Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Problem 76a

Chapter 17, Problem 76a

Assume that in an alternate universe, the laws of physics are very different from ours and that “ideal” gases behave as follows: At constant temperature, pressure is inversely proportional to the square of the volume.

Verified step by step guidance

Identify the relationship given in the problem: At constant temperature, pressure (P) is inversely proportional to the square of the volume (V). This can be expressed mathematically as \( P \propto \frac{1}{V^2} \), or equivalently \( P = k \cdot \frac{1}{V^2} \), where \( k \) is a proportionality constant.

Rearrange the equation to express the proportionality constant \( k \): \( k = P \cdot V^2 \). This shows that the product of \( P \) and \( V^2 \) remains constant for a given temperature.

To analyze a process involving this gas, such as a change in volume or pressure, use the relationship \( P_1 \cdot V_1^2 = P_2 \cdot V_2^2 \), where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume.

If additional information is provided, such as specific values for \( P_1 \), \( V_1 \), and either \( P_2 \) or \( V_2 \), substitute these values into the equation \( P_1 \cdot V_1^2 = P_2 \cdot V_2^2 \) to solve for the unknown variable.

Finally, ensure that the units of pressure and volume are consistent throughout the calculation, and interpret the result in the context of the alternate universe's gas behavior.

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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 describes the behavior of ideal gases through the equation PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. In typical scenarios, it assumes that pressure is inversely proportional to volume at constant temperature, but this question introduces a unique variation.

Inverse Proportionality

Inverse proportionality refers to a relationship where one variable increases as the other decreases. In this context, the pressure of the gas is inversely proportional to the square of the volume, meaning that if the volume increases, the pressure decreases at a rate proportional to the square of that volume, which is a deviation from the standard ideal gas behavior.

Alternate Universe Physics

The concept of alternate universe physics explores hypothetical scenarios where the fundamental laws of physics differ from our own. This question posits a universe where the behavior of gases diverges from the established principles, prompting a reevaluation of how we understand gas behavior under different conditions, particularly in terms of pressure and volume relationships.

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