The initial pressure and volume of a gas are $P$ and $V$ respectively. First,the gas is expanded to a volume of $9V$ by an isothermal process,and then it is compressed to a volume of $V$ by an adiabatic process. What is its final pressure (in $P$)? (Ratio of specific heat at constant pressure to constant volume $\gamma = \frac{3}{2}$)

An ideal monatomic gas of $n$ moles is taken through a cycle $W-X-Y-Z-W$ consisting of consecutive adiabatic and isobaric quasi-static processes,as shown in the schematic $V-T$ diagram. The volumes of the gas at $W, X,$ and $Y$ points are $64 \ cm^3, 125 \ cm^3,$ and $250 \ cm^3,$ respectively. If the absolute temperature of the gas $T_W$ at point $W$ is such that $nRT_W = 1 \ J$ ($R$ is the universal gas constant),then the amount of heat absorbed (in $J$) by the gas along the path $XY$ is $.....$

The slopes of isothermal and adiabatic curves are related as

Two cylinders $A$ and $B$ are fitted with pistons and contain equal amounts of a diatomic gas at $300 \ K$. The piston of cylinder $A$ is free to move,while the piston of cylinder $B$ is fixed. If the same amount of heat is supplied to each cylinder,the temperature of gas in $A$ increases by $30 \ K$. What is the increase in the temperature of gas in $B$ (in $K$)?

$540$ calories of heat converts $1$ cubic centimeter of water at $100\,^oC$ into $1671$ cubic centimeter of steam at $100\,^oC$ at a pressure of one atmosphere. Then the work done against the atmospheric pressure is nearly .......... $cal$.

One mole of an ideal monoatomic gas undergoes the following four reversible processes:
Step $1$: It is first compressed adiabatically from volume $8.0 \, m^{3}$ to $1.0 \, m^{3}$.
Step $2$: Then expanded isothermally at temperature $T_{1}$ to volume $10.0 \, m^{3}$.
Step $3$: Then expanded adiabatically to volume $80.0 \, m^{3}$.
Step $4$: Then compressed isothermally at temperature $T_{2}$ to volume $8.0 \, m^{3}$.
Then,$T_{1} / T_{2}$ is: