# What's the difference in a PV diagram that is curved versus one that is straight?

So what would the difference be between the graph above versus one that has the same inital and final points but the path is curved. I'm sure it has something to do with temperature, so does it mean temperature is constant? Or is there something else going on?

In an isotherm, which is a path through the $$PV$$ diagram where temperature $$T$$ is held constant, curved lines are formed. $$P$$ as a function of $$V$$ is given by the ideal gas law. $$PV = NkT \Rightarrow \boxed{P = \frac{NkT}{V}}$$ So, the isothermal curves are similar to the shape of $$y = 1/x$$ in the first quadrant.

However, other sorts of curves exist, such as adiabatic curves, and, also, really any other sort of curve you'd like. Curves through a $$PV$$ diagram don't need to be isotherms, though they certainly could be.

A straight-line path, like the one in your picture, is very obviously not an isotherm. Thus, going along such a path requires a changing temperature.

You have $$PV=NkT$$, and along the path, $$P=-aV+b$$, where I assume you know how to compute $$a$$ and $$b$$. Replacing the second into the first results in:

$$T=\frac{V(b-aV)}{Nk}$$

which means that, along the path, the temperature varies quadratically with the change in volume.

The only difference between a straight PV diagram versus a curved PV diagram is the work done in both cases (provided the final and initial points are same for both diagrams).

As you might know, the work done in such a case is the area under the curve of the PV diagram. So the work done will be more or less depending on the shape of the PV curve.

The temperature may vary in the process or it may remain constant. As @TrevorKafka has properly shown, an isotherm is the special PV curve where the temperature is maintained constant.

But to summarize, the important difference between the two cases is the work done. Hope this helps!