Carbon-14 Dating Numerical: Calculate the Age of Charcoal from Its Radioactive Activity

Given data:

Ratio of ¹⁴C to ¹²C in atmospheric CO₂ = 1.3 × 10⁻¹²

Half-life of ¹⁴C = 5730 years

Observed activity of the charcoal = 300 decays/min

Correct Answer: Approximately 4867 years

The tree from which the charcoal originated has been dead for approximately 4867 years. To solve this radiocarbon dating problem, we first calculate the original number of ¹⁴C atoms present when the tree was alive. From this value, we determine the initial radioactive activity of the carbon sample. Finally, the decrease in activity from the initial value to the observed value of 300 decays per minute is used to calculate the time elapsed since the death of the tree.

This problem is slightly more advanced than a standard Carbon-14 half-life question because the initial activity is not directly provided. It must first be calculated using the mass of carbon, the atmospheric ¹⁴C/¹²C ratio, and the radioactive decay constant.

Understanding the Given Information

The charcoal sample contains 36 g of carbon and currently produces 300 radioactive decays per minute. While the tree was alive, its carbon isotope ratio was assumed to be in equilibrium with atmospheric carbon. The given atmospheric ratio is:

¹⁴C/¹²C = 1.3 × 10⁻¹²

The half-life of Carbon-14 is:

t₁/₂ = 5730 years

The calculation requires three major steps. First, we determine the number of ¹²C atoms in 36 g of carbon. Second, we use the given isotope ratio to calculate the initial number of ¹⁴C atoms. Third, we calculate the initial activity and compare it with the present activity to determine the age of the charcoal.

Step-by-Step Calculation of the Age of the Charcoal

Step 1: Calculate the Number of Moles of Carbon

The charcoal contains 36 g of carbon. Taking the molar mass of ¹²C as 12 g mol⁻¹:

Number of moles = Mass/Molar mass

Therefore:

Number of moles = 36/12 = 3 mol

Thus, the charcoal sample contains approximately 3 moles of carbon atoms.

Step 2: Calculate the Number of ¹²C Atoms

One mole contains Avogadro’s number of particles:

Nₐ = 6.022 × 10²³ atoms mol⁻¹

Therefore, the number of ¹²C atoms in 3 moles is:

Number of ¹²C atoms = 3 × 6.022 × 10²³

Number of ¹²C atoms = 1.8066 × 10²⁴ atoms

Thus, the sample originally contained approximately:

N(¹²C) = 1.8066 × 10²⁴ atoms

Because ¹⁴C is present only in a tiny proportion, treating the 36 g sample as essentially ¹²C is an excellent approximation.

Step 3: Calculate the Initial Number of ¹⁴C Atoms

The atmospheric isotope ratio is given as:

¹⁴C/¹²C = 1.3 × 10⁻¹²

Therefore:

N₀(¹⁴C) = 1.3 × 10⁻¹² × N(¹²C)

Substituting the number of ¹²C atoms:

N₀(¹⁴C) = 1.3 × 10⁻¹² × 1.8066 × 10²⁴

Therefore:

N₀(¹⁴C) = 2.34858 × 10¹² atoms

Thus, when the tree was alive, the carbon corresponding to the present charcoal sample contained approximately:

2.35 × 10¹² atoms of ¹⁴C

Calculating the Decay Constant of Carbon-14

The radioactive decay constant is related to the half-life by the equation:

λ = 0.693/t₁/₂

Since the observed activity is given in decays per minute, it is convenient to express the half-life in minutes.

The half-life is:

5730 years

Using:

1 year = 365 days

1 day = 24 hours

1 hour = 60 minutes

the half-life in minutes is:

t₁/₂ = 5730 × 365 × 24 × 60

t₁/₂ = 3.011688 × 10⁹ minutes

Therefore:

λ = 0.693/(3.011688 × 10⁹)

λ ≈ 2.301 × 10⁻¹⁰ min⁻¹

This value represents the probability of decay per unit time for a ¹⁴C nucleus.

Calculating the Initial Activity of the Living Tree

Radioactive activity is related to the number of radioactive atoms by:

A = λN

At the time the tree died:

A₀ = λN₀

Substituting the calculated values:

A₀ = (2.301 × 10⁻¹⁰) × (2.34858 × 10¹²)

Therefore:

A₀ ≈ 540.5 decays/min

Thus, the carbon sample would have had an activity of approximately 540.5 decays per minute when the tree was alive.

The charcoal currently shows an activity of:

A = 300 decays/min

The decrease from approximately 540.5 decays/min to 300 decays/min occurred because Carbon-14 underwent radioactive decay after the death of the tree.

Using the Radioactive Activity Equation to Calculate Age

Radioactive activity decreases exponentially according to the equation:

A = A₀e⁻ˡᵗ

where A is the present activity, A₀ is the initial activity, λ is the decay constant, and t is the elapsed time.

For Carbon-14 dating, the equation can also be written in terms of half-life:

A/A₀ = (1/2)^(t/5730)

Substituting the values:

300/540.5 = (1/2)^(t/5730)

Therefore:

0.555 = (1/2)^(t/5730)

Taking natural logarithms:

ln(0.555) = (t/5730) ln(1/2)

Rearranging:

t = 5730 × [ln(0.555)/ln(0.5)]

This gives:

t ≈ 4867 years

Therefore, the tree has been dead for approximately 4867 years.

