Answer:
4.80×10³ years
Step-by-step explanation:
Let the original amount (N₀) of ²²⁶Rn = 1 g
Therefore,
12.5% of the original amount = 12.5% × 1 = 12.5/100 × 1 = 0.125 g
Next, we shall determine the number of half-lives that has elapse. This can be obtained as follow:
Original amount (N₀) = 1
Amount remaining (N) = 0.125 g
Number of half-lives (n) =?
N = 1/2ⁿ × N₀
0.125 = 1/2ⁿ × 1
0.125 = 1/2ⁿ
Cross multiply
0.125 × 2ⁿ = 1
Divide both side by 0.125
2ⁿ = 1/0.125
2ⁿ = 8
Express 8 in index form with 2 as the base
2ⁿ = 2³
n = 3
Thus, 3 half-lives has elapsed.
Finally, we shall determine the time taken for only 12.5% of the original sample of ²²⁶Rn to remain.
This can be obtained as follow:
Half-life (t½) = 1.60×10³ years
Number of half-lives (n) = 3
Time (t) =?
t = n × t½
t = 3 × 1.60×10³
t = 4.80×10³ years.
Thus, it will take 4.80×10³ years for 12.5% of the original sample of ²²⁶Rn to remain.