PV Diagrams

How to plot a PV diagram

Before we get into the details, let’s look at how to plot a PV diagram (the following information will become more apparent as you read through this explanation!). To begin your plot, you will need to find the solutions and relationships between the thermodynamic cycle. Here is a helpful list of how to plot your PV diagrams:

1. Identify the processes in the cycle. How many processes does the gas go through? Which ones are they?
2. Identify useful relationships between the variables. Look for relationships such as “the gas doubles its pressure”, “the gas decreases its temperature”, or “the gas maintains its volume”. This will give you helpful information on the direction of the process in the PV diagram. An example of this is when the cycle or process increases its volume – this means the arrow goes from left to right.
3. Look for keywords, such as compression, expansion, no heat transfer, etc. These will tell you in which direction your process goes. An example is when you read “a gas compresses at constant temperature” – this is an isothermal line that goes from a lower pressure to a higher pressure (bottom to top).
4. Calculate any variable that you need. In the states where you don’t have more information, you can use the gases laws to calculate variables you don’t know. The remaining variables can give you more information about the process and its direction.
5. Order your data and draw the cycle. Once you’ve identified all your processes and have the information on each variable, order them by state. For example, state 1 (p₁,V₁,T₁), state 2 (p₂,V₂,T₂), and so on. Finally, draw the lines that link all states using the processes you identified in step 1.

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.

The basics of PV diagrams

When it comes to drawing basic PV diagrams, there are specific rules you must follow:

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.
3. An arrow indicates the direction of the processes.

Creating PV diagrams for isothermal processes

Using the rules above, we can create diagrams for an isothermal process of expansion and compression.

• Diagram 3 (the top diagram in the set of diagrams below) shows isothermal expansion. In this case, the expansion comes with a decrease in pressure from p₁ to p₂ and a volume increase from V₁ to V₂.
• Diagram 3 (the bottom diagram in the set of diagrams below) shows isothermal compression, and the inverse process 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. In this case, adiabatic processes follow this equation:

\[p_1 V_1 ^{\gamma} = p_2 V_2^\gamma\]

Because of this equation, the processes form a much steeper curve (see the image below). In PV diagrams, the main difference between isothermals and adiabats (lines in adiabatic processes) is their steeper slope. In this process, expansion and compression follow the same behaviours as isothermals.

Creating PV diagrams for isometric and isobaric processes

Constant volume (isometric or isochoric) processes and constant pressure (isobaric) processes follow a straight line in PV diagrams. You can see these processes below.

Constant volume (isometric or isochoric) processes

In a process with constant volume (isometric or isochoric), lines will be straight, vertical lines (see diagram 6). There is no area below the lines in these cases, and the work is zero. The diagram 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.

Constant pressure (isobaric) processes

In a constant pressure (isobaric) process, lines will be straight, horizontal lines. In these cases, the area below the lines is regular, and we can calculate the work by multiplying the pressure by the volume change. In diagram 7, you can see 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).

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.

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.

In many problems that model combustion engines, turbomachinery, or even biological processes, it is customary to use thermal engines and thermodynamic diagrams and processes to simplify the represented objects.