Problem 61 - Pearson Physics

Question

A gas does a certain amount of work as it expands by a volume \(\Delta V\) at a pressure \(P\). If the pressure of the gas is increased, but the volume change remains the same, does the work done by the gas increase, decrease, or stay the same?

Step-by-Step Solution

Answer

The work done by the gas increases when pressure increases.

Step 1: Understanding the Relationship

Work done by a gas during expansion or compression at constant pressure is given by the formula: \[ W = P \times \Delta V \]where \( W \) is the work done, \( P \) is the pressure, and \( \Delta V \) is the change in volume.

Step 2: Analyzing Pressure Change

We need to consider how increasing the pressure \( P \) affects the work done. As per the equation \( W = P \times \Delta V \), if \( \Delta V \) remains constant and \( P \) increases, the value of \( W \) will increase since work is directly proportional to pressure.

Step 3: Conclusion Based on Analysis

Based on the analysis, when the pressure increases while the volume change \( \Delta V \) remains the same, the work done \( W \) by the gas increases.

Key Concepts

Work Done by Gas

When a gas expands, it can do work on its surroundings. This concept is fundamental in thermodynamics. The work done is the energy transferred when the gas changes volume.
At constant pressure, the work (\( W \)) done by the gas is calculated using the formula:

• \( W = P \times \Delta V \)

Where:

• \( P \) is the pressure exerted by the gas.
• \( \Delta V \) is the change in volume during expansion or compression.

Work is essentially a measure of energy. It is the area under the pressure-volume curve in a graph, which shows how the gas expands or contracts. This simple relationship implies that, for a constant change in volume, an increase in pressure will increase the work done by the gas. This principle is crucial in engines and refrigeration units, where controlled gas expansion and compression help extract usable work.

Pressure-Volume Relationship

The relationship between pressure and volume is a cornerstone of thermodynamics. It is often visualized via a P-V diagram, showcasing how these two variables interact.
At constant pressure,

• If the volume of a gas increases, the gas does work on the surroundings.
• If the volume decreases, the surroundings do work on the gas.

This relationship is expressed in various gas laws, such as Boyle's and Charles's laws, which help us understand how changes in conditions affect gases. For a constant pressure process, represented as a horizontal line on a P-V diagram, the area under the curve is equal to the work done by the gas. The larger the volume change, the more work is performed, given a set pressure. This interplay between pressure and volume is essential for devices like pistons and compressors.

Constant Pressure Process

A constant pressure process, also known as an isobaric process, is a thermodynamic process where the pressure remains unchanged.
In practical terms:

• The system can expand or contract without altering the pressure level.
• The work done solely depends on the volume change.

Such a process might occur during heating or cooling in an open container, where pressure equilibrates with the atmospheric pressure.
Because pressure is constant, calculations become straightforward:

• The formula for work simplifies as it is directly determined by the product of pressure and volume change.

This simplicity makes the constant pressure process a valuable concept in both theoretical and applied thermodynamics, particularly when considering energy transfer and engine cycles.