If one mole of an ideal gas goes through the process $A \rightarrow B$ and $B \rightarrow C$ as shown in the $P-V$ diagram. Given that $T_A = 400 \, K$ and $T_C = 400 \, K$. If $\frac{P_B}{P_A} = \frac{1}{5}$,then find the total heat supplied to the gas (in $J$). (in $.2$)

Write one process where work is converted into heat and heat is converted into work.

Consider the following statements:
$A$. Zeroth law of thermodynamics gives the concept of temperature.
$B$. First law of thermodynamics gives the concept of internal energy.
$C$. In isothermal expansion of an ideal gas, $\Delta Q \neq \Delta W$.
$D$. The product of intensive and extensive variables is extensive.
$E$. The ratio of any extensive variable to mass will be an extensive variable.
Choose the correct combination of statements from the options given below:

An ideal gas undergoes a cyclic thermodynamic process in different ways as shown in the corresponding $P-V$ diagrams in column $3$ of the table. Consider only the path from state $1$ to $2$. $W$ denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here $\gamma$ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is $n$.

Column $I$	Column $II$	Column $III$
$(I)$ $W_{1-2} = \frac{1}{\gamma-1}(P_2V_2 - P_1V_1)$	$(i)$ Isothermal	$(P)$ [Graph $P$]
$(II)$ $W_{1-2} = -P(V_2 - V_1)$	(ii) Isochoric	$(Q)$ [Graph $Q$]
$(III)$ $W_{1-2} = 0$	(iii) Isobaric	$(R)$ [Graph $R$]
$(IV)$ $W_{1-2} = -nRT \ln(\frac{V_2}{V_1})$	(iv) Adiabatic	$(S)$ [Graph $S$]

$(1)$ Which of the following options is the only correct representation of a process in which $\Delta U = \Delta Q - P \Delta V$?
$[A] (II) (iii) (P)$ $[B] (II) (iii) (R)$ $[C] (II) (iv) (S)$ $[D] (III) (iii) (P)$
$(2)$ Which one of the following options is the correct combination?
$[A] (III) (ii) (S)$ $[B] (II) (iv) (R)$ $[C] (II) (iv) (P)$ $[D] (IV) (ii) (S)$
$(3)$ Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in an ideal gas?
$[A] (III) (iv) (R)$ $[B] (I) (ii) (Q)$ $[C] (I) (iv) (Q)$ $[D] (I) (iv) (R)$

Match List-$I$ with List-$II$.

$A$. Isobaric	$I$. $\Delta Q = \Delta W$
$B$. Isochoric	$II$. $\Delta Q = \Delta U$
$C$. Adiabatic	$III$. $\Delta Q = 0$
$D$. Isothermal	$IV$. $\Delta Q = \Delta U + P \Delta V$

$\Delta Q = \text{Heat supplied}$,$\Delta W = \text{Work done by the system}$,$\Delta U = \text{Change in internal energy}$,$P = \text{Pressure of the system}$,$\Delta V = \text{Change in volume of the system}$. Choose the correct answer from the options given below:

The volume $V$ of a given mass of monoatomic gas changes with temperature $T$ according to the relation $V = KT^{2/3}$. The work done when temperature changes by $90\,K$ will be $xR$. The value of $x$ is $[R = \text{universal gas constant}]$