A sample of Bismuth-210 decayed to 33% of its original mass after 8 days

a) Find the half-life of this element

The answer is 5 days

(1/2) ⁿ = decimal amount remaining, where n i number of half-lives.

33% = 33/100 = 0.33

(1/2) ⁿ = 0.33

log ((1/2) ⁿ) = log (0.33)

n * log (1/2) = log (0.33)

n = log (0.33) / log (1/2) = log (0.33) / log (0.5) = - 0.4815/-0.3010 = 1.6

Number of half-lives (x) is quotient of total time elapsed (t) and length of half-life:

x = 1.6

t = 8 days

x = t/half-life

half-life = t/x = 8 days / 1.6 = 5