An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure). Initially the gas is at temperature $T _1$, pressure $P_1$ and volume $V_1$ and the spring is in its relaxed state. The gas is then heated very slowly to temperature $T_2$, pressure $P _2$ and volume $V _2$. During this process the piston moves out by a distance $x$. Ignoring the friction between the piston and the cylinder, the correct statement$(s)$ is(are)
$(A)$ If $V_2=2 V_1$ and $T_2=3 T_1$, then the energy stored in the spring is $\frac{1}{4} P_1 V_1$
$(B)$ If $V_2=2 V_1$ and $T_2=3 T_1$, then the change in internal energy is $3 P_1 V_1$
$(C)$ If $V_2=3 V_1$ and $T_2=4 T_1$, then the work done by the gas is $\frac{7}{3} P_1 V_1$
$(D)$ If $V_2=3 V_1$ and $T_2=4 T_1$, then the heat supplied to the gas is $\frac{17}{6} P_1 V_1$
IIT 2015, Medium
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$P \text { (pressure of gas) }=P_1+\frac{k x}{A}$
$W =\int PdV = P _1\left( V _2- V _1\right)+\frac{ kx ^2}{2}= P _1\left( V _2- V _1\right)+\frac{\left( P _2- P _1\right)\left( V _2- V _1\right)}{2}$
$\Delta U = nC _{ V } \Delta T =\frac{3}{2}\left( P _2 V _2- P _1 V _1\right)$
$Q = W +\Delta U$
$\text { Case I: } \Delta U =3 P _1 V _1, W =\frac{5 P _1 V _1}{4}, Q =\frac{17 P _1 V _1}{4}, U _{\text {spring }}=\frac{ P _1 V _1}{4} $
$\text { Case II: } \Delta U =\frac{9 P _1 V _1}{2}, W =\frac{7 P _1 V _1}{3}, Q =\frac{41 P _1 V _1}{6}, U _{\text {spring }}=\frac{ P _1 V _1}{3}$
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