$1$ mole of rigid diatomic gas performs a work of $Q/5$ when heat $Q$ is supplied to it. The molar heat capacity of the gas during this transformation is $\frac{xR}{8}.$ The value of $x$ is $\ldots \ldots \ldots .$ $[R =$ universal gas constant $]$

In the $p-V$ diagram below,the dashed curved line is an adiabat. For a process that is described by a straight line joining two points $X$ and $Y$ on the adiabat (solid line in the diagram),heat is (Hint: consider the variation in temperature from $X$ to $Y$ along the straight line)

The figure below shows two paths that may be taken by a gas to go from a state $A$ to a state $C.$ In process $AB,$ $400 \, J$ of heat is added to the system and in process $BC,$ $100 \, J$ of heat is added to the system. The heat absorbed by the system in the process $AC$ will be ...... $J$

Two cylinders $A$ and $B$ fitted with pistons contain an equal amount of an ideal diatomic gas at temperature $T$ $K$. The piston of cylinder $A$ 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 $dT_{A}$,then the rise in temperature of the gas in cylinder $B$ is (where $\gamma = \frac{C_{P}}{C_{V}}$):

$A$ student records $\Delta Q, \Delta U,$ and $\Delta W$ for a thermodynamic cycle $A \rightarrow B \rightarrow C \rightarrow A$. Certain entries are missing in the table below. Find the correct entry from the following options.

	$AB$	$BC$	$CA$
$\Delta W$	$40 \, J$		$30 \, J$
$\Delta U$		$50 \, J$
$\Delta Q$	$150 \, J$	$10 \, J$

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: