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 76b

Chapter 17, Problem 76b

Assume that in an alternate universe, the laws of physics are very different from ours and that “ideal” gases behave as follows: At constant pressure, the volume varies directly with the 2/3 power of the temperature.

Verified step by step guidance

Identify the relationship given in the problem: At constant pressure, the volume (V) varies directly with the 2/3 power of the temperature (T). This can be expressed mathematically as \( V \propto T^{\frac{2}{3}} \).

Introduce a proportionality constant \( k \) to replace the proportionality symbol, resulting in the equation \( V = k T^{\frac{2}{3}} \). This equation describes the behavior of the gas under the given conditions.

To determine the value of \( k \), you would need specific values for \( V \) and \( T \) at a particular state. If these values are provided, substitute them into the equation \( V = k T^{\frac{2}{3}} \) and solve for \( k \).

If the problem involves comparing two states of the gas (e.g., initial and final states), use the relationship \( \frac{V_1}{V_2} = \left( \frac{T_1}{T_2} \right)^{\frac{2}{3}} \). This equation is derived by dividing the expressions for \( V \) at two different temperatures while keeping pressure constant.

Substitute the known values for \( V_1 \), \( V_2 \), \( T_1 \), and \( T_2 \) into the equation \( \frac{V_1}{V_2} = \left( \frac{T_1}{T_2} \right)^{\frac{2}{3}} \) to solve for the unknown variable, whether it is a volume or a temperature.

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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 typically 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. Understanding this law is crucial for analyzing gas behavior under various conditions.

Volume-Temperature Relationship

In thermodynamics, the relationship between volume and temperature is often described by Charles's Law, which states that at constant pressure, the volume of a gas is directly proportional to its absolute temperature. In this alternate universe scenario, the volume varies with the 2/3 power of temperature, indicating a non-linear relationship that deviates from classical gas behavior, necessitating a new approach to understanding gas dynamics.

Non-Linear Relationships

Non-linear relationships occur when changes in one variable do not produce proportional changes in another variable. In the context of the question, the volume of the gas varies as the temperature raised to the 2/3 power, which means that as temperature increases, volume increases at a decreasing rate. This concept is essential for analyzing systems where traditional linear assumptions do not apply.

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