Question

One-eighth of a sample of 227 90 Th remains undecayed after 54 days. What is the half-life of this thorium isotope?

Answer 1

1/8 is half of 1/4 which is half of 1/2. Therefore we can say that 3 half lives passed so 54/3 is equal to 18. You can verify this by saying after one half life passes it has been 18 days and I have half of my sample left. After two it has been 36 days and I am left with 1/4 of my sample. After three half lives I have 1/8 of my sample left and it has been 54 days. Or you could put it in the equation\[\ln(amount\ at\ some\ time)=\ln(initial\ amount)- kt\]\[\ln(1/8)=ln(1)-kt\]\[{\ln(1/8)\over t} = k\] then put it in the half life equation \[t_{1/2} = {.693\over k}\]