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 56a

Chapter 19, Problem 56a

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 is the final temperature?

Verified step by step guidance

Step 1: Recognize that the process described is isothermal, meaning the temperature of the gas remains constant throughout the expansion. Therefore, the final temperature T₂ is equal to the initial temperature T₁.

Step 2: Recall the ideal gas law, which is given by the equation:

P V = n R T

, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature.

Step 3: Since the temperature remains constant (T₁ = T₂), the product of pressure and volume (P × V) must also remain constant during the isothermal process. This is expressed as:

P_{1} V_{1} = P_{2} V_{2}

Step 4: In this problem, the volume doubles, so

V_{2} = 2 V_{1}

. Substituting this into the equation for isothermal processes, we find that the pressure must adjust accordingly to maintain the equality.

Step 5: Conclude that the final temperature T₂ is the same as the initial temperature T₁, as the isothermal condition ensures no change in temperature. Thus, T₂ = T₁.

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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 is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. This law is crucial for understanding the behavior of gases under various conditions.

Isothermal Process

An isothermal process is a thermodynamic process that occurs at a constant temperature. For an ideal gas, this means that any expansion or compression of the gas occurs without a change in temperature, which implies that heat is exchanged with the surroundings to maintain thermal equilibrium. This concept is essential for analyzing the behavior of gases during expansion or compression.

Doubling Volume in Isothermal Expansion

When an ideal gas undergoes isothermal expansion and its volume doubles, the pressure of the gas decreases according to Boyle's Law, which states that pressure is inversely proportional to volume at constant temperature. In this scenario, since the temperature remains constant, the final state of the gas can be analyzed using the Ideal Gas Law to determine how the pressure and other properties change as the volume increases.

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Related Practice

Textbook Question

An ideal-gas process is described by p=cV^1/2, where c is a constant. Find an expression for the work done on the gas in this process as the volume changes from V₁ to V₂.

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Textbook Question

An ideal-gas process is described by p=cV^1/2, where c is a constant. 0.033 mol of gas at an initial temperature of 150°C is compressed, using this process, from 300 cm³ to 200 cm³. How much work is done on 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 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

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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Textbook Question

A 10-cm-diameter cylinder contains argon gas at 10 atm pressure and a temperature of 50°C. A piston can slide in and out of the cylinder. The cylinder's initial length is 20 cm. 2500 J of heat are transferred to the gas, causing the gas to expand at constant pressure. What are the final length of the cylinder?

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 is the final temperature?

Verified video answer for a similar problem:

Key Concepts

Ideal Gas Law

Isothermal Process

Doubling Volume in Isothermal Expansion

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 is the final temperature?