# What sort of thermodynamic process is this?

I have to calculate the work done in this thermodynamic cycle: LINK. But the bottom bit of the cycle is something I've never seen before. I guess the curve in $pV$ space is given by

$$\left( V- \frac{V_2 -V_1}{2} \right) ^2 + \left( p-p_1 \right) ^2 = \left( \frac{V_2 -V_1}{2} \right) ^2$$

Which is physically meaningless because of units...

• Seems to me like an exercise on integration. This is no thermodynamic proces for as far as I know.
– Nick
Jun 23 '13 at 21:53
• Suitable constants can probably be introduced to the equation to make it dimensionally self-consistent. Jun 23 '13 at 22:10
• Your equation isn't completely correct, since the curve is elliptic, not circular. That explains the dimensional problem. Jun 23 '13 at 22:11
• The correct equation (shifted to center on the origin) would be $V^2/V_0^2 + p^2/p_0^2 = 1$ where $V_0 = (V_2-V_1)/2$ and $p_0 = p_2-p_1$. But I think @Nick is right, this is most likely intended as an exercise mainly on integration. To give an example for which it is a little bit harder to find the work. Jun 23 '13 at 22:22

The work done in a reversible cyclic process equals $-\int PdV$ which is also equal to the negative area under the graph of that cyclic process.
From the looks of it, process $D \to A$ appears to be a semi-ellipse, whose area equals $1/2\pi ab$ , where a and b are lengths of semi-axes ($p_2-p_1$ and $(v_2-v_1)/2$).