Problem 2

Dating historical events using

^{210}\ce{Pb}

. Nathan Thompson, an early settler of Lord Howe Island, planted European deciduous trees; the year is unknown. Pollen of European oak and elm (which pollinate in their first year) accumulated in lake sediment together with radioactive

^{210}\ce{Pb}

(half-life = 22.0 years). In 1995 a sediment core was taken: oak/elm pollen first occurs at 50 cm depth, where the

^{210}\ce{Pb}

activity is 1.40 Bq kg⁻¹, versus 356 Bq kg⁻¹ at the top of the core. (2.1) In what year were the seeds planted? (2.2)

^{210}\ce{Pb}

is a daughter of

\ce{^{238}U}

, which stays in the earth's crust:

\ce{^{238}U} \rightarrow \ce{^{234}U} \rightarrow \ce{^{230}Th} \rightarrow \ce{^{226}Ra} \rightarrow \ce{^{222}Rn} \rightarrow (\ce{^{218}Po}, \ce{^{214}Bi}) \rightarrow \ce{^{210}Pb} \rightarrow \ce{^{206}Pb}

(stable). Which step explains how

^{210}\ce{Pb}

reaches rainwater while its parent

\ce{^{238}U}

remains in the crust?

Step 1 of 2: Counting half-lives

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n = \log_2\frac{356}{1.40} = \frac{\ln 254}{\ln 2} \approx 8\ \text{half-lives};\qquad t = 8 \times 22.0 = 176\ \mathrm{years}\ \Rightarrow\ 1995 - 176 = 1819

Analysis

Each half-life halves the activity: 356 → 178 → 89 → 44.5 → 22.2 → 11.1 → 5.6 → 2.8 → 1.4 Bq kg⁻¹ — exactly eight halvings reach the pollen horizon. The 50 cm layer was therefore deposited $176 \pm 2$ years before 1995, dating the planting to about 1819.