Assuming the velocity to be same , the wavelength of the waves associated with which of the following particles would be maximum ?

Options

A. An electron
B. A proton
C. An` alpha -` particle
D. A deutron

Answer

To solve the question regarding which particle has the maximum wavelength when their velocities are the same, we will use the de Broglie wavelength formula:

### Step-by-Step Solution:

1. **Understand the de Broglie Wavelength Formula**:
The de Broglie wavelength (\( \lambda \)) is given by the equation:
\[ \lambda = \frac{h}{mv} \]
where:
- \( \lambda \) = wavelength
- \( h \) = Planck's constant (\( 6.626 \times 10^{-34} \) Js)
- \( m \) = mass of the particle
- \( v \) = velocity of the particle

2. **Identify the Relationship**:
From the formula, we can see that the wavelength is inversely proportional to the mass of the particle when the velocity is constant:
\[ \lambda \propto \frac{1}{m} \]
This means that as the mass increases, the wavelength decreases.

3. **List the Masses of the Particles**:
We need to compare the masses of the particles in question. Let's denote the masses as follows:
- Mass of an electron (\( m_e \)) = \( 9.1 \times 10^{-31} \) kg
- Mass of a proton (\( m_p \)) = \( 1.67 \times 10^{-27} \) kg
- Mass of an alpha particle (\( m_{\alpha} \)) = \( 6.68 \times 10^{-27} \) kg
- Mass of a deuteron (\( m_d \)) = \( 3.35 \times 10^{-27} \) kg

4. **Compare the Masses**:
Now we can arrange the particles based on their masses:
- Electron: \( 9.1 \times 10^{-31} \) kg (lightest)
- Proton: \( 1.67 \times 10^{-27} \) kg
- Deuteron: \( 3.35 \times 10^{-27} \) kg
- Alpha particle: \( 6.68 \times 10^{-27} \) kg (heaviest)

5. **Determine the Maximum Wavelength**:
Since the wavelength is inversely proportional to mass, the particle with the smallest mass will have the largest wavelength. From our comparison, the electron has the smallest mass.

6. **Conclusion**:
Therefore, the particle with the maximum wavelength, assuming the velocity is the same, is the electron.

### Final Answer:
The electron has the maximum wavelength associated with it.