Question

One mole of monatomic ideal gas undergoes the thermodynamic process $a \rightarrow b$ as shown in the $P-V$ diagram. During this process, the maximum temperature will occur when gas volume is $\frac{m}{n} V_{0} .$ Find $(m+n)$.

Answer 1

Equation of $P V$ graph:
$P=-\left(\frac{P_{0}}{V_{0}}\right) V+3 P_{0}=m V+C$
$\left(m=-P_{0} / V_{0}\right.$ and $C=3 P_{0}$ )
Putting this in ideal gas equation $P V=n R T$, we get
$m V^{2}+C V=R T \quad(\because n=1)$
Differentiating above equation w.r.t. $V$, we get
$\Rightarrow 2 m V+C=R \frac{d T}{d V}$
For $T_{\max }, \frac{d T}{d V}=0 $
$\Rightarrow V=-\frac{C}{2 m}=\frac{3 V_{0}}{2}$