The age of a bone was to be determined by 14C dating. 14C has a half-life of 5,730 years, and a Geiger counter shows that the 14C in living tissue gives 15.2 cpm g-1 of carbon. If the same Geiger counter shows that the 14C in a sample of the bone gives 3.8 cpm g-1 of carbon, how old is the bone?
k = 0.693/t1/2
Then
ln(No/N) = kt
Use 15.2 for No
Use 3.8 for N
Substitute k fro above.
Solve for t.
To determine the age of the bone using 14C dating, we need to understand the relationship between the number of 14C atoms and the time it takes for half of them to decay.
Here's how you can calculate the age of the bone:
1. Start by finding the decay constant (λ) of 14C. The decay constant is given by the formula:
λ = ln(2) / half-life
In this case, the half-life of 14C is 5,730 years, so we can calculate the decay constant:
λ = ln(2) / 5730 = 0.000121
2. Now, determine the number of 14C atoms in the living tissue. The Geiger counter reading for living tissue is 15.2 cpm g-1 of carbon.
3. Next, calculate the number of 14C atoms in the bone sample using the Geiger counter reading of 3.8 cpm g-1 of carbon.
4. To convert the Geiger counter readings into the number of 14C atoms, divide the Geiger count (cpm) by the conversion factor (cpm g-1). This will give you the number of 14C atoms per gram of carbon.
5. Once you have the number of 14C atoms in the living tissue and the bone sample, you can use the decay equation:
N = N0 * e^(-λt)
where N is the number of 14C atoms at a given time, N0 is the initial number of 14C atoms, t is the time in years, and e is the base of the natural logarithm.
Rearrange the equation to solve for t:
t = (ln(N0/N)) / λ
Plug in the values from steps 2 and 3 to calculate the age of the bone.
t = (ln(N0 / N)) / λ
Note: N0 is the number of 14C atoms in living tissue, and N is the number of 14C atoms in the bone sample.
By following these steps and plugging in the appropriate values, you can determine the age of the bone using 14C dating.