Sodium-24 has a half-life of 15 hours. When will 4 mg of Sodium-24 remain if the original sample contains 64 mg?

Question

Sodium-24 has a half-life of 15 hours. When will 4 mg of Sodium-24 remain if the original sample contains 64 mg?

Answer 1

You can solve by using natural logs, but if we notice that everything is in powers of 2 , we can do it otherwise (we could use log to the base 2 here, but let's just work this without using logs)

I could have used fewer steps, but I spread out for clarity.

4 = 64 * (1/2)^t/15

1/16 = (1/2)^t/15

16 = 2^t/15

2⁴ = 2^t/15

2^4*15 = 2^t

t = 4*15

t = 60

Answer 2

4 = 64(0.5)^t/15

0.0625 = (0.5)^t/15

ln 0.0625 ≈ (t/15)(ln 0.5)

(15 ln 0.0625) / ln 0.5 = t