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The carbon−14 decay rate of a sample obtained from a young tree is 0.266 disintegration per second per gram of the sample. Another wood sample prepared from an object recovered at an archaeological excavation gives a decay rate of 0.178 disintegration per second per gram of the sample. What is the age of the object? (The half-life of carbon−14 is 5715 years.) × 10 years Enter your answer in scientific notation.

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Answer:

  • 3.30 × 10³ years

Explanation:

1) Carbon-14 disintegration rate of a young (alive) tree: 0.266 per second per gram.

2) Carbon-14 disintegration rate of a wood sample prepeared from an object recovered at an archaelogical excavation (dead matter): 0.178 per second per gram.

3) Ratio of decay: A/A₀ = 0.178 / 0.266 = 0.669

4) Half-life equation: A/A₀ = (1/2)ⁿ, where n is the number of half-lives since the object died.

5) Substitute and solve for n:

  • 0.669 = (1/2)ⁿ

[tex]nlog(1/2)=log(0.669)\\ \\n=\frac{log(0.669)}{log(1/2)}\\ \\n=0.578[/tex]

That means that 0.578 half-life has elapsed since the wood with which the object was created was dead matter.

6) Convert number of half-lives to years:

  • 0.578 half-life × 5715 years/half-life = 3,303 years = 3.30 × 10³ years.

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