A given mass of a gas expands from a state $A$ to the state $B$ by three paths $1, 2$ and $3$ as shown in $T-V$ indicator diagram. If $W_1, W_2$ and $W_3$ respectively be the work done by the gas along the three paths, then
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In a thermodynamics process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by $T \Delta X$, where $T$ is temperature of the system and $\Delta X$ is the infinitesimal change in a thermodynamic quantity $X$ of the system. For a mole of monatomic ideal gas
$X=\frac{3}{2} R \ln \left(\frac{T}{T_A}\right)+R \ln \left(\frac{V}{V_A}\right)$. Here, $R$ is gas constant, $V$ is volume of gas, $T_A$ and $V_A$ are constants.
The $List-I$ below gives some quantities involved in a process and $List-II$ gives some possible values of these quantities.
List-$I$ List-$II$ $(I)$ Work done by the system in process $1 \rightarrow 2 \rightarrow 3$ $(P)$ $\frac{1}{3} R T_0 \ln 2$ $(II)$ Change in internal energy in process $1 \rightarrow 2 \rightarrow 3$ $(Q)$ $\frac{1}{3} RT _0$ $(III)$ Heat absorbed by the system in process $1 \rightarrow 2 \rightarrow 3$ $(R)$ $R T _0$ $(IV)$ Heat absorbed by the system in process $1 \rightarrow 2$ $(S)$ $\frac{4}{3} RT _0$ $(T)$ $\frac{1}{3} RT _0(3+\ln 2)$ $(U)$ $\frac{5}{6} RT _0$ If the process carried out on one mole of monatomic ideal gas is as shown in figure in the PV-diagram with $P _0 V _0=\frac{1}{3} RT _0$, the correct match is,
$(1)$$I \rightarrow Q, II \rightarrow R , III \rightarrow P , IV \rightarrow U$
$(2)$ $I \rightarrow S , II \rightarrow R , III \rightarrow Q , IV \rightarrow T$
$(3)$ $I \rightarrow Q , II \rightarrow R , III \rightarrow S , IV \rightarrow U$
$(4)$ $I \rightarrow Q , II \rightarrow S , III \rightarrow R , IV \rightarrow U$
($2$) If the process on one mole of monatomic ideal gas is an shown is as shown in the $TV$-diagram with $P _0 V _0=\frac{1}{3} RT _0$, the correct match is
$(1)$ $I \rightarrow S, II \rightarrow T, III \rightarrow Q , IV \rightarrow U$
$(2)$ $I \rightarrow P , II \rightarrow R, III \rightarrow T , IV \rightarrow S$
$(3)$ $I \rightarrow P, II \rightarrow, III \rightarrow Q, IV \rightarrow T$
$(4)$ $I \rightarrow P, II \rightarrow R, III \rightarrow T, IV \rightarrow P$
Give the answer or quetion $(1)$ and $(2)$
