If the ratio of vapour density for hydrogen and oxygen is `1/(16)`, then under constant pressure the ratio of their r.m.s velocities will be

Options

A. `4/1`
B. `1/4`
C. `1/(16)`
D. `(16)/1`

Explanation

To solve the problem of finding the ratio of the root mean square (r.m.s) velocities of hydrogen and oxygen given the ratio of their vapor densities, we can follow these steps:

### Step 1: Understand the relationship between r.m.s velocity and density
The r.m.s velocity (v_rms) of a gas is given by the formula:
\[ v_{rms} = \sqrt{\frac{3P}{\rho}} \]
where \( P \) is the pressure and \( \rho \) is the density of the gas.

### Step 2: Set up the ratio of r.m.s velocities
Since the pressure is constant for both gases, we can express the ratio of their r.m.s velocities as:
\[ \frac{v_{rms, H_2}}{v_{rms, O_2}} = \sqrt{\frac{\rho_{O_2}}{\rho_{H_2}}} \]

### Step 3: Use the given vapor density ratio
We are given that the ratio of vapor densities for hydrogen (H₂) and oxygen (O₂) is:
\[ \frac{\rho_{H_2}}{\rho_{O_2}} = \frac{1}{16} \]
This means:
\[ \rho_{H_2} = \frac{1}{16} \rho_{O_2} \]

### Step 4: Substitute the density ratio into the r.m.s velocity ratio
Substituting the density ratio into our r.m.s velocity ratio gives:
\[ \frac{v_{rms, H_2}}{v_{rms, O_2}} = \sqrt{\frac{\rho_{O_2}}{\frac{1}{16} \rho_{O_2}}} = \sqrt{\frac{\rho_{O_2}}{\rho_{O_2}/16}} = \sqrt{16} = 4 \]

### Step 5: Conclusion
Thus, the ratio of the r.m.s velocities of hydrogen to oxygen is:
\[ \frac{v_{rms, H_2}}{v_{rms, O_2}} = 4 \]

### Final Answer:
The ratio of their r.m.s velocities is \( 4:1 \).

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