Three samples of the same gas $A, B$ and $C$ $(\gamma = 3/2)$ have initially equal volume. Now the volume of each sample is doubled. The process is adiabatic for $A$,isobaric for $B$,and isothermal for $C$. If the final pressures are equal for all three samples,the ratio of their initial pressures is:

What is the specific heat of a gas in an isothermal process and an adiabatic process?

An ideal gas is subjected to a cyclic process involving four thermodynamic states. The amounts of heat $(Q)$ and work $(W)$ involved in each of these states are:
$Q_1 = 6000 \ J, Q_2 = -5500 \ J, Q_3 = -3000 \ J, Q_4 = 3500 \ J$
$W_1 = 2500 \ J, W_2 = -1000 \ J, W_3 = -1200 \ J, W_4 = x \ J$
The ratio of the net work done by the gas to the total heat absorbed by the gas is $\eta$. The values of $x$ and $\eta$ respectively are:

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?
$[A] (III) (iv) (R)$ $[B] (I) (ii) (Q)$ $[C] (I) (iv) (Q)$ $[D] (I) (iv) (R)$

Two cylinders $A$ and $B$ fitted with pistons contain equal amounts of an ideal diatomic gas at $300 \ K$. The piston of $A$ is free to move,while that of $B$ is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise in temperature of the gas in $A$ is $30 \ K$,then the rise in temperature of the gas in $B$ is ..... $K$.

Consider one mole of helium gas enclosed in a container at initial pressure $P_1$ and volume $V_1$. It expands isothermally to volume $4 V_1$. After this,the gas expands adiabatically and its volume becomes $32 V_1$. The work done by the gas during isothermal and adiabatic expansion processes are $W_{\text{iso}}$ and $W_{\text{adia}}$,respectively. If the ratio $\frac{W_{\text{iso}}}{W_{\text{adia}}} = f \ln 2$,then $f$ is: