a piece of charcoal containing 25.0g of carbon is found in some remains of ancient city. The sample shows carbon 14 activity R of 250decays/min. How long has the tree from which this charcoal came from been dead?
To determine the age of the tree from which the charcoal sample came, we need to make use of the concept of half-life in radioactive decay.
The half-life of carbon-14 (C-14) is approximately 5730 years. This means that in 5730 years, half of the amount of C-14 in a sample will decay. By measuring the remaining activity of C-14, we can calculate the time it took for the tree to reach its current state.
First, we need to find the initial activity (R₀) of the charcoal when it was alive. We know the present activity (R) is 250 decays per minute, but we need to find R₀. The relationship between the activities and the amounts of C-14 can be described by the decay equation:
R = R₀ * (1/2)^(t / T)
Where R is the present activity, R₀ is the initial activity, t is the time in years, and T is the half-life of C-14.
To find R₀, we rearrange the equation:
R₀ = R * (2)^(t / T)
Now we can substitute the known values into the equation:
R₀ = 250 * (2)^(t / 5730)
Next, we need to find the number of moles (n) of carbon-14 in the charcoal sample when it was alive. We can use the atomic mass of carbon and the mass of carbon in the sample to calculate this:
n = mass of carbon / atomic mass of carbon
Since the mass of carbon in the sample is 25.0 grams and the atomic mass of carbon is 12.01 g/mol:
n = 25.0 g / 12.01 g/mol = 2.08 mol
Finally, we can use the formula for the activity of a radioactive sample:
R₀ = λ * N
Where R₀ is the initial activity, λ is the decay constant, and N is the number of radioactive atoms.
Rearranging the equation, we get:
λ = R₀ / N
Substituting the values:
λ = R₀ / (2.08 mol * 6.022 x 10^23 atoms/mol)
Now we know the decay constant, which is given by:
λ = ln(2) / T
Substituting T = 5730 years, we can solve for t, the time the tree has been dead:
t = (5730 years) * ln(2) / λ
Plugging in the value of λ we found earlier, we can calculate t.
To determine how long the tree has been dead, we can use the concept of radioactive decay and the half-life of carbon-14.
The half-life of carbon-14 is approximately 5730 years. It means that after each half-life, the amount of carbon-14 in the sample decreases by half.
We are given that the carbon-14 activity (R) is 250 decays/min, but we need to convert it to the number of carbon-14 atoms remaining in the sample.
The decay rate (lambda) can be calculated using the following formula:
lambda = R / N
where R is the activity and N is Avogadro's number (6.022 x 10^23 decays/mol).
Let's calculate the decay rate:
lambda = 250 decays/min / (6.022 x 10^23 decays/mol) ≈ 4.15 x 10^-22 decays/min/mol
Next, we need to determine the moles of carbon-14 in the sample. We can use the molar mass of carbon-14 (14.007 g/mol) to convert the mass of carbon (25.0 g) into moles:
moles of carbon-14 = 25.0 g / 14.007 g/mol ≈ 1.785 moles
Now, let's calculate the number of carbon-14 atoms in the sample by multiplying the moles by Avogadro's number:
number of carbon-14 atoms = 1.785 moles x (6.022 x 10^23 atoms/mol) ≈ 1.074 x 10^24 atoms
Since the half-life of carbon-14 is 5730 years, we can determine the number of half-lives that have occurred:
number of half-lives = (ln initial amount / ln 2)
number of half-lives = ln(1.074 x 10^24 atoms) / ln(2)
Now we can calculate the time using the formula:
time = number of half-lives x half-life
Let's calculate it step by step:
Step 1: Calculate the number of half-lives
number of half-lives = ln(1.074 x 10^24 atoms) / ln(2)
Step 2: Calculate the time
time = number of half-lives x half-life
Finally, we can substitute the value of the half-life (5730 years) to calculate the time in years:
time = number of half-lives x 5730 years
Please note that the calculation of the number of half-lives involves the natural logarithm (ln).