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:

Two cylinders $A$ and $B$ fitted with pistons contain equal number of moles of an ideal monoatomic gas at $400 \,K$. The piston of $A$ is free to move while that of $B$ is held fixed. The same amount of heat energy is given to the gas in each cylinder. If the rise in temperature of the gas in $A$ is $42 \,K$, what is the rise in temperature of the gas in $B$ (in $\,K$)? (Given $\gamma = 5/3$)

Which of the following statements is correct regarding the $p-V$ graph?
$1$. The slope of the $p-V$ graph in an isothermal process is $-\frac{p}{V}$.
$2$. The slope of the $p-V$ graph in an adiabatic process is $-\frac{p}{V}$.
$3$. The slope of the $p-V$ graph in an isochoric process is $-\frac{\gamma p}{V}$.
$4$. The slope of the $p-V$ graph in an isobaric process is zero.

Heat energy of $735\,J$ is given to a diatomic gas allowing the gas to expand at constant pressure. Each gas molecule rotates around an internal axis but does not oscillate. The increase in the internal energy of the gas will be $..........\,J$

An ideal gas with pressure $P$,volume $V$ and temperature $T$ is expanded isothermally to a volume $2V$ and a final pressure $P_i$. The same gas is expanded adiabatically to a volume $2V$,the final pressure is $P_a$. In terms of the ratio of the two specific heats for the gas $\gamma$,the ratio $\frac{P_i}{P_a}$ is

The initial state of a certain gas is $(P_i, V_i, T_i)$. It undergoes expansion until its volume becomes $V_f$. Consider the following two cases:
$(a)$ The expansion takes place at constant temperature.
$(b)$ The expansion takes place at constant pressure.
Plot the $P-V$ diagram for each case. In which of the two cases is the work done by the gas more?