Radioactive Dating — Question 1

Question 1

A carbon activity ratio gives an ideal closed-system age

A preserved organic sample has carbon-specific activity 3.00 counts per minute per gram under a fixed counting setup. A modern reference measured with identical efficiency has 12.0 counts per minute per gram. Use carbon-14 half-life 5730 y, equal initial isotope ratios and no contamination.

1. Derive the age from the activity ratio.

2. Predict the sample-to-reference ratio after a further 5730 y and explain why total sample mass is not needed.

Question 1 - Solution

A carbon activity ratio gives an ideal closed-system age

(a) Method and calculation

$$f=\mathcal A/\mathcal A_0=e^{-\lambda t}\Rightarrow\boxed{t=\frac{T_{1/2}}{\ln2}\ln(\mathcal A_0/\mathcal A)}.$$$$\boxed{f=3.00/12.0=.250,\quad t=2(5730)=11460\,\mathrm y}.$$

(b) Interpretation and further calculation

$$\boxed{f\prime=f/2=.125}.$$$$\mathcal A_{\mathrm{specific}}=\lambda N_{14}/m_C\Rightarrow\boxed{\text{carbon-mass normalization removes sample-size dependence}}.$$

Physical check

This is an ideal decay-model age. Calendar dating requires calibration of past atmospheric isotope ratios and appropriate reservoir corrections.