Exercises

1. The mean pressure, $\bar{p}$ , of a thermally insulated gas varies with volume according to the relation
   $$\bar{p} V^{ \gamma} = K,$$
   where $\gamma>1$ and $K$ are positive constants. Show that the work done by this gas in a quasi-static process in which the state of the gas evolves from an initial macrostate with pressure $\bar{p}_i$ and volume $V_i$ to a final macrostate with pressure $\bar{p}_f$ and volume $V_f$ is
   $$W_{if} = \frac{1}{\gamma-1}\left(\frac{\bar{p}_i}{V_i}-\frac{\bar{p_f}}{V_f}\right).$$

2. Consider the infinitesimal quantity
   $${\mathchar'26\mkern-11mud}F\equiv (x^{ 2}-y) dx + x dy.$$
   Is this an exact differential? If not, find the integrating factor that converts it into an exact differential.

3. A system undergoes a quasi-static process that appears as a closed curve in a diagram of mean pressure, $\bar{p}$ , versus volume, $V$ . Such a process is termed cyclic, because the system ends up in a final macrostate that is identical to its initial macrostate. Show that the work done by the system is given by the area contained within the closed curve in the $\bar{p}$ -$V$ plane.