How Old Is the Wood Sample Based on Carbon-14 Decay?

[plexus0208]

Homework Statement

Background info: The first order rate of nuclear decay of an isotope depends only upon the isotope, not its chemical form or temperature. The half-life for decay of carbon-14 is 5730 years. Assume that the amount of C-14 present in the atmosphere as CO2 and therefore in a living organism has been constant for the last 50,000 years. An ancient sample containing C-14 will show fewer disintegrations of the C-14 that is present than a modern sample because the concentration of C-14 is lower in the ancient sample.

If a 1.00 gram sample of wood found in an archaelogical site in Arizona underwent 7.90x10³
disintegrations in a given time period (e.g., 20 h) and a modern sample underwent 1.84x10⁴ disintegrations in the same time period, how old is the ancient sample?

Homework Equations

First order:
ln[A]_t = -kt + ln[A]_o
[A]_t = e^-kt[A]_o

ln(([A]_o/2)/[A]_o) = -kt_1/2 = ln(1/2)
or ln2 = kt_1/2 = 0.693

The Attempt at a Solution

kt_1/2 = 0.693
k = 0.693/5730 = 1.21x10^-4

ln[A]_t = -kt + ln[A]_o
ln[A]_t = ?
ln[A]_o = ?
Solve for t?
Is this the right equation to use?

 

[Borek]

kt_1/2 = 0.693
t_1/2 = 0.693/5730 = 1.21x10^-4

t_1/2 is given, simple mistake here.

Other than that go for

$$\frac {A_t} {A_0} = e^{-kt}$$

and it becomes almost simple plug and chug.

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[plexus0208]

What do I use for At and A0?
The number of disintegrations?

 

[Borek]

Yes. You are interested in ratio of activities.

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methods