To find the nuclear binding energy for uranium-238, we need to use Einstein's mass-energy equivalence formula:
[tex]\[ E = mc^2 \][/tex]
Given:
- Mass defect \( m = 3.2008 \times 10^{-27} \) kg
- Speed of light \( c = 3 \times 10^8 \) m/s
First, we substitute the given values into the formula:
[tex]\[ E = (3.2008 \times 10^{-27} \, \text{kg}) \times (3 \times 10^8 \, \text{m/s})^2 \][/tex]
Let's calculate \( c^2 \) first:
[tex]\[ c^2 = (3 \times 10^8 \, \text{m/s})^2 = 9 \times 10^{16} \, \text{m}^2/\text{s}^2 \][/tex]
Now, multiply the mass defect by \( c^2 \):
[tex]\[ E = 3.2008 \times 10^{-27} \, \text{kg} \times 9 \times 10^{16} \, \text{m}^2/\text{s}^2 \][/tex]
[tex]\[ E = 3.2008 \times 9 \times 10^{-27 + 16} \, \text{J} \][/tex]
[tex]\[ E = 28.8072 \times 10^{-11} \, \text{J} \][/tex]
[tex]\[ E = 2.88072 \times 10^{-10} \, \text{J} \][/tex]
So, the nuclear binding energy \( E \) is \( 2.88072 \times 10^{-10} \) joules.
Examining the given choices:
A. \( 0.28807 \times 10^{-12} \) joules
B. \( 2.8807 \times 10^{-12} \) joules
C. \( 2.8807 \times 10^{-10} \) joules
D. \( 2.8807 \times 10^{-8} \) joules
E. \( 3.2008 \times 10^{27} \) joules
The correct choice is:
C. [tex]\( 2.8807 \times 10^{-10} \)[/tex] joules
