After 102 years the 40 gm of substance are remain.
What is the half life of a radio active substance ?
Half-life - Means that half of a radio active substance will be gone by the time that half-life is over.
Now here we have -
Initial Weight of the plutonium-238 = 90g
Half life of the plutonium = 88 years
The relation between substance mass and time is given by -
ln(N(t)/N₀) = -λt
Where -
N(t) = mass at any time t
N₀ = initial mass
λ = constant
t = time
Now , we know that after 88 years the 45g of substance remain because 88years is the half life of plutonium. so that -
ln(45/90) = - λ × 88
ln(0.5) = - λ × 88
- 0.7 = - λ × 88
λ = 78 × 10⁻⁴
Now time when 40gm of substance remains -
ln(40/90) = - 78 × 10⁻⁴ × t
- 0.81 = - 78 × 10⁻⁴ × t
t = 1.02 × 10²
t = 102 years
So that after 102 years the 40 gm of substance are remain.
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