Question

In a heat engine, 3.00 mol of a monatomic ideal gas, initially at 4.00 atm of pressure, undergoes an isothermal expansion, increasing its volume by a factor of 9.50 at a constant temperature of \(650.0 \mathrm{K} .\) The gas is then compressed at a constant pressure to its original volume. Finally, the pressure is increased at constant volume back to the original pressure. (a) Draw a \(P V\) diagram of this three-step heat engine. (b) For each step of this process, calculate the work done on the gas, the change in internal energy, and the heat transferred into the gas. (c) What is the efficiency of this engine?

Short answer

Question: Calculate the efficiency of the heat engine in the given three-step process. Answer: The efficiency of the heat engine can be calculated using the formula: \(\eta = 1 - \frac{Q_c}{Q_h}\), where \(Q_c\) is the heat released during constant pressure compression (step 2, \(Q_{BC}\)) and \(Q_h\) is the heat absorbed during the isothermal expansion (step 1, \(Q_{AB}\)). To find the efficiency, substitute the values of \(Q_{BC}\) and \(Q_{AB}\) obtained from the previous steps in the formula and calculate the result.

Step-by-step solution

Draw PV-diagram

Sketch a PV-diagram with the three steps of the process: isothermal expansion, constant pressure compression, and constant volume pressure increase. In the diagram, label each point as follows: - Point A: Initial state (pressure: \(P_A = 4.00\ \mathrm{atm}\), volume: \(V_A = \frac{nRT}{P_A}\)) - Point B: After isothermal expansion (pressure: \(P_B = \frac{P_A}{9.50}\), volume: \(V_B = 9.50V_A\)) - Point C: After constant pressure compression (pressure: \(P_C = P_B\), volume: \(V_C = V_A\)) - Point D: After constant volume pressure increase (pressure: \(P_D = P_A\), volume: \(V_D = V_A\))

Calculate work done during each step

Now, we'll calculate the work done on the gas during each step of the process: 1. Isothermal expansion (A to B): \(W_{AB} = -nRT_A\ln{\frac{V_B}{V_A}}\) 2. Constant pressure compression (B to C): \(W_{BC} = -P_B\Delta V_{BC} = -P_B(V_C - V_B)\) 3. Constant volume pressure increase (C to D): \(W_{CD} = 0\)

Calculate the change in internal energy for each step

Next, we'll calculate the change in internal energy for each step of the process: 1. Isothermal expansion (A to B): \(\Delta U_{AB} = \frac{3}{2}nR\Delta T_{AB} = 0\) (since \(\Delta T_{AB} = 0\) due to isothermal process) 2. Constant pressure compression (B to C): \(\Delta U_{BC} = \frac{3}{2}nR\Delta T_{BC}\) 3. Constant volume pressure increase (C to D): \(\Delta U_{CD} = \frac{3}{2}nR\Delta T_{CD}\)

Calculate the heat transferred for each step

Now, we'll calculate the heat transferred into the gas during each step of the process: 1. Isothermal expansion (A to B): \(Q_{AB} = \Delta U_{AB} + W_{AB}\) 2. Constant pressure compression (B to C): \(Q_{BC} = \Delta U_{BC} + W_{BC}\) 3. Constant volume pressure increase (C to D): \(Q_{CD} = \Delta U_{CD} + W_{CD}\)

Calculate the efficiency of the heat engine

Finally, we'll calculate the efficiency of the engine using the formula: \(\eta = 1 - \frac{Q_c}{Q_h}\) Here, \(Q_c\) is the heat released during constant pressure compression (step 2, \(Q_{BC}\)) and \(Q_h\) is the heat absorbed during the isothermal expansion (step 1, \(Q_{AB}\)). So, the efficiency is: \(\eta = 1 - \frac{Q_{BC}}{Q_{AB}}\) Calculate the efficiency using the values obtained in previous steps.