if Helium atom and Hydrogen molecule are moving with the same velocity , their wavelength ratio will be

Options

A. `4:1`
B. `1:2:`
C. `2:1`
D. `1:4`

Explanation

To find the wavelength ratio of a helium atom and a hydrogen molecule moving with the same velocity, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Wavelength Formula**:
The wavelength (λ) of a particle is given by the formula:
\[ \lambda = \frac{H}{mv} \]
where:
- \(H\) is Planck's constant,
- \(m\) is the mass of the particle,
- \(v\) is the velocity of the particle.

2. **Identify Constants**:
Since both the helium atom and the hydrogen molecule are moving with the same velocity, we can say that \(v\) is constant for both. Also, Planck's constant \(H\) is a constant value.

3. **Determine Masses**:
- The mass of a hydrogen molecule (H₂) is approximately 2 atomic mass units (amu).
- The mass of a helium atom (He) is approximately 4 amu.

4. **Write the Wavelengths**:
- For helium (He):
\[ \lambda_{He} = \frac{H}{m_{He}v} = \frac{H}{4v} \]
- For hydrogen (H₂):
\[ \lambda_{H_2} = \frac{H}{m_{H_2}v} = \frac{H}{2v} \]

5. **Calculate the Wavelength Ratio**:
To find the ratio of the wavelengths, we can set up the following equation:
\[ \frac{\lambda_{H_2}}{\lambda_{He}} = \frac{\frac{H}{2v}}{\frac{H}{4v}} = \frac{4}{2} = 2 \]

6. **Final Ratio**:
Thus, the wavelength ratio of helium to hydrogen is:
\[ \frac{\lambda_{He}}{\lambda_{H_2}} = \frac{1}{2} \]
This means the ratio of the wavelengths is 1:2.

### Conclusion:
The wavelength ratio of a helium atom to a hydrogen molecule moving with the same velocity is 1:2.