# Optional Practical: Carnot cycle simulation (GeoGebra)

### Introduction

Plotting PV curves for isothermal and adiabatic changes is also covered in the worksheet PV diagram in geogebra but plotting the closed curves of a Carnot cycle is a bit clumsy so here is an alternative (better) method. In the previous example the curves are plotted by tracing a point whose position is defined by the equations for isothermal and adiabatic changes. In this example the lines will be plotted directly from the equations.

### Setting up the axis

To make the calculations simpler to follow we are going to base the simulation on 1.203 x 10^{-3} moles of ideal gas. This means that the constant nR = 10 kPacm^{3}K^{-1} . So if the pressure is in kPa and the volume cm^{3} then the temperature of the gas in Kelvin is given by T = PV/10. This will give you the same graphs as the ones given as examples in my book :-)

- Open a new file in GeoGebra and zoom in so that the range of the axis is from 0 - 500.
- Use the move graphics tool to shift the origin to the bottom left of the window.

The gas can exist anywhere on this graph. Different points will have different P, V and T. It might be interesting to show the Temperature for different points so we will make a cursor.

- Add a slider for P by clicking the slider tool and setting the name to P and range from 0 to 500.

- Add as slide for V from 0 to 500.
- In the input bar at the bottom of the window type the equation T = P*V/10. This will add the number T to the numbers in the algebra pane. T will change as you vary P and V.

To show these values on the graph we will draw lines representing P and V.

- Write y=P in the input bar, this will add a horizontal line to the graph.
- x=V will add a vertical line.
- Place a point at the intersection of the two lines.

- Display the temperature by using the text tool. Type "Temperature" then select "T" from the objects.

### Isothermal Changes

The equation for an isothermal change is PV = 10T so the equation of the line will be P = 10T/V First you need to make a slider for the temperature of the isothermal.

- Add a slider with name T1 (the temperature of isothermal change 1) from 0 to 2000
- Input I1(x)=10T1/x this will make a function which will be represented by a curve on the graph.
- Add a second slider for T2 and input a function I2(x) = 10T2/x
- You can change the colour of the lines by double clicking and going to "object options".

### Adiabatic changes

The equation for an adiabatic change is PV^{5}^{/3} = constant. The value of the constant will change the position of the curve to see how this happens its best to plot the line.

- Add a slider for the "adiabatic constant" k1 from 0 to 1000000.
- input A1(x) = k1/x^(5/3)
- Add a second slider and function for another adiabatic change.
- Try varying the constants to change the curves.

### Making a Carnot cycle

Arrange the isothermal and adiabatic transformations so that they form a closed loop like the one below, this is a Carnot cycle. You can add points to the intersecting lines to make it more clear.

To tidy this up a bit it is possible to arrange only draw the lines between these points, first you need to write an equation to define the x values of these intersections. For example B is the intersection between lines A1 and I2 you can find which intersect by hovering over the point of intersection. The x value for this point can be found by equating

k1/x^{5}^{/3} = 10T2/x

So x = (k1/10T2)^{3/2}

- Input the equation b = (k1/(10T2))^(3/2). This will define a number that represents the x value of b
- Define numbers for each intersection. It is convenient to give these numbers the same name as the points but in small case.
- To only show the line between the points you need to define a second function with limits defined by the x value of the points. So to draw a the I2 line between point B and C input: Function[I2,b,c].
- Define functions for each line segment then hide the original lines.

You should end up with something like the simulation below.

### Investigation

- Use the movable point to analyse the changes in PV and T around a cycle.
- Using your simulation work through the examples on page 442 -443.
- Solve problems 32 -34 on page 444 and check your answers with the simulation.
- Use your simulation to help solving exercises 35 - 37.

This one has all the energies and efficiency calculated: