Correct answer is (D)

The speed of sound in a gas is given by the formula:

$v=\sqrt{\frac{\gamma RT}{M}}$

where $v$ is the speed of sound, $\gamma$ is the adiabatic index, $R$ is the universal gas constant, $T$ is the temperature, and $M$ is the molar mass of the gas.

For diatomic gases, such as hydrogen (H₂) and oxygen (O₂), the adiabatic index $\gamma$ is the same, approximately equal to $\frac{7}{5}$, and the temperature is given as the same for both gases.

Let's denote the speed of sound in hydrogen as $v_{\text{H}}$ and in oxygen as $v_{\text{O}}$. The ratio of the speeds can be calculated as:

$\frac{v_{\text{H}}}{v_{\text{O}}}=\sqrt{\frac{M_{\text{O}}}{M_{\text{H}}}}$

The molar mass of hydrogen (H₂) is $2\text{g/mol}$, and the molar mass of oxygen (O₂) is $32\text{g/mol}$.

Now, we can calculate the ratio of the speeds:

$\frac{v_{\text{H}}}{v_{\text{O}}}=\sqrt{\frac{32}{2}}=\sqrt{16}=4$

This means the ratio of the speed of sound in hydrogen gas to the speed of sound in oxygen gas at the same temperature is $4:1$.

Therefore, the correct answer is $4:1$.

Correct answer is (B)

$v_{rms}=\sqrt{\frac{\alpha +5}{\alpha }}{v}_{avg}$

$\sqrt{\frac{3RT}{m}}=\sqrt{\frac{\alpha +5}{5}}\sqrt{\frac{8RT}{\pi m}}$

$\frac{3\times \pi }{8}=\frac{\alpha +5}{\alpha }$

$\frac{33}{28}=\frac{\alpha +5}{\alpha }$

$\alpha =28$

Correct answer is (D)

The rms speed of a gas molecule is

${\mathrm{V}}_{\mathrm{RMS}}=\sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}}$

${\mathrm{V}}_{\mathrm{RMS}}\propto \sqrt{\mathrm{T}}$

Correct answer is (A)

$T_i=200\mathrm{K}{v}_{\mathrm{rms}}\propto \sqrt{T}$

$T_f=800\mathrm{K}$

$\frac{V_i}{V_f}=\sqrt{\frac{T_i}{T_f}}=\sqrt{\frac{200}{800}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$

$V_f=2V_i$

Translational K.E. $=\left(\frac{3}{2}PV\right)$

Correct answer is (C)

At NTP, $T=298\mathrm{K}$

$\begin{array}{rl}{v}_{\mathrm{rms}}& =\sqrt{\frac{3RT}{M}}\\ & =\sqrt{\frac{3k{N}_A\times 298}{5\times {10}^{-17}\times N_A}}\end{array}$

$\simeq 15\mathrm{mm}/\mathrm{s}$