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23. The Second Law of Thermodynamics
The Carnot Cycle
12:06 minutes
Problem 20.19
Textbook Question
Textbook Question(III) A Carnot cycle, shown in Fig. 20–5, has the following conditions: Vₐ = 7.5 L , Vᵦ = 15.0L , T_H = 470°C and T_L = 260°C . The gas used in the cycle is 0.50 mol of a diatomic gas, γ = 1.4 . Calculate
(a) the pressures at a and b;
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Verified step by step guidance
1
Convert the temperatures from Celsius to Kelvin by adding 273.15 to each temperature. This gives T_H in Kelvin and T_L in Kelvin.
Use the ideal gas law, PV = nRT, to calculate the pressure at point a (P_a). You know the volume V_a, the number of moles n, and the temperature T_H in Kelvin. R is the ideal gas constant, 8.314 J/(mol·K).
Calculate the pressure at point b (P_b) using the ideal gas law again. This time, use the volume V_b, the same number of moles n, and the temperature T_L in Kelvin.
Check the consistency of your calculations by considering the adiabatic processes in the Carnot cycle. For an adiabatic process, P_1V_1^γ = P_2V_2^γ, where γ is the adiabatic index. Use this to cross-verify P_b calculated from the ideal gas law.
Review the calculated pressures at points a and b to ensure they are reasonable given the changes in volume and temperature, and the properties of the diatomic gas.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 efficiencies based on the temperatures of the hot and cold reservoirs.
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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. This law is fundamental in thermodynamics as it allows for the calculation of pressure and volume changes in gases under varying conditions. In the context of the Carnot cycle, it helps determine the pressures at different points in the cycle based on the given volumes and temperatures.
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Diatomic Gas Properties
Diatomic gases, such as nitrogen or oxygen, have specific heat capacities that differ from monatomic gases. The heat capacity ratio, denoted as γ (gamma), is the ratio of the specific heat at constant pressure (C_p) to that at constant volume (C_v). For diatomic gases, γ is typically around 1.4, which influences the calculations of work done and heat transfer during the adiabatic processes of the Carnot cycle.
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