Which of the following equations are correct?
$(A)$ $H = U + PV$
$(B)$ $G = H - TS$
$(C)$ $U = q + W$

$5 \, mol$ of an ideal gas at $100 \, K$ are allowed to undergo reversible compression till its temperature becomes $200 \, K$. If $C_v = 28 \, J \, K^{-1} \, mol^{-1}$,calculate $\Delta U$ and $\Delta pV$ for this process. $(R = 8.0 \, J \, K^{-1} \, mol^{-1})$

For the reaction $2H_{(g)} \to H_{2(g)}$,the signs of $\Delta H$ and $\Delta S$ are:

For a hypothetical reaction $A_{2(g)} + B_{2(g)} \rightleftharpoons 2AB_{(g)}$,at $200 \ K$,$\Delta_r G$ and $\Delta_r S$ are $20 \ kJ \ mol^{-1}$ and $-20 \ J \ K^{-1} \ mol^{-1}$ respectively. If $\Delta_r C_p = 20 \ J \ K^{-1} \ mol^{-1}$,find $\Delta_r H$ at $400 \ K$ in $kJ \ mol^{-1}$.

An ideal gas undergoes a reversible isothermal expansion from state $I$ to state $II$ followed by a reversible adiabatic expansion from state $II$ to state $III$. The correct plot$(s)$ representing the changes from state $I$ to state $III$ is(are)
($p$ : pressure,$V$ : volume,$T$ : temperature,$H$ : enthalpy,$S$ : entropy)

Consider the following statements:
$(A)$ Entropy of a perfect crystalline solid at absolute zero approaches zero.
$(B)$ For spontaneity of a reaction at constant temperature and pressure, $T\Delta S > \Delta H$ (where $\Delta H$ is positive).
Identify the correct answer from the options given below.