To find the nuclear binding energy of an atom given its mass defect, we use Einstein's famous equation [tex]\(E = mc^2\)[/tex], where [tex]\(E\)[/tex] is the energy, [tex]\(m\)[/tex] is the mass defect, and [tex]\(c\)[/tex] is the speed of light in a vacuum.
Here's a step-by-step solution:
1. Identify the given data:
- Mass defect ([tex]\(m\)[/tex]) = [tex]\(5.0446 \times 10^{-29}\, \text{kg}\)[/tex]
- Speed of light ([tex]\(c\)[/tex]) = [tex]\(3.00 \times 10^8\, \text{m/s}\)[/tex]
2. Write down the equation:
[tex]\[
E = mc^2
\][/tex]
3. Substitute the values into the equation:
- Mass defect ([tex]\(m\)[/tex]) is [tex]\(5.0446 \times 10^{-29}\, \text{kg}\)[/tex]
- Speed of light ([tex]\(c\)[/tex]) is [tex]\(3.00 \times 10^8\, \text{m/s}\)[/tex]
Thus, we get:
[tex]\[
E = (5.0446 \times 10^{-29}\, \text{kg}) \times (3.00 \times 10^8\, \text{m/s})^2
\][/tex]
4. Calculate the value:
- First, square the speed of light:
[tex]\[
(3.00 \times 10^8\, \text{m/s})^2 = 9.00 \times 10^{16}\, \text{m}^2/\text{s}^2
\][/tex]
- Then, multiply it by the mass defect:
[tex]\[
E = (5.0446 \times 10^{-29}\, \text{kg}) \times (9.00 \times 10^{16}\, \text{m}^2/\text{s}^2)
\][/tex]
- Perform the multiplication:
[tex]\[
E = 4.540140 \times 10^{-12}\, \text{J}
\][/tex]
5. Analyze the results:
The energy calculated is [tex]\(4.540140 \times 10^{-12}\, \text{J}\)[/tex].
6. Choose the correct answer:
Among the given options, the value that matches our calculated energy is:
[tex]\[
4.54 \times 10^{-12}\, \text{J}
\][/tex]
Therefore, the nuclear binding energy of the atom is [tex]\(\boxed{4.54 \times 10^{-12}\, \text{J}}\)[/tex].
