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:
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:
- 1 : 8
- 1 : 2
- 4 : 1
- 8 : 1
The Correct Option is D
Approach Solution - 1
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).
Approach Solution -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.
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.
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