One mole of an ideal gas undergoes a cyclic process,consisting of two isochores and two isobars. Temperatures at points $1$ and $3$ are $T_1$ and $T_3$ respectively. Find the work done by the gas over the cycle,if points $2$ and $4$ lie on the same isotherm.

The efficiency of a heat engine is $\eta$ and the coefficient of performance of a refrigerator is $\beta$. Then:

$A$ cyclic process is shown in the $p-v$ diagram. Which of the following statement$(s)$ is/are true?

The figure shows the $P-V$ plot of an ideal gas taken through a cycle $ABCDA$. The part $ABC$ is a semi-circle and $CDA$ is half of an ellipse. Then,
$(A)$ the process during the path $A \rightarrow B$ is isothermal
$(B)$ heat flows out of the gas during the path $B \rightarrow C \rightarrow D$
$(C)$ work done during the path $A \rightarrow B \rightarrow C$ is zero
$(D)$ positive work is done by the gas in the cycle $ABCDA$

$540$ 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 .......... $cal$.

$A$ container is filled with $20$ moles of an ideal diatomic gas at absolute temperature $T$. When heat is supplied to the gas,the temperature remains constant,but $8$ moles dissociate into atoms. The heat energy given to the gas is ......... (in $R T$)