Question

Figure shows the variation of internal energy "U" with the density "$\rho$" of one mole of ideal diatomic gas. Process BA is a part of a rectangular hyperbola. If the work done by gas in the process B $\to$ A is W joules. Find W ?

Answer 1

As given,
$U$-$\rho$ plot is a rectangular hyperbola.
$\therefore U\rho =constant$
$\therefore n{C}_{v}T\rho =constant$
$\therefore n\left(\frac{R}{T}\gamma -1\right)\rho =constant$
$\therefore n\frac{5}{2}RT\frac{M}{V}=constant$ (where $\rho =\frac{M}{V}$)
$\therefore \frac{5}{2}\left(nRT\right)\frac{M}{V}=\frac{5}{2}PV\frac{M}{V}=constant$ where we have substituted $PV=nRT$ from ideal gas equation
$\therefore \frac{5}{2}PM=constant$
$\therefore P=constant$
From the plot $\Delta U=7-2=5=\frac{5}{2}nR\Delta T={\int }_{V_1}^{V_2}PdV$
$\therefore W=\frac{5\times 2}{5}=2J$ since $n=1$