Question

If a system containing gases expands and pushes back a piston against a constant opposing pressure, what equation describes the work done on the system?

Short answer

The equation that describes the work done on the system is given by W = -P∆V.

Step-by-step solution

Understand the concept of work in thermodynamics

Work done by or on a system in the context of thermodynamics is generally expressed by the equation W = -P∆V, where W represents the work, P is the constant pressure, and ∆V is the change in volume. The negative sign indicates that work is done on the system when volume decreases.

Identify the process

In the given situation, the system is expanding, which means it is doing work against the surroundings. Since the pressure is constant, the equation for work done on the system should include the constant pressure opposing the expansion.

Apply the equation for work done

In this case, the work done on the system is expressed as W = -P∆V. We use a negative sign because work is done by the surroundings on the system when the system expands against the constant opposing pressure.

Key concepts

Work in Thermodynamics

In thermodynamics, work is a form of energy transfer that occurs when a force acts upon an object causing it to move. It is a key concept when studying energy changes within physical systems. Work can either be done on a system or by a system. When gases in a thermodynamic system expand and push against the boundaries of their container, such as a piston, they do work on the surroundings.

Mathematically, this is represented by the equation \( W = -P\triangle V \), where \( W \) stands for the work done on the system, \( P \) is the constant external pressure, and \( \triangle V \) is the change in volume of the system. This equation signifies that if the system expands \( (\triangle V > 0) \), work is done by the system (hence the negative sign in the equation), and if the system is compressed \( (\triangle V < 0) \), work is done on the system.

Constant Pressure

Constant pressure refers to a situation where the pressure exerted on or by a system remains unchanged despite variations in other conditions, such as volume or temperature. In many thermodynamic processes, especially those involving gases, it's common to assume that the pressure stays constant to simplify calculations.

This assumption is especially useful when dealing with gases that follow the ideal gas law, as it allows for the direct application of the work equation \( W = -P\triangle V \). The constant pressure condition is commonly encountered in processes like isobaric (constant-pressure) processes, which are essential for understanding various thermodynamic cycles and engine operations.

Expansion of Gases

The expansion of gases is a thermodynamic process involving an increase in the volume of a gaseous system. This phenomenon is central to various applications, from combustion engines to refrigeration cycles. During expansion, if the pressure is held constant, the gas does work on its surroundings as it pushes against the constraints of its container – for example, a piston in an engine cylinder.

As the gas expands, it must exert force over the distance moved by the piston, which constitutes mechanical work. An understanding of gas expansion is not only crucial for computing work in thermodynamic processes but also for interpreting the behavior of gases under different thermodynamic states and predicting the outcomes of real-world engineering systems.

Change in Volume

Change in volume, denoted as \( \triangle V \), is a crucial characteristic of many thermodynamic processes, particularly when analyzing the work done by or on a system involving gases. A positive change in volume indicates that the system has expanded, while a negative change suggests it has been compressed.

A variety of factors can affect the volume of gases, including temperature, pressure, and the amount of gas itself. According to Boyle's Law, for a fixed amount of gas at a constant temperature, the volume is inversely proportional to the pressure. The concept of volume change is integral to understanding work in thermodynamics as it is directly proportional to the work done in processes occurring at constant pressure, where the work done is directly calculated from the product of pressure and the change in volume.