The binding energy of an isotope of chlorine is 298 MeV. What is the mass defect of this chlorine nucleus in atomic mass units?

Options

A. 3.13u
B. 0.882u
C. 2.30u
D. 0.320u

Answer

To find the mass defect of the chlorine nucleus in atomic mass units (u), we can follow these steps:

### Step 1: Understand the relationship between binding energy and mass defect
The binding energy (BE) of a nucleus is related to its mass defect (Δm) by the equation:
\[ BE = Δm \cdot c^2 \]
where \(c\) is the speed of light.

### Step 2: Convert binding energy from MeV to Joules
The given binding energy is 298 MeV. We need to convert this into Joules using the conversion factor:
\[ 1 \text{ MeV} = 1.6 \times 10^{-13} \text{ Joules} \]
So,
\[ BE = 298 \text{ MeV} \times 1.6 \times 10^{-13} \text{ J/MeV} = 4.768 \times 10^{-11} \text{ J} \]

### Step 3: Use the binding energy to find mass defect
We can rearrange the equation from Step 1 to find the mass defect:
\[ Δm = \frac{BE}{c^2} \]
We know that the speed of light \(c\) is approximately \(3 \times 10^8 \text{ m/s}\). Therefore,
\[ c^2 = (3 \times 10^8)^2 = 9 \times 10^{16} \text{ m}^2/\text{s}^2 \]
Now substituting the values:
\[ Δm = \frac{4.768 \times 10^{-11} \text{ J}}{9 \times 10^{16} \text{ m}^2/\text{s}^2} = 5.297 \times 10^{-28} \text{ kg} \]

### Step 4: Convert mass defect from kg to atomic mass units (u)
To convert kilograms to atomic mass units, we use the conversion factor:
\[ 1 \text{ u} = 1.6605 \times 10^{-27} \text{ kg} \]
So,
\[ Δm \text{ (in u)} = \frac{5.297 \times 10^{-28} \text{ kg}}{1.6605 \times 10^{-27} \text{ kg/u}} \approx 0.319 \text{ u} \]

### Final Answer
The mass defect of the chlorine nucleus is approximately:
\[ Δm \approx 0.319 \text{ u} \]

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