A skull fragment found in 1936 at Baldwin Hills, California, was dated by 14C analysis. Approximately 100g of bone was cleaned and treated with 1 M HCl(aq) to destroy the mineral content of the bone. The bone protein was then collected, dried and pyrolyzed. The CO2 produced was collected and purified, and the ratio of 14C to 12C was measured. If the sample contained roughly 6.3 % of the 14C present in living tissue, how old was the skeleton? (For 14C, T1/2 = 5.73 × 103 years.)

Question

A skull fragment found in 1936 at Baldwin Hills, California, was dated by 14C analysis. Approximately 100g of bone was cleaned and treated with 1 M HCl(aq) to destroy the mineral content of the bone. The bone protein was then collected, dried and pyrolyzed. The CO2 produced was collected and purified, and the ratio of 14C to 12C was measured. If the sample contained roughly 6.3 % of the 14C present in living tissue, how old was the skeleton? (For 14C, T1/2 = 5.73 × 103 years.)

Answer 1

Use the half life to determine k.

k = 0.693/t(1/2).

Then use
ln(No/N)=kt
No = the number of atoms you started with. Since that isn't given, just assume as easy number like 100. Then N is what you have now which the problem lists as 6.3% of the number you started with. You know k, and that leaves t (time) as the only unknown.

Answer 2

To determine the age of the skeleton, we can use the concept of radioactive decay. Carbon-14 (14C) is a radioactive isotope of carbon that undergoes radioactive decay over time. By measuring the ratio of 14C to 12C in a sample, we can estimate its age.

In this case, the sample contained approximately 6.3% of the 14C present in living tissue.

First, let's define the equation for radioactive decay:

N = N0 * (1/2)^(t/T)

Where:
N0 is the initial amount of the radioactive isotope (in this case, the 14C present in living tissue),
N is the current amount of the radioactive isotope,
t is the time that has passed since the decay started, and
T is the half-life of the radioactive isotope (in this case, T = 5.73 × 10^3 years for carbon-14).

Using this equation, we can solve for t, the age of the skeleton:

6.3% = 100 * (1/2)^(t/T)

Dividing both sides by 100, we get:

0.063 = (1/2)^(t/T)

To solve for t, we can take the logarithm of both sides (base 1/2):

log2(0.063) = log2[(1/2)^(t/T)]

Using the logarithm properties, we have:

log2(0.063) = (t/T) * log2(1/2)

Since log2(1/2) = -1, we can simplify further:

log2(0.063) = -(t/T)

Finally, solving for t, we have:

t = - (T * log2(0.063))

Substituting the given value for T (5.73 × 10^3 years) into the equation, we can calculate the age of the skeleton.

t = - (5.73 × 10^3 years) * log2(0.063)
t ≈ 18789 years

Therefore, the skeleton is approximately 18,789 years old based on the 14C analysis.