Largest wavelength is associated with which one of the following . If all these travel with same velocity

Options

A. `CO_(2)`
B. `He`
C. `H_(2)`
D. `NO_(2)`

Answer

To solve the question regarding which particle has the largest wavelength when all are traveling at the same velocity, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Wavelength**:
The wavelength (λ) associated with a particle can be calculated using the de Broglie wavelength formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

2. **Relate Momentum to Mass and Velocity**:
Momentum (p) can be expressed as:
\[ p = mv \]
where \( m \) is the mass of the particle and \( v \) is its velocity. Substituting this into the wavelength formula gives:
\[ \lambda = \frac{h}{mv} \]

3. **Analyze the Given Condition**:
The problem states that all particles travel with the same velocity (v). Therefore, the wavelength is inversely proportional to the mass of the particle:
\[ \lambda \propto \frac{1}{m} \]
This means that as the mass increases, the wavelength decreases, and vice versa.

4. **Identify the Masses of the Given Particles**:
We need to compare the masses of the particles mentioned:
- Hydrogen (H): mass ≈ 1
- Helium (He): mass ≈ 4
- Carbon Dioxide (CO₂): mass ≈ 44
- Nitrogen Dioxide (NO₂): mass ≈ 46

5. **Determine the Particle with the Smallest Mass**:
Among the listed particles, hydrogen has the smallest mass (approximately 1). Since the wavelength is inversely proportional to mass, hydrogen will have the largest wavelength.

6. **Conclusion**:
Therefore, the particle associated with the largest wavelength is hydrogen.

### Final Answer:
The largest wavelength is associated with **Hydrogen (H)**.

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