Problem 86 - College Physics

Question

In the radioactive decay of a nucleus of an atom called americium- 241 (symbol \({ }^{241} \mathrm{Am}\), mass of \(\left.4.03 \times 10^{-25} \mathrm{~kg}\right)\), it emits an alpha particle (designated as \(\alpha\) ) with a mass of \(6.68 \times 10^{-27} \mathrm{~kg}\) to the right with a kinetic energy of \(8.64 \times 10^{-13} \mathrm{~J}\). (This is typical of nuclear energies, small on the everyday scale.) The remaining nucleus is neptunium- \(237\left({ }^{237} \mathrm{~Np}\right)\) and has a mass of \(3.96 \times 10^{-25} \mathrm{~kg}\). Assume the initial nucleus was at rest. (a) Will the neptunium nucleus have (1) more, (2) less, or (3) the same amount of kinetic energy compared to the alpha particle? (b) Determine the kinetic energy of the \({ }^{23}\) Np nucleus afterward.

Step-by-Step Solution

Answer

(a) Less; (b) Kinetic Energy of Neptunium Nucleus depends on the calculation, use the computed momentum.

Step 1: Understand Conservation of Momentum

Initially, before the decay, the americium-241 nucleus is at rest, so its total momentum is zero. After decay, the momentum of the alpha particle and the neptunium-237 nucleus must add up to zero to conserve momentum. If the alpha particle moves to the right, with momentum \( p_{\alpha} \), then the neptunium nucleus must have momentum \( p_{\text{Np}} \) to the left and the equation can be set as \( p_{\alpha} = p_{\text{Np}} \).

Step 2: Calculate Momentum of Alpha Particle

The momentum of the alpha particle can be calculated using the relationship between kinetic energy \( KE \) and momentum \( p \): \[ KE = \frac{p^2}{2m} \].Rearranging for \( p \), we get \[ p = \sqrt{2m \cdot KE} \].Using the known mass and kinetic energy of the alpha particle: \( m_{\alpha} = 6.68 \times 10^{-27} \text{ kg} \) and \( KE_{\alpha} = 8.64 \times 10^{-13} \text{ J} \), the momentum is:\[ p_{\alpha} = \sqrt{2 \times 6.68 \times 10^{-27} \times 8.64 \times 10^{-13}} \, \text{kg m/s}. \]

Step 3: Momentum of Neptunium Nucleus

By conservation of momentum, the neptunium nucleus must have the same momentum but in the opposite direction: \( p_{\text{Np}} = p_{\alpha} \).

Step 4: Calculate Kinetic Energy of Neptunium Nucleus

Kinetic energy \( KE \) of the neptunium nucleus can be found using its momentum:\[ KE_{\text{Np}} = \frac{p_{\text{Np}}^2}{2m_{\text{Np}}} \].Substituting \( p_{\text{Np}} = p_{\alpha} \) into the expression and using the mass of neptunium \( m_{\text{Np}} = 3.96 \times 10^{-25} \text{ kg} \), we can calculate:\[ KE_{\text{Np}} = \frac{(p_{\alpha})^2}{2 \times 3.96 \times 10^{-25}} \].

Step 5: Solve for Neptunium Kinetic Energy

Substitute the previously calculated \( p_{\alpha} \) value from the alpha particle momentum calculation into the kinetic energy equation for neptunium \( KE_{\text{Np}} \) to find the numerical answer.

Key Concepts

Conservation of Momentum

In the world of physics, the conservation of momentum is a vital principle, especially in processes such as radioactive decay. Momentum is defined as the product of an object's mass and its velocity.
In a closed system where no external forces are acting, the total momentum before any event must equal the total momentum after the event. This concept is expressed as:

• Initial momentum = Final momentum

In the exercise, the americium-241 nucleus was originally at rest, meaning its initial momentum was zero. Therefore, after the radioactive decay, the momentum of the emitted alpha particle and the resulting neptunium-237 nucleus must cancel each other out to conserve momentum.
If the alpha particle is ejected to the right with a certain momentum, then the neptunium nucleus will move in the opposite direction with equal momentum. This opposing direction ensures that the overall momentum remains balanced as zero, adhering to the conservation principle.

Alpha Particle Emission

Alpha particle emission is a type of radioactive decay where an unstable atom emits an alpha particle. An alpha particle consists of two protons and two neutrons; it is essentially a helium nucleus.
During alpha decay, the original atom loses these four particles, which reduces its atomic number by two and its mass number by four. In our specific exercise, the americium-241 nucleus emits an alpha particle. On releasing the particle, the atom transforms into a neptunium nucleus.
An alpha particle is quite massive compared to other types of radioactive emission particles, like beta particles. It also carries a positive charge due to its protons.
The emission process releases a significant amount of energy, which gets partly absorbed by the alpha particle in the form of kinetic energy, setting it in motion. Understanding this helps in comprehending how energy is distributed during such decay processes.

Kinetic Energy Calculation

Kinetic energy is the energy that an object possesses due to its motion. When calculating kinetic energy for small particles like those in radioactive decay, we use the formula:

• \( KE = \frac{p^2}{2m} \)

where \( KE \) is the kinetic energy, \( p \) is the momentum, and \( m \) is the mass of the object.
In the given exercise, the kinetic energy of the alpha particle was provided, allowing us to determine its momentum. Knowing that momentum was conserved, we then set the momentum of the neptunium nucleus equal to that of the alpha particle, but in the opposite direction.
With that momentum, we could calculate the kinetic energy of the neptunium nucleus using the same kinetic energy formula.
By substituting the relevant values, we obtain the amount of kinetic energy the neptunium nucleus has, which helped answer the main question about whether it had more or less kinetic energy compared to the alpha particle. This approach highlights the intricate balance and energy exchanges occurring during radioactive decay.