PV Diagrams

PV diagrams are a handy tool used in thermodynamics. They help us visualise the changes that occur to variables like heat, volume, internal energy, entropy, pressure, and temperature during a process. These diagrams show the relationship between these changes and the different stages of the process. You may have also heard them called p-V diagrams. It's important to note that the symbol for pressure is usually written as a small letter p in A-levels, but you may also see it written as a capital letter P. In this explanation, we use the small letter p. Just remember to stay consistent with whatever your teacher or textbook uses.

How to plot a PV diagram

To plot a PV diagram, you need to follow a few simple steps. First, identify how many processes the gas goes through and what they are. Next, look for relationships between the variables. For example, if the gas doubles its pressure or decreases its temperature, this will help you determine the direction of the process in the diagram. Look for keywords like compression, expansion, and no heat transfer to help you understand which way the process goes. You can also calculate any variable you need using gas laws. Once you have all the information, order your data and draw the cycle by linking all the states with the processes you identified earlier. Remember to label each state with its pressure, volume, and temperature.

Calculating work with PV diagrams

A valuable characteristic of PV diagrams and models of thermodynamic processes is their symmetry. One example of this symmetry is an isobaric process (constant pressure) with a volume expansion from state 1 to state 2. You can see this in diagram 1.

Because of the mechanical work definition, when calculating work done (as pressure per change in volume) in PV diagrams, you can easily calculate this as the area below the curve or process (if this is a straight line). For example, in an isobaric process, the work is equal to the pressure multiplied by the volume change.

Mechanical work is the amount of energy that is transferred by a force.

The basics of PV diagrams

When you look at a PV diagram, you'll notice that the y-axis represents pressure, while the x-axis represents volume. Pressure values increase from the bottom to the top of the diagram, while volume values increase from left to right. You'll also see arrows that indicate the direction of the processes. This helps you understand how the gas is changing as it goes through each process in the cycle.

Creating PV diagrams for isothermal processes

If we follow the rules mentioned earlier, we can create PV diagrams for different processes. For instance, Diagram 3 (the top diagram in the set of diagrams below) represents an isothermal expansion process. During this process, the gas undergoes a decrease in pressure from p1 to p2 and an increase in volume from V1 to V2. On the other hand, Diagram 3 (the bottom diagram in the set of diagrams below) shows an isothermal compression process, where the opposite occurs. The volume decreases from V1 to V2, and the pressure increases from p1 to p2.

For isothermals (isothermic process lines), larger temperatures will be further away from the origin. As the diagram below shows, temperature T2 is larger than temperature T1, which is represented by how far they are from their origin.

Creating PV diagrams for adiabatic processes

PV diagrams for adiabatic processes are similar to isothermal processes, but they follow a different equation. Adiabatic processes are marked by the equation:

PV^γ = constant

Where γ is the ratio of the specific heats (Cp/Cv) of the gas. Because of this equation, adiabatic processes form a much steeper curve in the PV diagram, as shown in the image below.

In PV diagrams, the main difference between isothermals and adiabats is their slope. Adiabatic lines are steeper because the gas is not allowed to exchange heat with its surroundings, so its temperature changes more rapidly with changes in volume or pressure. However, the behaviors of expansion and compression are the same as in isothermal processes.

Creating PV diagrams for isometric and isobaric processes

In PV diagrams, constant volume (isometric or isochoric) processes and constant pressure (isobaric) processes are represented by straight lines.

In a process with constant volume, lines will be straight and vertical (see diagram 6 below). The area below the lines in these cases zero, is also. The shows a process from state 1 to state 2 with increased pressure on the left and a process going in the opposite direction from state 1 to state 2 on the right.

[Insert constant volume PV diagram (Diagram 6) here]

In a constant pressure process, lines will be straight and horizontal. In these cases, the area below the lines is regular, and we can calculate the work done by multiplying the pressure by the volume change (see diagram 7 below). The diagram shows a process from state 1 to state 2 with increased volume (below) and a process going in the opposite direction from state 1 to state 2 (above).

In many processes, such as in isobaric ones, work can be negative. This occurs when the gas goes from a larger volume to a smaller one, as expressed in the equation below:

W = -PΔV

If Vf < Vi, then W is negative, indicating that work is done on the system rather than by the system.

In PV diagrams, constant volume processes are represented by straight, vertical lines, and constant pressure processes are represented by straight, horizontal lines.

