I have been studying isothermal expansion process, the heat change particulary for a real gas. On the internet is many solved problems about this, but it's always about changing volume from $v_i$ to $v_f$. But what if I want to study how the presure changes? Because volume and pressure are tied, but for real gas I cannot use the law stating $\;$$p_iv_i=$ const. explicitly.
The state equations for one mole of the real gas are:
$p = \frac {RT}{v-b} - \frac {a}{v^2}$ $\;$ and $\;$ $u = cRT - \frac{a}{v}$, $\;$ where $a$ and $b$ are constants.
Then for the volume change we have:
$\int_{v_i}^{v_f}\frac {RT}{v-b} - \frac {a}{v^2}dV = [RTln(v-b)+a/v]^{v_f}_{v_i}$