Problem 59 - Pearson Physics

Question

How is the work done in a constant-pressure process determined from a \(P V\) graph?

Step-by-Step Solution

Answer

Work is the area under the horizontal line on a PV graph and calculated as \(W = P \Delta V\).

Step 1: Understanding a PV graph

In a PV graph, the x-axis represents the volume \(V\) and the y-axis represents the pressure \(P\) of a system. Each point on the graph represents a possible state of the system with a specific pressure and volume.

Step 2: Identifying a Constant-Pressure Process

A constant-pressure process on a PV graph is depicted as a horizontal line. This is because the pressure \(P\) remains unchanged while the volume \(V\) changes.

Step 3: Interpreting Work Done in Thermodynamics

The work done by or on a system in a constant-pressure process can be represented by the formula \(W = P \Delta V\), where \(\Delta V\) is the change in volume.

Step 4: Visualizing Work on a PV Graph

On a PV graph, the work done in a constant-pressure process is the area under the horizontal line representing the process. This area is a rectangle whose height is the constant pressure \(P\) and base is the change in volume \(\Delta V\).

Step 5: Calculating the Work Done

To find the numerical value of the work done, calculate the area of the rectangle mentioned above using the formula \(W = P \Delta V\). This gives the work done by or on the system during the process.

Key Concepts

PV graph

A PV graph is a crucial tool in thermodynamics, providing a visual representation of the relationship between pressure and volume of a system. On this graph, the x-axis represents the volume ( V ) while the y-axis represents pressure ( P ).

Each point on the graph denotes a specific state of the system with distinct pressure and volume values. By studying these points and the curves or lines that connect them, we can gain insights into the various processes occurring within the system.

The slope and shape of the lines on a PV graph can describe different types of thermodynamic processes, allowing us to determine properties like whether the process is isothermal, adiabatic, or as in this case, isobaric. Understanding a PV graph is essential for analyzing any thermodynamic process involving pressure and volume changes.

constant-pressure process

In thermodynamics, a constant-pressure process, also known as an isobaric process, occurs when the pressure of the system remains constant as the volume changes.

On a PV graph, this process is depicted by a horizontal line, since no change in pressure occurs irrespective of the volume change.

Isobaric processes are common in many real-world scenarios, such as when a gas expands or compresses in a piston that allows the gas to exchange heat with its surroundings while maintaining constant pressure.

The main advantage of a constant-pressure process is the simplicity of the calculations involved, especially when determining work done, due to the unchanging pressure throughout the process.

work done

Work done in thermodynamics is a key concept that relates to the energy transferred from or to a system. In a constant-pressure process on a PV graph, the work done is calculated using the formula: \[ W = P \Delta V \]Where \( P \) is the constant pressure and \( \Delta V \) is the change in volume of the system.

This formula signifies that the work is equal to the area under the horizontal line representing the constant-pressure.

Calculating this area involves multiplying the pressure by the volume change, resulting in a simple rectangular area under the line. This straightforward approach makes analyzing work done in a constant-pressure process easy and intuitive.

thermodynamic process

A thermodynamic process involves the transformation of energy within a system through various means such as heat, work, or internal energy change. These processes are often depicted on PV graphs to monitor how the variables of pressure, volume, and temperature interact.

Each thermodynamic process has distinct characteristics depending on the path or line it follows on the PV graph, indicating whether it's constant-volume (isovolumetric), constant-pressure (isobaric), constant-temperature (isothermal), or adiabatic (no heat exchange).

Analyzing these processes helps in understanding the behavior of gases and fluids in engines, refrigerators, and even in atmospheric dynamics.

By recognizing the type of process, engineers can design more efficient systems and predict how changes in one variable will affect others, thus optimizing performance across various applications.