In thermodynamics, changes occur to variables like heat, volume, internal energy, entropy, pressure, and temperature. We can visualise these changes more easily by making diagrams, which show the relationship between these changes and the thermodynamic stages of a process. These unique diagrams are known as PV diagrams (pressure-volume diagrams).
You might also see PV diagrams written as p-V diagrams. Also, in A-levels, the symbol for pressure is typically p (small letter). However, you may also see the symbol P (capital letter). In this explanation, we have used p, but in many of our other explanations, P is used. Both are acceptable, but you must remain consistent in your choice (and follow what your textbook or teacher uses).
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:
- Identify the processes in the cycle. How many processes does the gas go through? Which ones are they?
- 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.
- 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).
- 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.
- 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.
Diagram 1. An advantage of PV diagrams is their symmetry. Manuel R. Camacho – StudySmarter Originals
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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.
Diagram 2. The
work done in PV diagrams is the area below the curve or straight line. Manuel R. Camacho – StudySmarter Originals
Mechanical work is the amount of energy that is transferred by a force.
The basics of PV diagrams
When it comes to drawing basic PV diagrams, there are specific rules you must follow:
- The y-axis represents the pressure, and the x-axis represents the volume.
- Increasing pressure values follow a down-to-up direction, and increasing volume values follow left to right.
- 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.
Diagram 3. Isothermal expansion is shown in the first part of the diagram, and isothermal compression is shown in the second part. Manuel R. Camacho – StudySmarter Originals
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.
Diagram 4. T2 is larger than T1. Manuel R. Camacho – StudySmarter Originals
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.
Diagram 5. In PV diagrams, the main difference between isothermals and adiabats is their steeper slope. Manuel R. Camacho – StudySmarter Originals
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).
Diagram 6. In a process with constant volume, lines are vertical. There is no area below the lines, and the work is zero. Manuel R. Camacho – StudySmarter Originals
Diagram 7. In a process with constant pressure, lines are horizontal. The area below the lines is regular, and work can be calculated by multiplying the pressure by the volume change. Manuel R. Camacho – StudySmarter Originals
In many processes (such as in isobaric ones), work can be negative. You can see this when the gas goes from a larger volume to a smaller one. This is expressed in the equation below. If Vf < Vi, then W is negative.
\[W = p(V_f - V_i)\]
- Constant volume = straight, vertical lines in PV diagram
- Constant pressure = straight, horizontal lines in PV diagram
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:
\[p_1V_1 = p_2V_2\]
We solve for V2 by replacing p2 with p1/2.
\[V_2 = \frac{p_1V_1}{\frac{p_1}{2}} = 2V_1\]
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.
Diagram 8. The increase of volume means the process goes from left to right. Manuel R. Camacho – StudySmarter Originals
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.
\(V_3 = 0.024 m^3\)
\(p_3 = p_1 \text{ and } p_3 > p_2\)
Figure 9. The volume stays the same (isometric or isochoric). Manuel R. Camacho – StudySmarter Originals
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.
Figure 10. The gas inside the piston goes back to its initial state through compression at constant pressure. Manuel R. Camacho – StudySmarter Originals
You can also find out how work and heat behave in the example above.
The heat is equal to the area below the curves or lines. In the example, only two lines have an area below the curve, and these represent the expansion of the piston (state 1 to state 2) and the compression of the piston (state 3 to state 1). The work will be equal to the difference in both areas.If we look at the heat, we can assume the gas is expanding, and this is work done by the gas on the piston. Thus, the gas is giving energy.
In processes 2 to 3, the gas increases its pressure in the piston. The only way this can happen is by introducing external energy into the gas. The molecules start moving rapidly, and the gas wants to expand, but it can’t. In this case, work is not done because the piston does not move (but we are giving energy to the gas).
In the process 3 to 1, we compress the gas without exerting pressure on it, and it decreases in volume. This can only be achieved by heat loss. Therefore, the gas is giving energy back, and at the same time, we give mechanical energy 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.
Diagram 11. Carnot cycle showing its two isobars and two isothermal lines. Manuel R. Camacho – StudySmarter Originals
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.
PV Diagrams - Key takeaways
- PV diagrams are a valuable tool to help us visualise thermodynamic relationships in a thermodynamic process.
- PV diagrams offer a simple way to calculate the heat by calculating the area below the horizontal curves or lines.
- PV diagrams are used for isothermal, adiabatic, isochoric, and isobaric processes.
- Adiabatic lines will be steeper than isothermal lines in a PV diagram.
- The temperature of the isothermal lines will be greater the further they are from the PV origin.
- Isochoric lines are also known as isometric or constant volume lines. They are vertical lines and have no area beneath them, meaning no work is done.
- Isobaric lines, also known as constant pressure lines, are horizontal lines. The work done below them equals the pressure multiplied by the difference between the initial and the final volume.
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