To determine how much energy was released during the nuclear change, we need to use the famous equation from Einstein's theory of relativity, which is [tex]\( E = mc^2 \)[/tex]. Here, [tex]\( E \)[/tex] represents the energy released, [tex]\( m \)[/tex] stands for the mass difference between the initial and final states, and [tex]\( c \)[/tex] is the speed of light (approximately [tex]\( 3 \times 10^8 \)[/tex] meters per second).
1. Calculate the mass difference:
- Initial mass: [tex]\( 2.3465 \times 10^{-27} \)[/tex] kg
- Final mass: [tex]\( 2.3148 \times 10^{-27} \)[/tex] kg
[tex]\[
\text{Mass difference} = 2.3465 \times 10^{-27} \, \text{kg} - 2.3148 \times 10^{-27} \, \text{kg}
\][/tex]
[tex]\[
\text{Mass difference} = 3.17 \times 10^{-29} \, \text{kg}
\][/tex]
Here, we have rounded the mass difference for simplicity.
2. Calculate the energy released using the mass difference and the speed of light [tex]\( c \)[/tex]:
[tex]\[
E = (3.17 \times 10^{-29} \, \text{kg}) \times (3 \times 10^8 \, \text{m/s})^2
\][/tex]
Evaluate the speed of light squared:
[tex]\[
(3 \times 10^8 \, \text{m/s})^2 = 9 \times 10^{16} \, (\text{m}^2/\text{s}^2)
\][/tex]
Now multiply this by the mass difference:
[tex]\[
E = (3.17 \times 10^{-29} \, \text{kg}) \times (9 \times 10^{16} \, (\text{m}^2/\text{s}^2))
\][/tex]
Thus, the energy released [tex]\( E \)[/tex]:
[tex]\[
E = 2.853 \times 10^{-12} \, \text{J}
\][/tex]
Comparing this result to the provided options:
1. [tex]\( 2.85 \times 10^{-12} \, \text{J} \)[/tex]
2. [tex]\( 2.08 \times 10^{-10} \, \text{J} \)[/tex]
3. [tex]\( 2.11 \times 10^{-10} \, \text{J} \)[/tex]
4. [tex]\( 8.56 \times 10^{-4} \, \text{J} \)[/tex]
The energy released during the nuclear change is [tex]\( 2.85 \times 10^{-12} \, \text{J} \)[/tex], which matches the first answer provided.
Therefore, the correct answer is:
[tex]\[
\boxed{2.85 \times 10^{-12} \, \text{J}}
\][/tex]
