The half-life of 226Ra is 1.60x103 years. How long until only 12.5% of the original sample of 226Ra remains?

4.80×10³ years

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.