Answer:
Energy released = 18.985 J
Explanation:
The exponential decay of radioactive substance is given by -
N(t) = N₀ [tex]e^{-kt}[/tex]
where
N₀ = initial quantity
k = decay constant
Half life, [tex]t_{1/2} = \frac{ln 2}{k}[/tex]
⇒[tex]k = \frac{ln 2}{t_{1/2} }[/tex]
Given,
N₀ = 12.5 + 3 = 15.5 × 10⁻⁶ gm
[tex]t_{1/2}[/tex] = 32.6 + 18 = 50.6 × 10⁶ years
So,
[tex]k = \frac{ln 2}{50.6 * 10^{6} }[/tex] = 1.361 × 10⁻⁸ year⁻¹
Now,
N(1) = 15.5 × 10⁻⁶ [tex]e^{-1.361*10^{-8} *1}[/tex]
= 15.49999978904
Now,
Substance decayed = N₀ - N(t)
= 15.5 × 10⁻⁶ - 15.49999978904 × 10⁻⁶
= 21.095 × 10⁻¹⁷ kg
⇒Δm = 21.095 × 10⁻¹⁷ kg
So,
Energy released = Δmc²
= 21.095 × 10⁻¹⁷ × 3 ×10⁸ × 3 × 10⁸
= 189.855 ×10⁻¹
= 18.985 J
⇒Energy released = 18.985 J
