Question 100: Thermodynamics & KTG

Question

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 ___________.

Accepted Answer

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

Thermodynamics & KTG

Choose the correct answer: