Two moles of an ideal monoatomic gas are allowed to expand adiabatically and reversibly from $300 \ K$ to $200 \ K$. The work done in the process will be $..... \ kJ$.

For the reaction:
$CH_{4(g)} + 2O_{2(g)} \to CO_{2(g)} + 2H_2O_{(l)}$;
$\Delta_r H_{300} = -212.7 \ Kcal/mol$.
The absolute value of $\Delta_r U_{300}^o$ for this reaction is ....... $Kcal/mol$ $(R = 2 \ cal/mol-K)$.

For an ideal gas, consider only $P-V$ work in going from an initial state $X$ to the final state $Z$. The final state $Z$ can be reached by either of the two paths shown in the figure. Which of the following choice(s) is (are) correct? [take $\Delta S$ as change in entropy and $w$ as work done].
(A) $\Delta S_{X \to Z} = \Delta S_{X \to Y} + \Delta S_{Y \to Z}$
(B) $w_{X \to Z} = w_{X \to Y} + w_{Y \to Z}$
(C) $w_{X \to Y \to Z} = w_{X \to Y} + w_{Y \to Z}$
(D) $\Delta S_{X \to Y \to Z} = \Delta S_{X \to Y}$

For the complete combustion of ethene,$C_2H_{4(g)} + 3O_{2(g)} \rightarrow 2CO_{2(g)} + 2H_2O_{(l)}$,the amount of heat produced as measured in a bomb calorimeter is $1406 \ kJ \ mol^{-1}$ at $300 \ K$. The minimum value of $T \Delta S$ needed to reach equilibrium is $(-)....... \ kJ$. (Nearest integer) Given: $R = 8.3 \ J \ K^{-1} \ mol^{-1}$

The molar heat capacity for an ideal gas at constant pressure is $20.785 \ J \ K^{-1} \ mol^{-1}$. The change in internal energy is $5000 \ J$ upon heating it from $300 \ K$ to $500 \ K$. The number of moles of the gas is [Nearest integer] (Given: $R = 8.314 \ J \ K^{-1} \ mol^{-1}$)

Which of the following relations is correct?