In the isothermal expansion of $10\,g$ of gas from volume $V$ to $2V$,the work done by the gas is $575\,J$. What is the root mean square speed of the molecules of the gas at that temperature (in $m/s$)?

Two closed containers of equal volume are filled with air at pressure $P_0$ and temperature $T_0$. Both are connected by a narrow tube. If one of the containers is maintained at temperature $T_0$ and the other at temperature $T$,then the new pressure in the containers will be:

If the heat required to increase the rms speed of $4$ moles of a diatomic gas from $v$ to $\sqrt{3} v$ is $83.1 \ kJ$,then the initial temperature of the gas is (Universal gas constant $R = 8.31 \ J \ mol^{-1} \ K^{-1}$) (in $^{\circ} C$)

The cross-sectional area of a piston is $8 \times 10^{-3} \, m^2$. The gas has a temperature of $300 \, K$,a pressure of $1.0 \times 10^5 \, Pa$,and a volume of $2.4 \times 10^{-3} \, m^3$. When the gas is heated by a heater,the piston moves by $0.1 \, m$. If the spring constant is $8000 \, N/m$,find the final temperature in $K$.

Statement $(I)$: Gas thermometers are less sensitive than liquid thermometers.
Statement $(II)$: The ratio of the universal gas constant and Avogadro's number is called Boltzmann's constant.
Statement $(III)$: The density of a given mass of a gas at constant pressure is inversely proportional to its absolute temperature.
The correct option among the following is:

$A$ very tall vertical cylinder is filled with a gas of molar mass $M$ under isothermal conditions at temperature $T$. The density and pressure of the gas at the base of the container are $\rho_0$ and $p_0$,respectively. Choose the correct statement$(s)$.