Nuclear Physics - AP Physics 1 Exam Question

Question:

An unstable isotope of uranium (²³⁸U) undergoes α-decay and produces a new isotope. The initial nucleus has a mass of 238 atomic mass units (u) and a charge of +92e. The ejected α-particle has a mass of 4u and a charge of +2e. Determine the mass and charge of the new nucleus after the α-decay.

Answer:

Given:

• Mass of initial nucleus ($m_{\text{initial}}$) = 238u
• Charge of initial nucleus ($q_{\text{initial}}$) = +92e
• Mass of ejected α-particle ($m_{\text{α-particle}}$) = 4u
• Charge of ejected α-particle ($q_{\text{α-particle}}$) = +2e

The reaction can be represented as: U²³⁸ $\rightarrow$ Y + α

Since mass and charge are conserved in nuclear reactions, we can write the conservation equations as:

• Mass Conservation: $m_{\text{initial}}$ = $m_{\text{new nucleus}}$ + $m_{\text{α-particle}}$
• Charge Conservation: $q_{\text{initial}}$ = $q_{\text{new nucleus}}$ + $q_{\text{α-particle}}$

Now solving the equations:

1. Mass Conservation Equation:

$238u = m_{\text{new nucleus}} + 4u$

Subtracting 4u from both sides:

$m_{\text{new nucleus}} = 238u - 4u$

$m_{\text{new nucleus}} = 234u$

The mass of the new nucleus is 234u.

2. Charge Conservation Equation:

$92e = q_{\text{new nucleus}} + 2e$

Subtracting 2e from both sides:

$q_{\text{new nucleus}} = 92e - 2e$

$q_{\text{new nucleus}} = 90e$

The charge of the new nucleus is +90e.

Therefore, after the α-decay, the mass of the new nucleus is 234 atomic mass units (u) and its charge is +90 elementary charges (e).