A 25.0 gram piece of charcoal is found in some ruins of an ancient city. The sample shows a 14C activity, R, of 250 decays/min. How long (in years) has the tree this charcoal came from been dead?

I think I have the use the equation: R=lamda*No*e^(-lamda*t) but I am not sure how I can use the 25 grams of charcoal in this equation. I have already converted 250 decays/min. into decays/year.

To determine how long the tree has been dead, you can use the equation for radioactive decay.

The equation you mentioned, R = λ * N₀ * e^(-λt), is indeed the correct equation to use. Let's break it down:

- R represents the activity rate, which is the number of decays per unit time.
- λ (lambda) is the decay constant, a characteristic property of the radioactive isotope. It represents the probability of decay per unit time.
- N₀ is the initial number of radioactive nuclei.
- t is the time elapsed since the death of the tree.

In this case, the given information includes:
- R = 250 decays/min
- We want to find the time, t.

To incorporate the mass of the charcoal into the equation, we need to recognize that the initial number of radioactive nuclei (N₀) is proportional to the mass of the sample. The proportional constant is given by the specific activity of the radioactive isotope.

The specific activity (A) is defined as the decay rate per unit mass of the sample. In this case, the specific activity of 14C is 13.6 decays/min/g. Therefore, we can calculate the initial number of radioactive nuclei (N₀) using the mass of the charcoal (m) and the specific activity (A):

N₀ = A * m

N₀ = 13.6 (decays/min/g) * 25.0 (g)

Now we have the value of N₀, so we can rearrange the equation and solve for t:

R = λ * N₀ * e^(-λt)

250 (decays/year) = λ * [13.6 (decays/min/g) * 25.0 (g)] * e^(-λt)

To convert the decay rate from decays/min to decays/year, you need to multiply by the number of minutes in a year (assuming 365 days in a year and 24 hours in a day, thus 365 * 24 * 60 = 525,600 minutes per year).

λ = 0.693 / half-life

For 14C, the half-life is about 5730 years. Therefore:

λ = 0.693 / 5730

Now, plug in the values and solve for t to find the time since the tree has been dead.