$A$ $15\,g$ mass of nitrogen gas is enclosed in a vessel at a temperature $27\,^{\circ}C.$ The amount of heat transferred to the gas,so that the rms velocity of the molecules is doubled,is about ...... $kJ$ [Take $R = 8.3\,J/K\,mole$ ]

$2 \text{ moles}$ of a monatomic gas requires heat energy $Q$ to be heated from $30^{\circ} C$ to $40^{\circ} C$ at constant volume. The heat energy required to raise the temperature of $4 \text{ moles}$ of a diatomic gas from $28^{\circ} C$ to $33^{\circ} C$ at constant volume is

An oxygen cylinder of volume $30 \ L$ has $18.20 \ \text{moles}$ of oxygen. After some oxygen is withdrawn from the cylinder, its gauge pressure drops to $11 \ \text{atm}$ at a temperature of $27^{\circ} \text{C}$. The mass of the oxygen withdrawn from the cylinder is nearly equal to (Given: $R = \frac{100}{12} \ \text{J mol}^{-1} \text{K}^{-1}$, molecular mass of $O_2 = 32 \ \text{g/mol}$, $1 \ \text{atm} = 1.01 \times 10^5 \ \text{Pa}$) (in $\text{kg}$)

Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$. Statement $I$: Change in internal energy of a system containing $n$ mole of ideal gas can be written as $\Delta U = nC_v(T_f - T_i) = \frac{nR}{\gamma - 1}(T_f - T_i)$,where $\gamma = C_p/C_v, T_i = $ initial temperature,$T_f = $ final temperature. Statement $II$: Relation between degree of freedom $f$ and $\gamma(= C_p/C_v)$ is $\gamma = 1 + \frac{2}{f}$. Choose the correct answer from the options given below.

One mole of an ideal diatomic gas is taken through the cycle as shown in the figure.
$1 \rightarrow 2$: isochoric process
$2 \rightarrow 3$: straight line on $P-V$ diagram
$3 \rightarrow 1$: isobaric process
The average molecular speed of the gas in the states $1, 2$ and $3$ are in the ratio

$A$ light container having a diatomic gas enclosed within is moving with velocity $v$. The mass of the gas is $M$ and the number of moles is $n$. The kinetic energy of the gas with respect to the ground is: