Molecular weight of oxygen is 32 and that of hydrogen is 2. The ratio of the rms velocity of hydrogen molecule at 400K to rms velocity of oxygen molecule at 900 K is given by

Options

A. `2/3`
B. `5/3`
C. `8/3`
D. `10/3`

Explanation

To find the ratio of the root mean square (rms) velocity of hydrogen molecules at 400 K to the rms velocity of oxygen molecules at 900 K, we can use the formula for rms velocity:

\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

where:
- \(R\) is the universal gas constant,
- \(T\) is the temperature in Kelvin,
- \(M\) is the molar mass of the gas in kg/mol.

### Step 1: Write the formula for rms velocity of hydrogen and oxygen

For hydrogen (\(H_2\)):
\[ v_{rms, H_2} = \sqrt{\frac{3RT_{H_2}}{M_{H_2}}} \]

For oxygen (\(O_2\)):
\[ v_{rms, O_2} = \sqrt{\frac{3RT_{O_2}}{M_{O_2}}} \]

### Step 2: Substitute the known values

Given:
- Molecular weight of hydrogen, \(M_{H_2} = 2 \, \text{g/mol} = 0.002 \, \text{kg/mol}\)
- Molecular weight of oxygen, \(M_{O_2} = 32 \, \text{g/mol} = 0.032 \, \text{kg/mol}\)
- Temperature of hydrogen, \(T_{H_2} = 400 \, \text{K}\)
- Temperature of oxygen, \(T_{O_2} = 900 \, \text{K}\)

### Step 3: Write the ratio of rms velocities

The ratio of the rms velocities is given by:
\[ \frac{v_{rms, H_2}}{v_{rms, O_2}} = \frac{\sqrt{\frac{3RT_{H_2}}{M_{H_2}}}}{\sqrt{\frac{3RT_{O_2}}{M_{O_2}}}} \]

### Step 4: Simplify the ratio

The \(3R\) terms cancel out:
\[ \frac{v_{rms, H_2}}{v_{rms, O_2}} = \sqrt{\frac{T_{H_2}}{M_{H_2}} \cdot \frac{M_{O_2}}{T_{O_2}}} \]

Substituting the values:
\[ = \sqrt{\frac{400 \, \text{K}}{0.002 \, \text{kg/mol}} \cdot \frac{0.032 \, \text{kg/mol}}{900 \, \text{K}}} \]

### Step 5: Calculate the expression inside the square root

Calculating the terms:
\[ = \sqrt{\frac{400 \times 0.032}{0.002 \times 900}} \]

Calculating the numerator and denominator:
- Numerator: \(400 \times 0.032 = 12.8\)
- Denominator: \(0.002 \times 900 = 1.8\)

Now substituting back:
\[ = \sqrt{\frac{12.8}{1.8}} = \sqrt{7.1111} \approx 2.67 \]

### Step 6: Final ratio calculation

To find the exact ratio:
\[ = \sqrt{\frac{12.8}{1.8}} = \sqrt{\frac{128}{18}} = \sqrt{\frac{64}{9}} = \frac{8}{3} \]

Thus, the ratio of the rms velocity of hydrogen to that of oxygen is:
\[ \frac{v_{rms, H_2}}{v_{rms, O_2}} = \frac{8}{3} \]

### Final Answer
The ratio of the rms velocity of hydrogen molecules at 400 K to that of oxygen molecules at 900 K is \( \frac{8}{3} \).

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