Alternative Calculation Using the Decay Constant Formula

The same answer can be obtained directly from:

A = A₀e⁻ˡᵗ

Rearranging the equation:

A₀/A = eˡᵗ

Taking the natural logarithm:

ln(A₀/A) = λt

Therefore:

t = (1/λ) ln(A₀/A)

Using the decay constant in years:

λ = 0.693/5730 year⁻¹

and substituting:

t = (5730/0.693) ln(540.5/300)

The activity ratio is:

540.5/300 ≈ 1.8017

Therefore:

t = (5730/0.693) × ln(1.8017)

t ≈ 4867 years

This confirms the answer obtained by the half-life form of the radioactive decay equation.

Why Must the Initial Carbon-14 Activity Be Calculated First?

In many simple radiocarbon dating questions, both the initial and present amounts of ¹⁴C are given directly. In this question, however, only the present activity is given.

The initial activity must therefore be reconstructed from the information about the living tree. While alive, the tree continuously exchanged carbon with the atmosphere through photosynthesis. Its ¹⁴C/¹²C ratio was therefore assumed to match the atmospheric value given in the question.

The calculation follows this sequence:

Mass of carbon → Number of ¹²C atoms → Number of ¹⁴C atoms → Initial activity → Age of charcoal

Understanding this sequence is essential because directly using 300 decays/min without calculating the initial activity would not provide enough information to determine the age.

Why Does Carbon-14 Activity Decrease After the Tree Dies?

While a tree is alive, it continuously absorbs carbon dioxide during photosynthesis. This carbon includes both stable carbon isotopes and a very small quantity of radioactive ¹⁴C.

As long as the tree remains alive, carbon exchange with the environment continues. After death, however, the tree no longer absorbs atmospheric carbon dioxide. No fresh Carbon-14 enters its tissues.

The existing ¹⁴C atoms continue to decay, causing the total number of radioactive atoms and the measured activity to decrease with time.

Therefore:

Living tree → continuous carbon exchange

Dead tree → carbon exchange stops

After death → ¹⁴C decreases by radioactive decay

The amount of Carbon-14 remaining therefore acts as a natural radioactive clock.

Relationship Between Radioactive Activity and Number of Carbon-14 Atoms

Radioactive activity is defined as the number of nuclear decays occurring per unit time. It is given by:

A = λN

where A is the activity, λ is the decay constant, and N is the number of radioactive atoms.

This equation shows that activity is directly proportional to the number of ¹⁴C atoms present. If the number of radioactive atoms decreases, the activity also decreases in exactly the same proportion.

Therefore:

A/A₀ = N/N₀

This is why radioactive activity can be used directly in the Carbon-14 decay equation. The ratio of present activity to initial activity behaves exactly like the ratio of the present number of ¹⁴C atoms to the initial number.

Why the Age Is Less Than One Carbon-14 Half-Life

The half-life of Carbon-14 is 5730 years. After one complete half-life, the activity would fall to exactly half of its initial value.

The initial activity calculated for the sample is approximately:

A₀ = 540.5 decays/min

After one half-life, the expected activity would be:

540.5/2 = 270.25 decays/min

However, the measured activity is:

300 decays/min

Since 300 decays/min is greater than 270.25 decays/min, less than one complete half-life has elapsed.

Therefore, the calculated age must be less than 5730 years. The answer of approximately 4867 years is consistent with this expectation.

Significance of the ¹⁴C to ¹²C Ratio

The given atmospheric ratio:

¹⁴C/¹²C = 1.3 × 10⁻¹²

means that Carbon-14 is extremely rare compared with Carbon-12. For every enormous number of ¹²C atoms, only a tiny number of ¹⁴C atoms are present.

Despite this low abundance, ¹⁴C is useful for dating because it is radioactive and decays at a predictable rate. Carbon-12 is stable and provides the reference against which the initial Carbon-14 abundance can be estimated.

In this question, the isotope ratio allows us to calculate the original number of radioactive ¹⁴C atoms in the 36 g carbon sample.

Complete Calculation at a Glance

The 36 g carbon sample contains:

36/12 = 3 mol of carbon

Therefore, the number of ¹²C atoms is:

3 × 6.022 × 10²³ = 1.8066 × 10²⁴ atoms

Using the atmospheric ratio:

N₀(¹⁴C) = 1.3 × 10⁻¹² × 1.8066 × 10²⁴

N₀(¹⁴C) = 2.34858 × 10¹² atoms

The Carbon-14 decay constant is:

λ = 0.693/(5730 × 365 × 24 × 60)

λ ≈ 2.301 × 10⁻¹⁰ min⁻¹

Therefore, the initial activity was:

A₀ = λN₀

A₀ ≈ 540.5 decays/min

Using the present activity of 300 decays/min:

300/540.5 = (1/2)^(t/5730)

Solving for t:

t ≈ 4867 years

Final Answer

The original 36 g carbon sample contained approximately 2.35 × 10¹² atoms of ¹⁴C when the tree was alive. This corresponds to an initial activity of approximately 540.5 decays per minute.

The present activity is only 300 decays per minute. Applying the radioactive decay equation with a Carbon-14 half-life of 5730 years gives:

Age of charcoal ≈ 4867 years

Therefore, the tree from which the charcoal came has been dead for approximately:

Correct Answer: 4867 years