The half-life of carbon-14 is 5,730 years. An artifact produces 8.8 disintegrations of 14C per minute per gram of carbon in the sample. Estimate the age of this sample assuming that its original radioactivity was 15.3 disintegrations per minute per gram of carbon.

Approximately 4574.86 years

Explanation:

Hello,

To find the age of this sample, we should first of all convert the disintegration per minute to per year so that we can work on the same unit as our half life (T½), then we can find the disintegration constant and use it to find the year of the artifact.

Data;

T½ = 5730 years

Initial rate of radioactivity (No) = 15.3 disintegration per minute.

Current rate of radioactivity (N) = 8.8 disintegration per minute.

1 year = 525600 minutes

1 mins = 8.8 disintegration

525600mins = N disintegration

N = (525600 × 8.8) / 1

N = 4625280

1 mins = 15.3 disintegration

525600 mins = No

No = 8041680

But T½ = In2 / λ

λ = In2 / T½

λ = 0.693 / 5730

λ = 1.209×10⁻⁴ (this is the disintegration constant)

We can now find the how old the artifact is using our disintegration constant and other parameters.

In(N÷No) = -λt

In[4625280 / 8041680] = -(1.209×10⁻⁴ × t)

In[0.57516] = -1.209×10⁻⁴t

-0.5531 = -1.209×10⁻⁴ t

Solve for t

t = 0.5531 / 1.209×10⁻⁴

t = 4574.86 years

The artifact is approximately 4574.86 years

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