Figure E19.819.8 shows a pVpV-diagram for an ideal gas in which its absolute temperature at bb is one-fourth of its absolute temperature at aa. Did heat enter or leave the gas from aa to bb? How do you know?

Ch 19: The First Law of Thermodynamics

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 19: The First Law of Thermodynamics

Problem 8d

Chapter 19, Problem 8d

Figure E $19.8$ shows a $p V$ -diagram for an ideal gas in which its absolute temperature at $b$ is one-fourth of its absolute temperature at $a$ . Did heat enter or leave the gas from $a$ to $b$ ? How do you know?

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Identify the process: The pV-diagram shows a horizontal line from point a to point b, indicating an isobaric process (constant pressure).

Use the ideal gas law: For an ideal gas, the relationship between pressure (P), volume (V), and temperature (T) is given by the equation $ PV = nRT $, where n is the number of moles and R is the ideal gas constant.

Relate temperatures at points a and b: According to the problem, the temperature at b ($ T_b $) is one-fourth of the temperature at a ($ T_a $). Therefore, $ T_b = \frac{1}{4} T_a $.

Analyze the change in volume: Since the process is isobaric and the temperature decreases, the volume must also decrease to maintain the relationship $ PV = nRT $. This is consistent with the diagram showing a decrease in volume from a to b.

Determine heat transfer: In an isobaric process, the change in internal energy ($ \Delta U $) is related to the heat added to the system (Q) and the work done by the system (W) by the first law of thermodynamics: $ \Delta U = Q - W $. Since the temperature decreases, $ \Delta U $ is negative, indicating that heat must have left the system (Q is negative).

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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 (p), volume (V), and temperature (T) of an ideal gas through the equation PV = nRT, where n is the number of moles and R is the ideal gas constant. This law helps predict how a gas will behave under varying conditions of pressure and temperature, making it essential for understanding gas behavior in thermodynamic processes.

Thermodynamic Processes

Thermodynamic processes describe how a system changes from one state to another, often involving heat transfer. In this case, the transition from point a to point b on the pV diagram indicates a change in volume while pressure remains constant, which is characteristic of an isobaric process. Understanding these processes is crucial for determining whether heat is absorbed or released.

Heat Transfer

Heat transfer refers to the movement of thermal energy from one object or system to another due to a temperature difference. In the context of the question, analyzing the change in temperature from point a to point b helps determine if heat entered or left the gas. If the temperature decreases, it indicates that heat has left the system, while an increase would suggest heat has entered.

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Textbook Question

Five moles of an ideal monatomic gas with an initial temperature of $127$ °C expand and, in the process, absorb $1500$ J of heat and do $2100$ J of work. What is the final temperature of the gas?

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Textbook Question

A gas undergoes two processes. In the first, the volume remains constant at $0.200$ m³ and the pressure increases from $2.00 \times 10^{5}$ Pa to $5.00 \times 10^{5}$ Pa. The second process is a compression to a volume of $0.120$ m³ at a constant pressure of $5.00 \times 10^{5}$ Pa. Find the total work done by the gas during both processes.

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Textbook Question

The process $a b c$ shown in the $p V$ -diagram in Fig. E $19.11$ involves $0.0175$ mol of an ideal gas. What was the lowest temperature the gas reached in this process? Where did it occur?

In Fig. $19.7$ a, consider the closed loop $1 right arrow 3 right arrow 2 right arrow 4 right arrow 1$ . This is a cyclic process in which the initial and final states are the same. Find the total work done by the system in this cyclic process, and show that it is equal to the area enclosed by the loop.

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Textbook Question

A gas undergoes two processes. In the first, the volume remains constant at $0.200$ m³ and the pressure increases from $2.00$\times$10^5$ Pa to $5.00$\times$10^5$ Pa. The second process is a compression to a volume of $0.120$ m³ at a constant pressure of $5.00$\times$10^5$ Pa. In a $p V$ -diagram, show both processes.

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