A sample of bismuth 210 decayed to 33 of its original mass a

A sample of bismuth -210 decayed to 33% of its original mass after 8 days. Find the half-life of this element.

Solution

The basic equation for 1st order reactions (radioactive decay is 1st order) is-
ln(Ao/A) = kt
where Ao = original amount. A = amount after time t and k = reaction rate constant

You know that 100% went to 33% ant t = 8 days. So-
ln(100/33) = k * 8
k = 1.1087 / 8 = 0.1386 day^-1

At the half life Ao = 100% and A = 50%
ln(100/50) = 0.1386 * t
t = half life = 0.693 / 0.1386 = 5 days