To determine how much mass is lost through radioactive decay when [tex]\(1.8 \times 10^{15}\)[/tex] joules (J) of energy are released, we can use the famous equation from Einstein's theory of relativity, which relates energy and mass:
[tex]\[ E = mc^2 \][/tex]
Where:
- [tex]\( E \)[/tex] is the energy released.
- [tex]\( m \)[/tex] is the mass lost.
- [tex]\( c \)[/tex] is the speed of light in a vacuum, approximately [tex]\( 3 \times 10^8 \)[/tex] meters per second (m/s).
Rearranging this equation to solve for mass ([tex]\( m \)[/tex]) gives us:
[tex]\[ m = \frac{E}{c^2} \][/tex]
Now, let's substitute the given values into this equation:
1. The energy released [tex]\( E = 1.8 \times 10^{15} \)[/tex] J.
2. The speed of light [tex]\( c = 3 \times 10^8 \)[/tex] m/s.
Substitute these values:
[tex]\[ m = \frac{1.8 \times 10^{15}}{(3 \times 10^8)^2} \][/tex]
Calculate the square of the speed of light:
[tex]\[ (3 \times 10^8)^2 = 9 \times 10^{16} \][/tex]
Now substitute this back into the equation:
[tex]\[ m = \frac{1.8 \times 10^{15}}{9 \times 10^{16}} \][/tex]
Divide the numerator by the denominator:
[tex]\[ m = \frac{1.8}{9} \times 10^{15 - 16} \][/tex]
[tex]\[ m = 0.2 \times 10^{-1} \][/tex]
[tex]\[ m = 0.02 \, \text{kg} \][/tex]
Therefore, the mass lost through radioactive decay when [tex]\( 1.8 \times 10^{15} \)[/tex] J of energy are released is:
A. 0.02 kg
