$CO_2$ $(O-C-O)$ is a triatomic gas. The mean kinetic energy of one gram of the gas will be (where $N$ is Avogadro's number,$k$ is Boltzmann's constant,and the molecular weight of $CO_2 = 44$).

$A$ piston is slowly pushed into a metal cylinder containing an ideal gas. Which of the following statements is/are incorrect?

The speed of sound in an ideal gas at a given temperature $T$ is $v$. The rms speed of gas molecules at that temperature is $v_{\text{rms}}$. The ratio of the velocities $v$ and $v_{\text{rms}}$ for helium and oxygen gases are $X$ and $X^{\prime}$,respectively. Then,$\frac{X}{X^{\prime}}$ is equal to

Two thermally insulated vessels $1$ and $2$ contain air at temperatures $(T_1, T_2)$,volumes $(V_1, V_2)$,and pressures $(P_1, P_2)$. If the valve connecting them is opened,what will be the equilibrium temperature of the system?

$4.0 \, g$ of a gas occupies $22.4 \, L$ at $NTP$. The specific heat capacity of the gas at constant volume is $5.0 \, J K^{-1} mol^{-1}$. If the speed of sound in this gas at $NTP$ is $952 \, m s^{-1}$,then the heat capacity at constant pressure is .... $J K^{-1} mol^{-1}$ (Take gas constant $R = 8.3 \, J K^{-1} mol^{-1}$)

The average translational energy and the rms speed of molecules of a sample of oxygen gas at $300 \ K$ are $6.21 \times 10^{-21} \ J$ and $484 \ m/s$ respectively. The corresponding values at $600 \ K$ are nearly (assuming ideal gas behaviour):