To determine how much mass is lost through radioactive decay when [tex]\(1.8 \times 10^{15} \, \text{J}\)[/tex] of energy are released, we can use Einstein's famous equation from his theory of relativity:
[tex]\[ E = mc^2 \][/tex]
Here's the step-by-step process to solve the problem:
1. Identify the given values:
- Energy released ([tex]\(E\)[/tex]): [tex]\(1.8 \times 10^{15} \, \text{J}\)[/tex]
- Speed of light ([tex]\(c\)[/tex]): [tex]\(3 \times 10^8 \, \text{m/s}\)[/tex] (a constant value)
2. Rearrange the equation to solve for mass ([tex]\(m\)[/tex]):
[tex]\[ m = \frac{E}{c^2} \][/tex]
3. Substitute the given values into the equation:
[tex]\[ m = \frac{1.8 \times 10^{15} \, \text{J}}{(3 \times 10^8 \, \text{m/s})^2} \][/tex]
4. Calculate the mass:
[tex]\[ m = \frac{1.8 \times 10^{15}}{9 \times 10^{16}} \][/tex]
Simplify the expression inside the fraction:
[tex]\[ m = \frac{1.8}{9} \times \frac{10^{15}}{10^{16}} \][/tex]
Simplify further:
[tex]\[ m = 0.2 \times 10^{-1} \][/tex]
Convert to simpler form:
[tex]\[ m = 0.02 \, \text{kg} \][/tex]
So, the mass lost through radioactive decay when [tex]\(1.8 \times 10^{15} \, \text{J}\)[/tex] of energy are released is 0.02 kg.
Therefore, the correct answer is:
C. 0.02 kg
