Optional Practical: Carnot cycle simulation (GeoGebra)
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 kPacm3K-1 . So if the pressure is in kPa and the volume cm3 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.
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".
The equation for an adiabatic change is PV5/3 = 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/x5/3 = 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.
- 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: