Which of the following relations are correct?
$(A)$ $\Delta U = q + p \Delta V$
$(B)$ $\Delta G = \Delta H - T \Delta S$
$(C)$ $\Delta S = \frac{q_{rev}}{T}$
$(D)$ $\Delta H = \Delta U - \Delta nRT$
Choose the most appropriate answer from the options given below:

Identify the correct statement$(s)$:

The enthalpy of dilution of $4 \ M \ HCl$ to $2 \ M \ HCl$ is $-2.5 \ kJ/mol$. Find the enthalpy change when $500 \ mL$ of $4 \ M \ HCl$ is diluted to $2 \ M \ HCl$ (in $kJ$).

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Statement $II$: In a constant volume process, no work is produced and all the heat withdrawn goes into the chaotic motion and is reflected by a temperature increase of the ideal gas.

$1 \ mol$ of $NH_3$ $(\gamma = 4/3)$ gas at $27 \ ^\circ C$ is expanded under reversible adiabatic conditions to make the volume $8$ times. Calculate the work done in $cal$.

$6 \ g$ of graphite is burnt in a bomb calorimeter at $25^{\circ} C$ and $1 \ atm$ pressure. The temperature of water increased from $25^{\circ} C$ to $31^{\circ} C$. If $\Delta H$ of this reaction is $-248 \ kJ \ mol^{-1}$,find out $C_V$ (in $kJ \ K^{-1}$) of the bomb calorimeter.