Given below are two statements:
Statement $I$: For an ideal gas, heat capacity at constant volume is always greater than the heat capacity at constant pressure.
Statement $II$: In a constant volume process, no work is produced and all the heat withdrawn goes into the chaotic motion and is reflected by a temperature increase of the ideal gas.

Match the following:

$(a)$ Entropy of vaporization	$(1)$ Decreases
$(b)$ $K$ for spontaneous process	$(2)$ Always has a $(+)$ value
$(c)$ Crystalline solid state	$(3)$ Has minimum entropy
$(d)$ $\Delta U$ for adiabatic expansion of an ideal gas	$(4)$ $\frac{\Delta H_{vap}}{T_b}$

$10 \, mol$ of an ideal gas expands isothermally and reversibly from a pressure of $10 \, atm$ to $1 \, atm$ at $300 \, K$. What is the largest mass (in $kg$) which can be lifted through a height of $100 \, m$ by the energy obtained in this process (in $, kg$)?

The enthalpies of combustion of $S_{(s)}$ and $H_{2(g)}$ are $-300 \ kcal \ mol^{-1}$ and $-290 \ kcal \ mol^{-1}$ respectively. Given the following reactions:
$SO_{3(g)} + H_2O_{(l)} \rightarrow H_2SO_{4(l)}$; $\Delta H = -130 \ kcal \ mol^{-1}$
$SO_{2(g)} + 1/2 O_{2(g)} \rightarrow SO_{3(g)}$; $\Delta H = -100 \ kcal \ mol^{-1}$
$S_{(s)} + O_{2(g)} \rightarrow SO_{2(g)}$; $\Delta H = -300 \ kcal \ mol^{-1}$
$H_{2(g)} + 1/2 O_{2(g)} \rightarrow H_2O_{(l)}$; $\Delta H = -290 \ kcal \ mol^{-1}$
The enthalpy of formation of $H_2SO_{4(l)}$ is:

Identify the incorrect statements from the following.
$I$. For adiabatic process,$\Delta U = w_{ad}$
$II$. Enthalpy is an intensive property
$III$. For the process,$H_2O_{(l)} \rightarrow H_2O_{(s)}$,the entropy increases

$A$ liquid confined inside an adiabatic container is taken from state $1$ to state $2$ by a single-stage process as shown in the $P-V$ diagram. Then,$\Delta H$ is: