Ch 19: Work, Heat, and the First Law of Thermodynamics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 19: Work, Heat, and the First Law of Thermodynamics

Problem 57b

Chapter 19, Problem 57b

5.0 g of nitrogen gas at 20°C and an initial pressure of 3.0 atm undergo an isobaric expansion until the volume has tripled. How much heat energy is transferred to the gas to cause this expansion?

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Determine the number of moles of nitrogen gas using the ideal gas law: \( PV = nRT \). Rearrange to solve for \( n \): \( n = \frac{PV}{RT} \). Use the given pressure \( P = 3.0 \; \text{atm} \), the initial temperature \( T = 20^{\circ}C = 293 \; \text{K} \), and the gas constant \( R = 0.0821 \; \text{L·atm/(mol·K)} \). Convert the mass of nitrogen gas (5.0 g) to moles if needed.

Recognize that the process is isobaric (constant pressure). For an isobaric process, the heat energy transferred \( Q \) is given by \( Q = nC_p\Delta T \), where \( C_p \) is the molar specific heat capacity at constant pressure for nitrogen gas (\( C_p = 29.1 \; \text{J/(mol·K)} \) for diatomic gases like nitrogen).

Calculate the final temperature \( T_f \) after the volume has tripled. Since the process is isobaric, the relationship between volume and temperature is \( \frac{V_f}{V_i} = \frac{T_f}{T_i} \). Use \( V_f = 3V_i \) and solve for \( T_f \): \( T_f = 3T_i \).

Determine the temperature change \( \Delta T \) using \( \Delta T = T_f - T_i \). Substitute the initial temperature \( T_i = 293 \; \text{K} \) and the calculated \( T_f \).

Substitute the values of \( n \), \( C_p \), and \( \Delta T \) into the equation \( Q = nC_p\Delta T \) to calculate the heat energy transferred to the gas during the expansion.

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Key Concepts

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

Isobaric Process

An isobaric process is a thermodynamic process in which the pressure remains constant while the volume and temperature of the gas change. In this scenario, as the nitrogen gas expands, it does so at a constant pressure of 3.0 atm, which simplifies the calculations for heat transfer since the pressure does not vary.

First Law of Thermodynamics

The First Law of Thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In the context of the isobaric expansion, the heat energy transferred to the gas can be calculated by considering both the work done during the expansion and the change in internal energy.

Heat Transfer in Gases

In an isobaric process, the heat transfer (Q) can be calculated using the formula Q = nCpΔT, where n is the number of moles, Cp is the specific heat capacity at constant pressure, and ΔT is the change in temperature. This relationship is crucial for determining how much heat energy is required to achieve the desired expansion of the gas.

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Related Practice

Textbook Question

0.10 mol of nitrogen gas follow the two processes shown in FIGURE P19.58. How much heat is required for each?

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Textbook Question

5.0 g of nitrogen gas at 20°C and an initial pressure of 3.0 atm undergo an isobaric expansion until the volume has tripled. What amount of heat energy is transferred from the gas as its pressure decreases?

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Textbook Question

5.0 g of nitrogen gas at 20°C and an initial pressure of 3.0 atm undergo an isobaric expansion until the volume has tripled. What are the gas volume and temperature after the expansion?

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Textbook Question

n moles of an ideal gas at temperature T₁ and volume V₁ expand isothermally until the volume has doubled. In terms of n, T₁, and V₁, what are the heat energy transferred to the gas?

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Textbook Question

5.0 g of nitrogen gas at 20°C and an initial pressure of 3.0 atm undergo an isobaric expansion until the volume has tripled. What is the gas pressure after the decrease?

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Textbook Question

n moles of an ideal gas at temperature T₁ and volume V₁ expand isothermally until the volume has doubled. In terms of n, T₁, and V₁, what are the work done on the gas?

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