To determine the nuclear binding energy of an atom with a given mass defect, we use Einstein’s famous equation [tex]\( E = mc^2 \)[/tex]. Here’s a step-by-step breakdown of the calculation:
1. Identify the given data:
- Mass defect ([tex]\( m \)[/tex]) = [tex]\( 1.643 \times 10^{-28} \)[/tex] kg
- Speed of light ([tex]\( c \)[/tex]) = [tex]\( 3.00 \times 10^8 \)[/tex] m/s
2. Write down the formula:
The equation for energy is [tex]\( E = mc^2 \)[/tex].
3. Square the speed of light:
[tex]\[
c^2 = (3.00 \times 10^8)^2 = 9.00 \times 10^{16} \, (\text{m}^2/\text{s}^2)
\][/tex]
4. Multiply the mass defect by the squared speed of light to find the binding energy:
[tex]\[
E = (1.643 \times 10^{-28}) \times (9.00 \times 10^{16})
\][/tex]
5. Perform the multiplication:
- Multiply the numerical coefficients:
[tex]\[
1.643 \times 9.00 = 14.787
\][/tex]
- Combine the powers of ten:
[tex]\[
10^{-28} \times 10^{16} = 10^{-28 + 16} = 10^{-12}
\][/tex]
- Putting it all together:
[tex]\[
E = 14.787 \times 10^{-12} \, \text{J}
\][/tex]
6. Convert the result into scientific notation:
[tex]\[
E = 1.4787 \times 10^{-11} \, \text{J}
\][/tex]
7. Compare the result with the provided choices:
The choices are:
- [tex]\( 1.83 \times 10^{-48} \)[/tex] J
- [tex]\( 4.93 \times 10^{-20} \)[/tex] J
- [tex]\( 1.48 \times 10^{-11} \)[/tex] J
- [tex]\( 5.48 \times 10^{45} \)[/tex] J
The value [tex]\( 1.4787 \times 10^{-11} \)[/tex] J is very close to the choice [tex]\( 1.48 \times 10^{-11} \)[/tex] J.
8. Conclude the correct answer:
The nuclear binding energy of the atom with a mass defect of [tex]\( 1.643 \times 10^{-28} \, \text{kg} \)[/tex] is [tex]\(\boxed{1.48 \times 10^{-11} \, \text{J}}\)[/tex].
