer 1

Work done - a) isothermal expansion dw = pdV w = integral dw = integral^V_V_pdV pV = nRT_h = integral^V_2_V_1 nRT_nV dy W_1 = 2.303 nRT_n log V_2/V_1 Adiabatic expansion w_2 = integral^V_3_v_2 pdv p_2 V_2^r = P_3 V_3^r = k = k times integral^V_3_V_2 dv/v^r = k(V^-r + 1/1 - r)V_2_v_1 = k/1 - r(1/V_3^r - 1 - 1/1/2^r - 1)k = P_2 V_2^r \r\n W_2 = 1/1 - r (P_3 V_3 - P_2 V_2) e) isothermal compression N_3 = integral^V_4V_3 pdv = -nRT_c ln V_3/V_4 = nRT_c ln V_4/V_3 = 2.303 nRT_c ln V_4/V_3 d) Adiabatic compression w_4 = integral^V_1_v_4 pdv P_4 V_4^r = P_1 V_1^r = K = k times integral^V_1 V_4 dv/v^r = k/1 - r(1/v_1^r - 1 - 1/v_4^r - 1) W_4 = 1/1 - r(P_1 V_1 - P_4 V_4) total work done implies w = W_1 + W_2 + W_3 + W_4 = nRT_h middot ln V_2/V_1 - nRT_c ln V_3/V_4