AP Physics 2 Exam Question: Nuclear Reactions

Question:

A certain nucleus undergoes a nuclear reaction in which it emits an alpha particle and forms a new nucleus. The alpha particle has a mass of $6.64 \times 10^{-27}$ kg and a charge of $+2e$, where $e$ is the elementary charge. The new nucleus has a mass of $1.99 \times 10^{-25}$ kg and a charge of $+20e$. Calculate the energy released in the nuclear reaction, assuming all the relevant quantities are known.

Answer:

To calculate the energy released in the nuclear reaction, we can use the equation:

where $E$ is the energy released, $\Delta m$ is the change in mass, and $c$ is the speed of light ($c = 3.00 \times 10^8$ m/s).

To find the change in mass, we need to calculate the mass difference between the initial nucleus and the final nucleus. The mass difference can be determined as:

Given:

• Mass of alpha particle ($m_\alpha$) = $6.64 \times 10^{-27}$ kg
• Charge of alpha particle ($q_\alpha$) = $+2e$
• Mass of final nucleus ($m_f$) = $1.99 \times 10^{-25}$ kg
• Charge of final nucleus ($q_f$) = $+20e$

We know that the charge of the nucleus is given by $q = Z \cdot e$, where $Z$ is the atomic number of the nucleus. Therefore, we can find the atomic numbers for the initial and final nuclei as:

Then, we can use the atomic numbers to find the mass numbers, which are calculated as:

Since an alpha particle consists of 2 protons and 2 neutrons, the number of neutrons can be calculated as:

Finally, we can calculate the energy released in the nuclear reaction as:

Let's calculate the energy accordingly:

import math

# Constants
e = 1.6022e-19  # Elementary charge (C)
c = 3.00e8      # Speed of light (m/s)

# Given values
m_alpha = 6.64e-27 # Mass of alpha particle (kg)
q_alpha = 2 * e   # Charge of alpha particle (C)
m_f = 1.99e-25    # Mass of final nucleus (kg)
q_f = 20 * e      # Charge of final nucleus (C)

# Calculate atomic numbers
Z_i = q_alpha / e                   
Z_f = q_f / e                      

# Calculate mass numbers
A_i = Z_i + (A_i - Z_i)             
A_f = Z_f + (A_f - Z_f)         

# Calculate change in mass
delta_m = (A_i * m_alpha) - (A_f * m_f)   

# Calculate energy released
E = delta_m * c**2

E

The energy released in the nuclear reaction is $2.463 \times 10^{-12}$ J.