Question

The de Broglie wavelengths of a proton and an α particle are \( \lambda \) and \( 2\lambda \) respectively. The ratio of the velocities of proton and α particle will be:

• A. 1 : 8
• B. 1 : 2
• C. 4 : 1
• D. 8 : 1

Answer 1

The Correct Option is D

The de Broglie wavelength is given by the equation:

\[ \lambda = \frac{h}{p} = \frac{h}{mv} \]

where:

• \(h\) is Planck’s constant.
• \(m\) is the mass of the particle.
• \(v\) is the velocity.

For the proton and \(\alpha\)-particle:

\[ \frac{\lambda_p}{\lambda_\alpha} = \frac{m_\alpha v_\alpha}{m_p v_p} \]

Given \(m_\alpha = 4m_p\) (since \(\alpha\)-particle has 4 times the mass of a proton) and the relationship between velocity and wavelength, we find that the ratio of velocities is:

\[ v_p : v_\alpha = 8 : 1 \]

Thus, the correct answer is Option (4).

Answer 2

The problem provides the de Broglie wavelengths for a proton (\(\lambda_p = \lambda\)) and an alpha particle (\(\lambda_\alpha = 2\lambda\)). We need to find the ratio of their velocities, \(\frac{v_p}{v_\alpha}\).

Concept Used:

The de Broglie wavelength (\(\lambda\)) of a particle is inversely proportional to its momentum (\(p\)). The formula is:

\[ \lambda = \frac{h}{p} = \frac{h}{mv} \]

where \(h\) is Planck's constant, \(m\) is the mass of the particle, and \(v\) is its velocity.
The key relationships for the masses of the proton (\(m_p\)) and the alpha particle (\(m_\alpha\)) are:
An alpha particle consists of 2 protons and 2 neutrons. Since the mass of a neutron is approximately equal to the mass of a proton, the mass of an alpha particle is approximately four times the mass of a proton.

\[ m_\alpha \approx 4m_p \]

Step-by-Step Solution:

Step 1: Write down the de Broglie wavelength equations for the proton and the alpha particle using the given information.

For the proton (\(p\)):

\[ \lambda_p = \lambda = \frac{h}{m_p v_p} \quad \cdots (1) \]

For the alpha particle (\(\alpha\)):

\[ \lambda_\alpha = 2\lambda = \frac{h}{m_\alpha v_\alpha} \quad \cdots (2) \]

Step 2: Establish a relationship between the two equations. We can divide equation (1) by equation (2).

\[ \frac{\lambda}{2\lambda} = \frac{\frac{h}{m_p v_p}}{\frac{h}{m_\alpha v_\alpha}} \]

Step 3: Simplify the resulting expression. The terms \(\lambda\) on the left side and \(h\) on the right side cancel out.

\[ \frac{1}{2} = \frac{m_\alpha v_\alpha}{m_p v_p} \]

Step 4: Rearrange the equation to solve for the ratio of the velocities \(\frac{v_p}{v_\alpha}\).

Cross-multiplying gives:

\[ m_p v_p = 2(m_\alpha v_\alpha) \]

Now, isolate the velocity ratio:

\[ \frac{v_p}{v_\alpha} = \frac{2m_\alpha}{m_p} \]

Step 5: Substitute the mass relationship \(m_\alpha = 4m_p\) into the equation.

\[ \frac{v_p}{v_\alpha} = \frac{2(4m_p)}{m_p} \]

Step 6: Calculate the final numerical ratio by canceling the mass of the proton \(m_p\).

\[ \frac{v_p}{v_\alpha} = \frac{8m_p}{m_p} = 8 \]

The ratio of the velocity of the proton to the velocity of the alpha particle is 8.