Question

Atomic masses of two oxygen isotopes \(^{16}_{8}\text{O}\) and \(^{18}_{8}\text{O}\) are \(15.99491\ \text{u}\) and \(17.99916\ \text{u}\), respectively, where \(\text{u}\) is the atomic mass unit. Masses of proton and neutron are given by \(1.00727\ \text{u}\) and \(1.00866\ \text{u}\), respectively. The speed of light is \(c\). What is the difference between the binding energies of \(^{18}_{8}\text{O}\) and \(^{16}_{8}\text{O}\) nuclei in units of \(\text{u } c^2\)?

• A. \(0.01307\)
• B. \(2.00425\)
• C. \(0.99559\)
• D. \(3.01291\)

Hint:

Notice that the proton terms cancel out completely when subtracting the binding energies.
The difference is simply the mass of the two extra neutrons minus the actual difference in the isotopic masses.
This significantly reduces the arithmetic load.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the difference between the binding energies of the oxygen isotopes \(^{18}_{8}\text{O}\) and \(^{16}_{8}\text{O}\) using the given isotopic and nucleon masses.

Step 2: Key Formula or Approach:

• Binding energy of a nucleus \(^{A}_{Z}\text{X}\).:
\[ \text{BE} = [Z m_p + (A - Z) m_n - M(^{A}_{Z}\text{X})] c^2 \]

• Difference in binding energy:
\[ \Delta \text{BE} = \text{BE}(^{18}_{8}\text{O}) - \text{BE}(^{16}_{8}\text{O}) \]

Step 3: Detailed Explanation:

• For \(^{16}_{8}\text{O}\).:
Protons, \(Z = 8\); Neutrons, \(N_1 = 16 - 8 = 8\).
\[ \text{BE}(^{16}_{8}\text{O}) = [8 m_p + 8 m_n - M(^{16}_{8}\text{O})] c^2 \]
• For \(^{18}_{8}\text{O}\).:
Protons, \(Z = 8\); Neutrons, \(N_2 = 18 - 8 = 10\).
\[ \text{BE}(^{18}_{8}\text{O}) = [8 m_p + 10 m_n - M(^{18}_{8}\text{O})] c^2 \]
• Subtracting the two equations to find the difference \(\Delta \text{BE}\).:
\[ \Delta \text{BE} = \left[ (8 m_p + 10 m_n - M(^{18}_{8}\text{O})) - (8 m_p + 8 m_n - M(^{16}_{8}\text{O})) \right] c^2 \] \[ \Delta \text{BE} = \left[ 2 m_n - (M(^{18}_{8}\text{O}) - M(^{16}_{8}\text{O})) \right] c^2 \]
• Substituting the numerical values given in the problem:
\[ 2 m_n = 2 \times 1.00866 = 2.01732\ \text{u} \] \[ M(^{18}_{8}\text{O}) - M(^{16}_{8}\text{O}) = 17.99916 - 15.99491 = 2.00425\ \text{u} \]
• Now calculate the difference:
\[ \Delta \text{BE} = [2.01732 - 2.00425]\ c^2 = 0.01307\ \text{u } c^2 \]

Step 4: Final Answer:
The difference between the binding energies is \(0.01307\ \text{u } c^2\).