PV diagram problems and solutions

PV diagrams simplify the work done and make it easier to represent changes in gas. We can make an easy example of this following a thermodynamic cycle.

A piston expands during an isothermal process from state 1 to state 2 with a volume of 0.012m3. During the process, its pressure on the gas decreases from p1 to p2 by half. Later, the piston follows an isometric process (constant volume), which expands its pressure to its initial value. It then goes back to its original state via an isobaric state. Draw and calculate the values of pressure and volume.

Step 1

First, we need to calculate the value for the volume at state 2. An isothermal process follows Boyle’s law, so we use the following equation:

We solve for V2 by replacing p2 with p1/2. This means that the volume V2 at state 2 is now 0.024m3. This value will be to the right of the original V1 value, as you can see in the image below. In the first step, the volume increase means the process goes left to right. The volume increase also decreases the pressure inside the piston from p1 to p2.

Step 2

We know this process follows an isometric relationship where it reaches the same pressure as before. In the second step, the volume stays the same (isometric or isochoric), increasing the pressure inside the piston from p2 to p3, where p3 is equal to p1. This means the variables are now V3=V2 and p3=p1.

Step 3

This means our next state will be at the same horizontal line as state 1 and the same vertical line as state 2. The following process is an isobaric process, which takes the gas inside the piston to the same original state 1. In this case, as we are at the same horizontal line as process 1, connecting the process is the last step.

Understanding Work and Heat in the Example

• Heat is equal to the area below the curves or lines in PV diagrams
• Two lines have an area below the curve in the example, representing the expansion of the piston (state 1 to state 2) and the compression of the piston (state 3 to state 1)
• Work will be equal to the difference in both areas
• Gas is expanding, and this is work done by the gas on the piston, thus giving energy
• Processes 2 to 3: gas increases its pressure in the piston and external energy is introduced into the gas
• Work is not done because the piston does not move, but energy is given to the gas
• Process 3 to 1: gas is compressed without exerting pressure on it and decreases in volume
• This can only be achieved by heat loss, and the gas is giving energy back while mechanical energy is given to the piston to compress it.

PV diagrams and thermodynamic cycles

Many engines or turbine systems can be idealised by following a series of thermodynamic processes. Some of these include the Brayton cycle, Stirling cycle, Carnot cycle, Otto cycle, or Diesel cycle. You can see the PV diagrams of the Carnot cycle below.

• PV diagrams are a valuable tool to visualise thermodynamic relationships in a process
• Heat can be calculated by finding the area below the horizontal curves or lines in PV diagrams
• PV diagrams are used for isothermal, adiabatic, isochoric, and isobar processes
• Adiabatic lines are steeper than isothermal lines in a PV diagram
• The of isothermal lines increases the further they are from the PV origin
• Isochoric lines are vertical lines with no area beneath them, meaning no work is done
• Isobaric lines are horizontal lines and the work done below them equals pressure multiplied by the difference between the initial and final volume.

PV Diagrams

How do you plot a PV diagram?

Here’s how you plot a PV diagram: identify the processes in the cycle, identify useful relationships between the variables, look for keywords that give you useful information, calculate any variable that you need, order your data, and then draw the cycle.

Which PV diagram represents the correct process path?

In PV diagrams, each point shows what state the gas is in. Whenever a gas undergoes a thermodynamic process, its state will change, and this path (or process) is mapped out in the PV diagram. When plotting a PV diagram, there are basic rules to follow so that you plot the correct process path. These are the rules: (1) the y-axis represents the pressure, and the x-axis represents the volume; (2) increasing pressure values follow a down-to-up direction, and increasing volume values follow left to right; and (3) an arrow indicates the direction of the processes.

How do you work out a PV diagram?

When it comes to working out and drawing a basic PV diagram there are specific rules you must follow. These are: (1) the y-axis represents the pressure, and the x-axis represents the volume; (2) increasing pressure values follow a down-to-up direction, and increasing volume values follow left to right; and (3) an arrow indicates the direction of the processes.

What is a PV diagram in physics?

A PV diagram in physics is a diagram used to represent the thermodynamic stages of a process. PV diagrams identify processes such as isobaric, isochoric, isothermal, and adiabatic processes.

What is a PV diagram with an example?

A PV diagram is a diagram used to represent the thermodynamic stages of a process. An example is an isobaric process (constant pressure). In an isobaric process, lines will be straight, horizontal lines.