Oxygen and hydrogen gas are at same temperature and pressure. And the oxygen molecule has 16 times the mass of hydrogen molecule. Then the ratio of their rms speed is

Options

A. 2
B. `1//4`
C. 4
D. 16

Explanation

To solve the problem of finding the ratio of the root mean square (rms) speeds of oxygen and hydrogen gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- We know that the oxygen molecule has a mass that is 16 times greater than that of the hydrogen molecule. Let's denote the mass of a hydrogen molecule as \( m_H \) and the mass of an oxygen molecule as \( m_O \).
- Therefore, we can write:
\[ m_O = 16 m_H \]

2. **Formula for Root Mean Square Speed:**
- The formula for the root mean square speed (\( v_{rms} \)) of a gas is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas.

3. **Setting Up the Ratio:**
- Since both gases are at the same temperature and pressure, we can set up the ratio of their rms speeds:
\[ \frac{v_{rms, O}}{v_{rms, H}} = \sqrt{\frac{M_H}{M_O}} \]

4. **Substituting the Masses:**
- Substituting \( M_O = 16 M_H \) into the equation, we get:
\[ \frac{v_{rms, O}}{v_{rms, H}} = \sqrt{\frac{M_H}{16 M_H}} = \sqrt{\frac{1}{16}} = \frac{1}{4} \]

5. **Final Result:**
- Therefore, the ratio of the root mean square speeds of oxygen to hydrogen is:
\[ \frac{v_{rms, O}}{v_{rms, H}} = \frac{1}{4} \]

### Conclusion:
The ratio of the root mean square speeds of oxygen to hydrogen is \( \frac{1}{4} \).