Problem 103 - Master Resource Book in JEE Main Physics

Question

After absorbing a slowly moving neutron of mass \(M_{N}\) comonecular a nucleus of mass \(M\) breaks into two nuclei of masses \(m_{1}\) and \(5 m_{1}\left(6 m_{1}=M+m_{N}\right)\), respectively. If the de-Broglie wavelength of the nucleus with mass \(m\), is \(\lambda\), then de-Broglie wavelength of the other nucleus will be (a) \(25 \lambda\) (b) \(5 \lambda\) (c) \(\frac{\lambda}{5}\) (d) \(\lambda\)

Step-by-Step Solution

Answer

The de-Broglie wavelength of the other nucleus is \(5 \lambda\) (Option b).

Step 1: Understand the Problem

We have a nucleus of mass \(M\) that absorbs a neutron of mass \(M_N\) and then breaks into two smaller nuclei with masses \(m_1\) and \(5m_1\). We are given that \(6m_1 = M + M_N\). We need to find the de-Broglie wavelength of the nucleus with mass \(5m_1\), knowing the de-Broglie wavelength \(\lambda\) of the nucleus with mass \(m_1\).

Step 2: Apply Conservation of Momentum

Since the initial neutron is moving slowly, the initial momentum is approximately zero. After the reaction, momentum conservation gives: \(m_1 \cdot v_1 = 5m_1 \cdot v_2\), where \(v_1\) and \(v_2\) are the velocities of the masses \(m_1\) and \(5m_1\) respectively.

Step 3: Calculate Velocities Ratio

From \(m_1 \cdot v_1 = 5m_1 \cdot v_2\), cancel \(m_1\) to get \(v_1 = 5v_2\). This equation shows that the velocity of mass \(m_1\) is five times that of mass \(5m_1\).

Step 4: Use de-Broglie Wavelength Formula

The de-Broglie wavelength \(\lambda\) is given by \(\lambda = \frac{h}{mv}\), where \(h\) is Planck's constant, \(m\) the particle's mass, and \(v\) its velocity. For mass \(m_1\), \(\lambda = \frac{h}{m_1 \cdot v_1}\). For mass \(5m_1\), it will be \(\lambda_{5m_1} = \frac{h}{5m_1 \cdot v_2}\).

Step 5: Compare de-Broglie Wavelengths

Using \(v_1 = 5v_2\), we find \(\frac{h}{5m_1 \cdot v_2} = \frac{h}{m_1 \cdot (v_1/5)} = \frac{5h}{m_1 \cdot v_1}\). Hence, \(\lambda_{5m_1} = 5 \lambda\), where \(\lambda\) is the wavelength of the mass \(m_1\).

Key Concepts

Momentum Conservation

When analyzing nuclear reactions, such as the absorption of a neutron by a nucleus followed by a breakup into smaller nuclei, momentum conservation is a key principle.

• Initially, we consider the entire system, including the nucleus and neutron, which is at rest or moving slowly.
• Due to this, the total initial momentum is practically zero.
• According to the law of momentum conservation, the total momentum before and after the reaction must be equal.

This means that the combined momentum of the two resultant nuclei must also equal zero.
For the problem, we represented these using the equation: \[ m_1 \cdot v_1 = 5m_1 \cdot v_2 \]Where:

• \( m_1 \) is the mass of the first nucleus with velocity \( v_1 \).
• \( 5m_1 \) is the mass of the second nucleus with velocity \( v_2 \).

By balancing these quantum entities' momenta, we find out how the velocities of each nucleus are related. Thus, even if one subparticle moves faster, the overall integrity of motion is conserved.

Nuclear Fission

Nuclear fission, the process evident in this exercise, involves a nucleus splitting into smaller parts.
This reaction is significant not only in theoretical physics but also in practical applications such as nuclear reactors.

• The nucleus initially captures a neutron, which facilitates its split into two smaller nuclei.
• The fission process involves huge energy changes, owing to the mass defect and the principle of energy conservation.

Given the condition \( 6m_1 = M + M_N \), it shows the total mass before and after the reaction, factoring in the neutron and the original nucleus.
This equation illustrates how the original components match the product of the fission process, disregarding energy emissions.
Such understanding allows students to see how mass-energy equivalences manifest in atomic rearrangements.

Velocity Ratio

The velocity ratio between particles in a nuclear reaction reflects key insights into their motion post-interaction.
In this problem, after applying the conservation of momentum, we derived the equation:

• \[ v_1 = 5v_2 \]

This shows that the velocity of the nucleus with mass \( m_1 \) is five times that of the nucleus with mass \( 5m_1 \).
Understanding this ratio is crucial as it indicates how differently the two nuclei moved post-fission.

• Since the mass \( 5m_1 \) is greater, it moves slower to balance the momentum equalities.
• The velocity ratio emphasizes the differences in kinetic energies due to mass variations.

After grasping velocity interactions, calculating de-Broglie wavelengths becomes simpler, as velocity is a deciding factor in wave-particle duality experiments.