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Ch. 20 - Second Law of Thermodynamics
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 20 - Second Law of ThermodynamicsProblem 76a
Chapter 20, Problem 76a

1.00 mole of an ideal monatomic gas at STP first undergoes an isothermal expansion so that the volume at b is 2.5 times the volume at a (Fig. 20–25). Next, heat is extracted at a constant volume so that the pressure drops. The gas is then compressed adiabatically back to the original state. Calculate the pressures at b and c.
Graph showing pressure vs. volume for an ideal gas, illustrating isothermal and adiabatic processes between states a, b, and c.

Verified step by step guidance
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Step 1: Understand the problem and identify the key processes. The gas undergoes three thermodynamic processes: (1) isothermal expansion, (2) constant volume cooling, and (3) adiabatic compression. We need to calculate the pressures at points b and c. Start by noting that the gas is ideal and monatomic, and the initial conditions are at STP (Standard Temperature and Pressure). At STP, the pressure \( P_a \) is 1 atm, and the temperature \( T_a \) is 273.15 K.
Step 2: Use the ideal gas law \( PV = nRT \) to relate the pressure and volume during the isothermal expansion. Since the process is isothermal, the temperature remains constant, and the relationship \( P_a V_a = P_b V_b \) holds. Given that \( V_b = 2.5 V_a \), solve for \( P_b \) using \( P_b = \frac{P_a V_a}{V_b} \). Substitute the known values of \( P_a \) and \( V_b \) in terms of \( V_a \).
Step 3: For the constant volume cooling process, the volume remains constant at \( V_b \), and the pressure drops from \( P_b \) to \( P_c \). Use the ideal gas law again, \( P_c = \frac{nRT_c}{V_b} \). Since \( V_b \) is constant, the pressure ratio \( \frac{P_c}{P_b} = \frac{T_c}{T_b} \) can be used. Determine \( T_c \) in terms of \( T_b \) if additional information is provided or leave it in terms of the ratio.
Step 4: For the adiabatic compression back to the original state, use the adiabatic condition \( PV^\gamma = \text{constant} \), where \( \gamma = \frac{C_p}{C_v} \) for a monatomic gas is \( \frac{5}{3} \). Relate \( P_c, V_c \) to \( P_a, V_a \) using \( P_c V_c^\gamma = P_a V_a^\gamma \). Since the gas returns to its original state, \( V_c = V_a \), and solve for \( P_c \).
Step 5: Combine the results from the above steps to calculate \( P_b \) and \( P_c \). Ensure that the units are consistent throughout the calculations (e.g., pressure in atm, volume in liters, temperature in Kelvin). If numerical values are required, substitute them into the derived formulas to find the pressures at points b and c.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 for understanding the behavior of gases under various conditions, such as isothermal and adiabatic processes. In this scenario, it will help calculate the pressures at points b and c after the gas undergoes changes in volume and temperature.
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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 changing the internal energy. For an ideal gas, this implies that the product of pressure and volume remains constant (PV = constant). In this question, the isothermal expansion leads to a change in volume, which directly affects the pressure at point b.
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Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings, leading to changes in pressure and temperature due solely to work done on or by the system. For an ideal gas, the relationship between pressure and volume during adiabatic processes can be described by the equation PV^γ = constant, where γ is the heat capacity ratio. This concept is crucial for determining the pressure at point c after the gas is compressed adiabatically.
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