To find the nuclear binding energy for uranium-238, you would use the famous equation from Einstein's theory of relativity, which is [tex]\( E = mc^2 \)[/tex].
Here is a detailed, step-by-step solution:
1. Identify the given values:
- Mass defect ([tex]\(m\)[/tex]) = [tex]\(3.2008 \times 10^{-27}\)[/tex] kilograms.
- Speed of light ([tex]\(c\)[/tex]) = [tex]\(3 \times 10^8\)[/tex] meters per second.
2. Substitute the given values into the equation [tex]\( E = mc^2 \)[/tex]:
- [tex]\( E = (3.2008 \times 10^{-27} \, \text{kg}) \times (3 \times 10^8 \, \text{m/s})^2 \)[/tex].
3. Calculate the speed of light squared:
- [tex]\( c^2 = (3 \times 10^8 \, \text{m/s})^2 = 9 \times 10^{16} \, \text{m}^2/\text{s}^2 \)[/tex].
4. Multiply the mass defect by [tex]\( c^2 \)[/tex]:
- [tex]\( E = 3.2008 \times 10^{-27} \, \text{kg} \times 9 \times 10^{16} \, \text{m}^2/\text{s}^2 \)[/tex].
5. Perform the multiplication:
- [tex]\( E = 3.2008 \times 9 \times 10^{-27 + 16} \, \text{kg} \cdot \text{m}^2/\text{s}^2 \)[/tex].
- [tex]\( E = 28.8072 \times 10^{-11} \, \text{J} \)[/tex].
6. Adjust the scientific notation to match the form of the possible answers:
- [tex]\( E = 2.88072 \times 10^{-10} \, \text{J}\)[/tex].
Therefore, the correct answer is:
C. [tex]\(2.8807 \times 10^{-10}\)[/tex] joules.
