Problem 38 - Advanced Engineering Mathematics

Question

The pressure \(P\) exerted by an enclosed ideal gas is given by \(P=k(T / V)\), where \(k\) is a constant, \(T\) is temperature, and \(V\) is volume. Find: (a) the rate of change of \(P\) with respect to \(V\), (b) the rate of change of \(V\) with respect to \(T\), and (c) the rate of change of \(T\) with respect to \(P\).

Step-by-Step Solution

Answer

(a) \(-\frac{kT}{V^2}\), (b) \(\frac{k}{P}\), (c) \(\frac{V}{k}\).

Step 1: Identify Given Formula

We are given the formula: \( P = k \left( \frac{T}{V} \right) \), which relates the pressure \(P\), temperature \(T\), and volume \(V\) with a constant \(k\).

Step 2: Differentiating P with Respect to V

To find \( \frac{dP}{dV} \), differentiate \( P = k \frac{T}{V} \) with respect to \( V \). Using the constant \(kT\) and power rule: \( \frac{dP}{dV} = -kT \frac{1}{V^2} \).

Step 3: Differentiating V with Respect to T

Write \( V \) as a function of \( T \): \( V = k \frac{T}{P} \). Differentiate with respect to \( T \) to find \( \frac{dV}{dT} \): \( \frac{dV}{dT} = \frac{k}{P} \).

Step 4: Differentiating T with Respect to P

Rearrange the original equation to express \( T \): \( T = \frac{PV}{k} \). Differentiate this with respect to \( P \) to find \( \frac{dT}{dP} \): \( \frac{dT}{dP} = \frac{V}{k} \).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental concept in chemistry and physics that relates the pressure, volume, and temperature of an ideal gas. An ideal gas is a hypothetical gas that perfectly follows this law, allowing scientists to predict how changes in one variable will affect the others. The Ideal Gas Law equation is given by\[ PV = nRT \]where:

• \( P \) represents the pressure of the gas,
• \( V \) is the volume the gas occupies,
• \( n \) is the number of moles of the gas,
• \( R \) is the universal gas constant,
• and \( T \) is the temperature in Kelvin.

The version of the formula provided in the exercise simplifies this relationship by using a constant \( k \), and explains how pressure changes as a function of temperature and volume.

Rate of Change

In calculus, the rate of change is a measure of how one quantity changes in relation to another. When applied to the Ideal Gas Law, understanding the rate of change helps us determine how pressure, volume, or temperature will change when one of the other variables is adjusted. For example:

• The rate of change of pressure \( P \) with respect to volume \( V \), calculated as \( \frac{dP}{dV} \), can show how pressure decreases as volume increases, given by \( -kT \frac{1}{V^2} \).
• The rate of change of volume \( V \) with temperature \( T \), calculated as \( \frac{dV}{dT} \), indicates how the gas expands or contracts when the temperature changes, given by \( \frac{k}{P} \).
• The rate of change of temperature \( T \) with pressure \( P \), calculated as \( \frac{dT}{dP} \), represents how temperature changes when pressure is altered, expressed as \( \frac{V}{k} \).

These derivatives are crucial for scientific predictions in thermodynamics and other applications, providing concrete numbers to what can otherwise be abstract relationships.

Enclosed Gas Systems

In an enclosed gas system, a gas is sealed within a container that does not allow for the exchange of gas with the surroundings. The specific conditions of the gas, including its temperature, volume, and pressure, are governed by the Ideal Gas Law. Enclosed systems are crucial for laboratory experiments where strict control over gas variables is needed.
The exercise you are studying considers how these variables interrelate within such a confined space. By examining an enclosed system, predictions about gas behavior are accurate and reliable due to the lack of outside influence. The understanding of these systems is essential in designing engines, refrigeration units, and other devices that depend on the confinement of gases for efficient operation. By differentiating among the factors of an enclosed gas system, scientists and engineers can devise strategies for energy efficiency and system stability.