Example 2.77. Calculate the work $A$ performed during the isothermal ( $T=$ const) expansion of an ideal gas (pressure $P$ is related to the absolute temperature $T$ and volume $V$, mass $m$ and molar mass $\mu$ and the gas constant $R$, by the law $\left.P V=\frac{m}{\mu} R T\right)$.
Solution. $d A=P(V) d V$. Hence,

$$
A=\int_{V_{1}}^{V_{2}} P(V) d V=\frac{m}{\mu} R T \int_{V_{1}}^{V_{2}} \frac{d V}{V}=\frac{m}{\mu} R T \ln \left(\frac{V_{2}}{V_{1}}\right)
$$