An ideal monatomic gas of n moles is taken through a cycle \(W X Y Z W\) consisting of consecutive adiabatic and isobaric quasi-static processes, as shown in the schematic \(V-T\) diagram. The volume of the gas at \(W, X\) and \(Y\) points are, \(64 \mathrm{~cm}^3, 125 \mathrm{~cm}^3\) and \(250 \mathrm{~cm}^3\), respectively. If the absolute temperature of the gas \(T_W\) at the point \(W\) is such that \(n R T_W=1 \mathrm{~J}\) ( R is the universal gas constant), then the amount of heat absorbed (in J ) by the gas along the path \(X Y\) is \(\qquad\)

\(\begin{aligned}
& \mathrm{nRT}_{\mathrm{W}}=\mathrm{P}_{\mathrm{W}} \mathrm{~V}_{\mathrm{W}}=1 \mathrm{~J} \\
& \mathrm{P}_{\mathrm{W}}=\frac{1}{64} \times 10^6 \mathrm{~Pa}
\end{aligned}\)
For WX process
\(\begin{aligned}
& P_X V_X^Y=P_W V_W^y \\
& \Rightarrow P_X=P_W\left(\frac{V_W}{V_X}\right)^y
\end{aligned}\)
amount of heat absorbed in XY process
\(\mathrm{Q} =\mathrm{nCP} \Delta \mathrm{T}=\mathrm{n} \times \frac{5}{2} \mathrm{R} \times\left[\mathrm{T}_{\mathrm{Y}}-\mathrm{T}_{\mathrm{X}}\right] \quad\) \(\left[\text { For monoatomic gas } \mathrm{C}_{\mathrm{P}}=\frac{5 \mathrm{R}}{2}\right] \)
\( \mathrm{Q} =\frac{5}{2}\left[\mathrm{nRT}_{\mathrm{Y}}-\mathrm{nRT} \mathrm{T}_{\mathrm{X}}\right] \)
\( =\frac{5}{2}\left[\mathrm{P}_{\mathrm{Y}} \mathrm{V}_{\mathrm{Y}}-\mathrm{P}_{\mathrm{X}} \mathrm{V}_{\mathrm{X}}\right] \)
\( =\frac{5}{2} \mathrm{P}_{\mathrm{X}}\left[\mathrm{V}_{\mathrm{Y}}-\mathrm{V}_{\mathrm{X}}\right] \quad[\because \quad \mathrm{P}_{\mathrm{X}}=\mathrm{P}_{\mathrm{Y}} ;\) \(\text { Isobaric process }]\)
\( =\frac{5}{2} \times \mathrm{P}_{\mathrm{W}} \times\left[\frac{\mathrm{V}_{\mathrm{W}}}{\mathrm{V}_{\mathrm{X}}}\right]^{\mathrm{y}}\left[\mathrm{V}_{\mathrm{Y}}-\mathrm{V}_{\mathrm{X}}\right]\)
Putting values :
\(Q=1.6\) Joule