An archeological artifact was subjected to radiocarbon dating. The artifact showed a carbon-14 decay rate of 13.8 disintegrations/ min per gram of carbon. Carbon-14 has a half-life of 5715 years, and currently living organisms decay at the rate of 15.3 disintegrations/ min per gram of carbon. What is the approximate age of the artifact.
I know that you need to use the 1st integrated rate law, but how do you find t? Thank you!
To find the approximate age of the artifact, we can use the first-order integrated rate law equation for radioactive decay:
ln(N₀/N) = kt
Where:
N₀ = Initial amount of carbon-14 in the artifact
N = Amount of carbon-14 remaining after time t
k = Decay constant of carbon-14
In this case, the decay rate given for the artifact is 13.8 disintegrations per minute per gram of carbon. To convert this to a decay constant, we divide by the number of disintegrations per minute per gram of carbon for currently living organisms, which is 15.3.
k = (13.8/15.3) = 0.902
Next, we need to determine the fraction of carbon-14 remaining after time t. Since carbon-14 has a half-life of 5715 years, we use the half-life equation to find this fraction:
N/N₀ = (1/2)^(t/T1/2)
Where:
N₀ = Initial amount of carbon-14 (which is 100% since it's the starting point)
N = Amount of carbon-14 remaining (which is a fraction of the initial amount)
t = Time passed in years
T1/2 = Half-life of carbon-14 (which is 5715 years)
Equating the fraction N/N₀ to the decay constant k, we have:
k = (1/2)^(t/T1/2)
Taking the natural logarithm of both sides, we get:
ln(k) = (t/T1/2)ln(1/2)
Now we can solve for time t:
t = (ln(k))/(ln(1/2)) * T1/2
Substituting the values:
t = (ln(0.902))/(ln(1/2)) * 5715
Calculating this, we find:
t ≈ (0.0977) * 5715 ≈ 559.79 years
Therefore, the approximate age of the artifact is around 559.79 years.