Two cylinders of equal size are filled with an equal amount of ideal diatomic gas at room temperature. Both cylinders are fitted with pistons. In cylinder $A$,the piston is free to move (isobaric process),while in cylinder $B$,the piston is fixed (isochoric process). When the same amount of heat is supplied to both cylinders,the temperature of the gas in cylinder $A$ rises by $30\, K$. What will be the rise in temperature of the gas in cylinder $B$ (in $;K$)?

Two samples $A$ and $B$ of gas are initially at the same pressure and temperature. They are compressed from volume $V$ to $\frac{V}{2}$ ($A$ isothermally and $B$ adiabatically). The relation between the pressure of gas $A$ $(P_A)$ and the pressure of gas $B$ $(P_B)$ is:

For an ideal gas,which of the following statements is correct?

Read the following statements:
$A.$ When the small temperature difference between a liquid and its surroundings is doubled,the rate of loss of heat of the liquid becomes twice.
$B.$ Two bodies $P$ and $Q$ having equal surface areas are maintained at temperatures $10^{\circ}C$ and $20^{\circ}C$. The thermal radiation emitted in a given time by $P$ and $Q$ are in the ratio $1:1.15$.
$C.$ $A$ Carnot engine working between $100 K$ and $400 K$ has an efficiency of $75\%$.
$D.$ When the small temperature difference between a liquid and its surroundings is quadrupled,the rate of loss of heat of the liquid becomes twice.
Choose the correct answer from the options given below:

The slopes of isothermal and adiabatic curves are related as

An ideal gas expands from volume $V_1$ to $V_2$. This may be achieved by either of the three processes: isobaric,isothermal,and adiabatic. Let $\Delta U$ be the change in internal energy of the gas,$Q$ be the quantity of heat added to the system,and $W$ be the work done by the system. Identify which of the following statements is false for $\Delta U$?