$List I$ describes thermodynamic processes in four different systems. $List II$ gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.
| $List-I$ | $List-II$ |
| ($I$) $10^{-3} kg$ of water at $100^{\circ} C$ is converted to steam at the same temperature, at a pressure of $10^5 Pa$. The volume of the system changes from $10^{-6} m ^3$ to $10^{-3} m ^3$ in the process. Latent heat of water $=2250 kJ / kg$. | ($P$) $2 kJ$ |
| ($II$) $0.2$ moles of a rigid diatomic ideal gas with volume $V$ at temperature $500 K$ undergoes an isobaric expansion to volume $3 V$. Assume $R=8.0 Jmol ^1 K^{-1}$. | ($Q$) $7 kJ$ |
| ($III$) On mole of a monatomic ideal gas is compressed adiabatically from volume $V=\frac{1}{3} m^3$ and pressure $2 kPa$ to volume $\frac{v}{8}$ | ($R$) $4 kJ$ |
| ($IV$) Three moles of a diatomic ideal gas whose molecules can vibrate, is given $9 kJ$ of heat and undergoes isobaric expansion. | ($S$) $5 kJ$ |
| ($T$) $3 kJ$ |
Which one of the following options is correct?
IIT 2022, Advanced
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$(I)\Delta U =\Delta Q -\Delta W$
$=\left\{\left(10^{-3} \times 2250\right)-\frac{10^5\left(10^{-1}-10^{-4}\right)}{10^5}\right\} VJ$
$=(2.25-0.0999) VJ$
$=(2.1501) kJ$
$(II)$
$\Delta U = nC \Delta T$
$=\frac{5}{2} \pi R_{ T }$
$=\frac{5}{2} \cdot(0.2)(8)(1500-500) J$
$=4 kJ$
$(III)$
$P_1 V_2=P_2 V_2^2$
$\Rightarrow 2\left(\frac{1}{3}\right)^{s, 1}=P_2\left(\frac{1}{24}\right)^s$
$\Rightarrow P_2=64 kPa$
$\Delta U=n C_2 \Delta T=\frac{3}{2} \cdot\left(P_2 V_2-P_1 V_1\right)$
$=\frac{3}{2}\left(64 \times \frac{1}{24}-2 \times \frac{1}{3}\right) kJ$
$=3 VJ$
$(IV)\Delta U = HC C _{ V } \Delta T$
$= n \cdot \frac{7}{2} RAT$
$=\frac{7}{9} \Delta Q$
$=7 kJ$
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