The ratio of speed of sound in hydrogen gas to the speed of sound in oxygen gas at the same temperature is:

We need to find the ratio of the speed of sound in hydrogen gas to that in oxygen gas at the same temperature.

In an ideal gas, the speed of sound is given by the relation:

$$v = \sqrt{\frac{\gamma R T}{M}}$$

Here $$\gamma$$ denotes the adiabatic index, $$R$$ is the universal gas constant, $$T$$ represents the absolute temperature, and $$M$$ stands for the molar mass of the gas.

Because both hydrogen ($$H_2$$) and oxygen ($$O_2$$) are diatomic gases, they share the same adiabatic index $$\gamma = \frac{7}{5}$$. Consequently, at an identical temperature $$T$$, the ratio of their sound speeds depends solely on their molar masses, leading to

$$\frac{v_{H_2}}{v_{O_2}} = \sqrt{\frac{M_{O_2}}{M_{H_2}}}$$

By substituting the molar masses $$M_{H_2} = 2\ \mathrm{g/mol}$$ and $$M_{O_2} = 32\ \mathrm{g/mol}$$ into the above expression, we obtain

$$\frac{v_{H_2}}{v_{O_2}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4$$

Hence, the ratio of the speed of sound in hydrogen to that in oxygen is 4 : 1, which corresponds to Option B.