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 $\frac{17}{6} P_1 V_1$

Graph $A-B$ is an adiabatic curve. Choose the correct statement

An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding $P$ $V$ diagrams in column $3$ of the table. Consider only the path from state $1$ to $2 . W$ denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here $\gamma$ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is $n$.

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($1$) Which of the following options is the only correct representation of a process in which $\Delta U=\Delta Q-P \Delta V$ ?

$[A] (II) (iv) (R)$    $[B] (II) (iii) (P)$    $[C] (II) (iii) (S)$   $[D] (III) (iii) (P)$

($2$)  Which one of the following options is the correct combination?

$[A] (III) (ii) (S)$    $[B] (II) (iv) (R)$   $[C] (II) (iv) (P)$   $[D] (IV) (ii) (S)$

($3$) Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in an ideal gas?

$[A] (III) (iv) (R)$  $[B] (I) (ii)$ $(\mathrm{Q})$   $[C] (IV) (ii) (R)$    $[D] (I) (iv) (Q)$

The temperature of a hypothetical gas increases to $\sqrt 2 $ times when compressed adiabatically to half the volume. Its equation can be written as

Consider one mole of helium gas enclosed in a container at initial pressure $P_1$ and volume $V_1$. It expands isothermally to volume $4 V_1$. After this, the gas expands adiabatically and its volume becomes $32 V_1$. The work done by the gas during isothermal and adiabatic expansion processes are $W_{\text {iso }}$ and $W_{\text {adia, }}$ respectively. If the ratio $\frac{W_{\text {iso }}}{W_{\text {adia }}}=f \ln 2$, then $f$ is. . . . . . . .

Why does a gas get heated on compression ?