A wooden artifact from an ancient tomb contains 45% of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years.

Question

A wooden artifact from an ancient tomb contains 45% of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years.

Answer 1

An easy way to calculate these problems is to use the simple expression

fraction remaining = 0.5ⁿ where n is the number of half lives that have elapsed.

Fraction remaining = 0.45 thus, you have 0.45 = 0.5ⁿ and solving for n we have...

log 0.45 = n log 0.5

-0.347 = -0.301n

n = 1.15 half lives have elapsed

1.15 half lives x 5730 yrs/half life = 6590 years

Answer 2

While not exact, a reasonable approximation is that if it takes 5730 years for 50% to decay, another 5% would take 573 years for a total of 6303 years.

For a more accurate value, we have to use the radioactive decay law:

N = N₀e^-λt, where N is the radioactive amount after t seconds, N₀ is the original amount, e is Euler's constant and λ is the decay constant for the substance. By substituting 50 for N, 100 for N₀ and the value of 5730 years in seconds for t, we can compute the value of λ for Carbon-14, then plug that value back into the formula with 45 as the value of N to derive the t value for decay of 55% of the original amount. (Good luck with that...personally, I'd be satisfied with the approximation.)