How much energy is released if a sample loses 0.025 kg mass through radioactive decay?

A. [tex]$5.6 \times 10^{13} J$[/tex]
B. [tex]$2.25 \times 10^{15} J$[/tex]
C. [tex]$7.5 \times 10^6 J$[/tex]
D. [tex]$3.6 \times 10^{18} J$[/tex]



Answer :

To determine the amount of energy released when a sample loses 0.025 kg of mass through radioactive decay, we can use Einstein's well-known equation for the relationship between mass and energy:

[tex]\[ E = mc^2 \][/tex]

where:
- [tex]\( E \)[/tex] is the energy released.
- [tex]\( m \)[/tex] is the mass loss, which is 0.025 kg in this case.
- [tex]\( c \)[/tex] is the speed of light in a vacuum, which is approximately [tex]\( 3 \times 10^8 \)[/tex] meters per second (m/s).

Let's break down the steps:

1. Identify the mass loss ([tex]\( m \)[/tex]): 0.025 kg.
2. Identify the speed of light ([tex]\( c \)[/tex]): [tex]\( 3 \times 10^8 \)[/tex] m/s.
3. Substitute these values into the equation [tex]\( E = mc^2 \)[/tex]:

[tex]\[ E = 0.025 \, \text{kg} \times (3 \times 10^8 \, \text{m/s})^2 \][/tex]

4. Calculate the square of the speed of light:

[tex]\[ (3 \times 10^8 \, \text{m/s})^2 = 9 \times 10^{16} \, \text{m}^2/\text{s}^2 \][/tex]

5. Multiply the mass loss by the squared speed of light to find the energy released:

[tex]\[ E = 0.025 \, \text{kg} \times 9 \times 10^{16} \, \text{m}^2/\text{s}^2 \][/tex]
[tex]\[ E = 0.225 \times 10^{16} \, \text{J} \][/tex]
[tex]\[ E = 2.25 \times 10^{15} \, \text{J} \][/tex]

Therefore, the energy released when the sample loses 0.025 kg of mass is:

[tex]\( \boxed{2.25 \times 10^{15} \, \text{J}} \)[/tex]

So, the correct answer is option B. [tex]\( 2.25 \times 10^{15} \)[/tex] J.