One mole of monatomic gas undergoes a Carnot cycle with T_H = 350°C and T_L = 210°C. The initial pressure is 8.8 atm. During the isothermal expansion, the volume doubles. Find the values of the pressure and volume at the points a, b, c, and d of Fig. 20–5.

검증된 단계별 안내

Step 1: Convert the given temperatures from Celsius to Kelvin. Use the formula T(K) = T(°C) + 273.15. For T_H = 350°C, calculate T_H in Kelvin. Similarly, for T_L = 210°C, calculate T_L in Kelvin.

Step 2: Identify the key points in the Carnot cycle (a, b, c, d) and their corresponding processes: (a to b) isothermal expansion at T_H, (b to c) adiabatic expansion, (c to d) isothermal compression at T_L, and (d to a) adiabatic compression. Use the given information that the volume doubles during the isothermal expansion (a to b).

Step 3: For the isothermal expansion (a to b), use the ideal gas law, PV = nRT, to calculate the pressure at point b. Since the volume doubles, V_b = 2V_a. The temperature remains constant at T_H, so P_a * V_a = P_b * V_b. Solve for P_b.

Step 4: For the adiabatic expansion (b to c), use the adiabatic condition for a monatomic gas: P * V^(γ) = constant, where γ = 5/3 for a monatomic gas. Use the known values of P_b, V_b, and the relationship to find P_c and V_c. Note that the temperature at point c is T_L.

Step 5: For the isothermal compression (c to d) and adiabatic compression (d to a), repeat similar calculations. For the isothermal compression, use P_c * V_c = P_d * V_d, where the temperature is T_L. For the adiabatic compression, use the adiabatic condition again to find P_a and V_a, ensuring the cycle closes back to the initial state.

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13m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Carnot Cycle

The Carnot cycle is a theoretical thermodynamic cycle that provides the maximum possible efficiency for a heat engine operating between two temperature reservoirs. It consists of four reversible processes: two isothermal (constant temperature) and two adiabatic (no heat exchange). Understanding this cycle is crucial for analyzing the performance of real engines and calculating work done and heat transfer during the cycle.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of an ideal gas through the equation PV = nRT. In this context, it is essential for determining the state variables of the gas at different points in the Carnot cycle, especially during isothermal and adiabatic processes. This law allows us to calculate changes in pressure and volume as the gas undergoes transformations.

Isothermal Process

An isothermal process occurs at a constant temperature, meaning that any heat added to the system is used to do work rather than change the internal energy. For an ideal gas, this implies that the product of pressure and volume remains constant (PV = constant). In the context of the Carnot cycle, understanding isothermal expansion and compression is vital for calculating the work done and the changes in pressure and volume at specific points in the cycle.