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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).
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:
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.
When it comes to drawing basic PV diagrams, there are specific rules you must follow:
Using the rules above, we can create diagrams for an isothermal process of expansion and compression.
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.
PV diagrams for adiabatic processes are similar. In this case, adiabatic processes follow this equation:
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.
Constant volume (isometric or isochoric) processes and constant pressure (isobaric) processes follow a straight line in PV diagrams. You can see these processes below.
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.
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Which variables change in a thermodynamic process?
Temperature, volume, pressure, entropy, work, and internal energy.
Is there a way to visualise changes in thermodynamic variables?
Yes, you can visualise changes in thermodynamic variables with PV diagrams.
What represents the area below the curve in a PV diagram?
The work done.
What represents the axes in a PV diagram?
The pressure and the volume.
In PV diagrams, how do you know in which direction a process goes?
According to the arrow.
What is the difference between an isothermal line and an adiabatic line?
The adiabatic line is steeper.
Is it true that for larger temperatures, isothermal lines are closer to the origin?
False, isothermals with larger temperatures are further from the origin.
If the pressure stays the same in a process from state 1 to state 2, then the line that represents this in a PV diagram is _____?
A horizontal line.
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