$A$ hot air balloon with a payload rises in the air. Assume that the balloon is spherical in shape with a diameter of $11.7 \, m$ and the mass of the balloon and the payload (without the hot air inside) is $210 \, kg$. The temperature and pressure of the outside air are $27^{\circ} C$ and $1 \, atm = 10^5 \, N/m^2$,respectively. The molar mass of dry air is $30 \, g/mol$. The temperature of the hot air inside is close to .......... $^{\circ} C$. [The gas constant,$R = 8.31 \, J K^{-1} mol^{-1}$]

The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting,movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and $1$ moles of an ideal gas,respectively. In the left compartment,the piston is attached by a spring with spring constant $k$ and natural length $\frac{2L}{5}$. In thermodynamic equilibrium,the piston is at a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions,if the pressure in the right compartment is $P = \frac{kL}{A} \alpha$,then the value of $\alpha$ is:

The ratio of the speed of sound in a monatomic gas at $27^{\circ} C$ and the root-mean-square (rms) speed of the molecules of the same gas at a temperature of $127^{\circ} C$ is

According to the kinetic theory of gases,which of the following statements is $WRONG$?

$N$ moles of a diatomic gas in a cylinder are at a temperature $T$. Heat is supplied to the cylinder such that the temperature remains constant but $n$ moles of the diatomic gas get converted into monoatomic gas. What is the change in the total kinetic energy of the gas?

Read the given statements and decide which is/are correct on the basis of the kinetic theory of gases:
$(I)$ Energy of one molecule at absolute temperature $T = 0 \ K$ is zero.
$(II)$ $r.m.s.$ speeds of different gases are the same at the same temperature.
$(III)$ For one gram of all ideal gases,kinetic energy is the same at the same temperature.
$(IV)$ For one mole of all ideal gases,mean kinetic energy is the same at the same temperature.