A sample of charcoal from an archaeological site contains 55 g of carbon and decays at a rate of .877 Bq. How old is it?

ok I am totally lost, I don't know where to start

To calculate the age of the charcoal sample, we can use the concept of radioactive decay. Carbon-14 (C-14) is a radioactive isotope that is commonly used for dating archaeological samples. It decays at a known rate, which allows us to determine the age of the sample.

The decay of C-14 can be described by the following equation:

N(t) = N₀ * e^(-λt)

Where:
N(t) = the current amount of C-14 in the sample
N₀ = the initial amount of C-14 in the sample
λ = the decay constant
t = time in years

In this case, we are given that the current amount of carbon in the sample is 55 g, and it is decaying at a rate of 0.877 Bq (Becquerels).

Step 1: Determine the initial amount of C-14 (N₀).

The initial amount of C-14 can be calculated based on the assumption that the amount of C-14 in living organisms is constant and in equilibrium with the environment. This is known as the "carbon-14 activity of modern standard" (Aₒ) and is approximately 15.3 disintegrations per minute (dpm) per gram of carbon.

So, in the given sample, the initial amount of C-14 (N₀) can be calculated as follows:

N₀ = (0.877 Bq / 15.3 dpm) * 55 g

Step 2: Calculate the decay constant (λ).

The decay constant (λ) is a measure of the rate at which the radioactive isotope decays. For C-14, the decay constant is approximately 1.21 x 10^-4 years⁻¹.

Step 3: Calculate the age of the sample (t).

We can rearrange the decay equation to solve for time (t):

t = (-1 / λ) * ln(N(t) / N₀)

Using the values we have, we can now calculate the age of the sample:

t = (-1 / (1.21 x 10^-4 years⁻¹)) * ln(0.877 Bq / N₀)

Substituting the value of N₀ from step 1:

t = (-1 / (1.21 x 10^-4 years⁻¹)) * ln(0.877 Bq / ((0.877 Bq / 15.3 dpm) * 55 g))

After performing the calculations, you should be able to determine the age of the charcoal sample.

No worries! I can help you break down the problem and guide you on how to solve it step by step.

To determine the age of the charcoal sample, we can use the concept of carbon dating. Carbon-14, a radioactive isotope of carbon, decays over time in a predictable way. The rate at which it decays is measured in Becquerel (Bq).

The decay rate mentioned in the question (.877 Bq) represents the number of carbon-14 atoms that decay per second.

Now, there are several variables involved in carbon dating calculations, but we can simplify the equation to solve for the age of the sample.

The general equation for carbon dating is:
N(t) = N_0 * e^(-λt)

Where:
N(t) is the current amount of carbon-14 in the sample,
N_0 is the initial amount of carbon-14 in the sample,
e is the base of the natural logarithm (approximately 2.71828),
λ (lambda) is the decay constant of carbon-14 (approximately 0.693/5730 years), and
t represents the age of the sample in years.

To solve for t, we need the initial amount of carbon-14 in the sample (N_0). From the question, we are given the mass of carbon (55g), and we know that carbon-14 is a specific isotope of carbon. We can assume that the proportion of carbon-14 to normal carbon (carbon-12) in the atmosphere is constant.

Carbon-14 makes up about 1 part per trillion (1/10^12) of the carbon in the atmosphere, and carbon-12 makes up the majority.

To determine the initial amount of carbon-14 in the sample, we can use the fact that 1 mol of carbon-12 weighs approximately 12 grams.

Let's calculate the number of moles of carbon-14 in the sample:
(55g carbon) / (12g/mol of carbon-12) = 4.58 mol of carbon-12

Since the proportion of carbon-14 to carbon-12 is 1/10^12, the moles of carbon-14 can be determined as:
(4.58 mol of carbon-12) / (1 x 10^12) = 4.58 x 10^(-12) mol of carbon-14

Now that we have N_0, we can use the decay equation to calculate the age (t) of the sample.

Substituting the values into the equation:
N(t) = (4.58 x 10^(-12)) * e^(-λt)

We are given the decay rate (.877 Bq), so N(t) can be expressed as the decay rate multiplied by the age of the sample (t):
N(t) = (.877 Bq) * t

Now we can set up the equation:
(.877 Bq) * t = (4.58 x 10^(-12)) * e^(-λt)

To solve for t, we need to use numerical methods, such as iteration or approximation, since there is no analytical solution to this equation.

Alternatively, you can use online carbon dating calculators or specialized software specifically designed for this purpose, which will automatically solve the equation and provide you with the age of the sample based on the given variables.

Remember that this explanation provides a general understanding of the process involved in carbon dating and how to approach similar problems.