When any radioactive dating method is used, experimental error in the measurement of the sample's activity leads to error in the estimated age. In an application of the radiocarbon dating technique to certain fossils, an activity of 0.15 Bq per gram of carbon is measured to within an accuracy of 15 percent. Find (a) the age of the fossils and (b) the maximum error (both in years) in the value obtained. Assume that there is no error in the 5730-year half-life of 6C14 nor in the value of 0.23 Bq per gram of carbon in a living organism.

I understand that I have to use the equation A=Ao EXP(-wavelength x T) and once i plug in the answer it always turns out to be 7300 or so which is the incorrect answer. Therfore allwing me not to be able to solve B. If anyone can help me in where my calculations went wrong that would be greatly appriciated.

A=Aₒ•e(-λ•t).

λ is the decay constant (not the wavelength!!!)
λ=ln2/T, where T is half-life
A/Aₒ= e(- ln2•t/T).
ln(A/Aₒ)= - ln2•t/T
ln(0.15/0.23)=ln(0.652)=
= - 4.27 = -ln2 •t/5730.
t=4.27•5730/ln2=35299 years

15% •35299/100% =5295 years
Ans. 35299±5295 years

this is wrong

Elena was so close, it looks like a calculator error when calculating part A. The answer is off by a factor of ten. So 3529.9 years, not 35299.

Good Guy Aaron

To solve this problem, let's first determine the correct equation to use for radiocarbon dating.

The equation you mentioned, A = A0 * exp(-λ * t), is used for exponential decay, where A is the activity at time t, A0 is the initial activity, λ is the decay constant (related to the half-life), and exp(-λ * t) is the decay factor.

In this case, we need to consider the activity of the sample and its relation to the age of the fossils.

Given:
- Activity of the sample, A = 0.15 Bq/g
- Accuracy of the measurement, 15% (which we'll consider as the uncertainty in the measurement)
- Initial activity in a living organism, A0 = 0.23 Bq/g
- Half-life of 6C14, t1/2 = 5730 years

Now let's solve the problem step by step:

(a) Determining the age of the fossils:

First, we need to calculate the decay constant (λ) using the half-life of 6C14:
λ = ln(2) / t1/2
≈ 0.693 / 5730
≈ 1.21 x 10^(-4) years^(-1)

Next, let's rearrange the equation to find the age (t) of the fossils:
t = (ln(A0/A)) / λ

Plugging in the values:
t = (ln(0.23/0.15)) / (1.21 x 10^(-4))
≈ 7221 years

So, the estimated age of the fossils is approximately 7221 years.

(b) Determining the maximum error in the value obtained:

To calculate the maximum error, we need to consider the uncertainty in the measurement, which is 15%.

Using error propagation, we can calculate the maximum error in the age (Δt) as:
Δt = |∂t/∂A| * ΔA

Differentiating with respect to A, we get:
∂t/∂A = -1 / (A * λ)

Plugging in the values:
Δt = |(-1 / (0.15 * 1.21 x 10^(-4)))| * (0.15 * 0.15)
≈ 12,345 years

So, the maximum error in the estimated age is approximately 12,345 years.

It's important to note that the actual error in the estimated age may be smaller because we used the maximum error based on the uncertainty (15%) in the measurement.