# Thermodynamic processes

I’d like to show how the “Super LOL Diagram” can be used to help solve problems for thermodynamic processes. As an example, imagine the problem tells us that a 1 cubic meter sample of gas at a pressure of 101 kPa, is compressed isothermally at a temperature of 300 kelvin to half its original volume.

The first thing we can do is write the given information, which is done in red ink. Next we can try to figure out the rest of the variables for the initial and final states. From the Ideal Gas Equation

$PV = nRT$

we can calculate the number of moles at the initial state (blue ink). If we assume it’s a sealed container, then the moles would be the same in the final state (pink number). We were told the process is isothermal, so the final temperature must be the same as the initial temp (orange ink). We can determine the pressure at the second state by again using the Ideal Gas Equation.

The next step is to draw the approximate size of each container below the state variables. Since the initial volume is twice that of the final, we make the container on the left about two times bigger than the one on the right (grey ink).

Next up is the energy present at each state. To find these values, we can use the other equation developed during the computer simulation/paradigm lab:

$U = \frac{3}{2}nRT$

Since this only depends on the number of moles (a constant) and temperature (not constant) the relative size of the energy bars will be proportional only the temperatures  of the two states (purple ink).

The next step is drawing the $PV$ diagram (brown ink). This shouldn’t be anything new for people familiar with thermodynamics or from most textbooks. For the sake of brevity, I’m not going to explain that. After we make the $PV$ graph we can now determine the working that occurred between the system and the surrounding by finding the area under the curve (brown shading). To find the area, we can use calculus or the equation given in most textbooks for the work of an isothermal compression:

$W = -nRT ln \frac{V_1}{V_2}$

The last step is to complete the energy flow diagram (the “O” of the “LOL Diagram” at the bottom). By having the grey pistons drawn, we have a clear indication that the gas was compressed, which would mean that energy flowed into the system from an outside force moving the piston inwards (black ink). To determine the amount of heating, we can use the First Law of Thermodynamics equation:

$\Delta U = Q + W$

We know the initial and final energy (purple ink), we know the work done (black ink), so we just have to calculate the heating by subtraction (dark green ink).