Problem 24 - Calculus

Question

The radioactive isotope carbon- 14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to stable carbon- 12 at a rate proportional to the amount of carbon-14 present, with a half-life of 5730 years. Suppose \(C(t)\) is the amount of carbon- 14 present at time \(t.\) (a) Find the value of the constant \(k\) in the differential equation \(C^{\prime}=-k C.\) (b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained \(91 \%\) of the amount of carbon-14 contained in freshly made cloth of the same material. \(^{8}\) How old is the Shroud of Turin, according to these data?

Step-by-Step Solution

Answer

(a) \( k \approx 1.2097 \times 10^{-4} \text{ year}^{-1} \). (b) The Shroud of Turin is about 765 years old.

Step 1: Understand the Differential Equation

The decay of carbon-14 is modeled using the differential equation \( C'(t) = -kC(t) \), where \( k \) is a positive constant representing the rate of decay. Our initial task is to determine \( k \) using the given half-life of carbon-14, which is 5730 years.

Step 2: Relate Half-Life to the Constant k

The half-life of a radioactive substance is the time required for half of the substance to decay. For carbon-14, the half-life \( T_{1/2} \) is given as 5730 years. The relationship between the decay constant \( k \) and the half-life is given by the equation \( T_{1/2} = \frac{\ln(2)}{k} \).

Step 3: Calculate k

Substitute the value of the half-life into the formula to find \( k \):\[ k = \frac{\ln(2)}{5730} \approx 1.2097 \times 10^{-4} \text{ year}^{-1}. \]

Step 4: Formulate the Exponential Decay Model

The amount of carbon-14 at any time \( t \) can be expressed as \( C(t) = C_0 e^{-kt} \), where \( C_0 \) is the initial amount of carbon-14 and \( t \) is the time in years since death.

Step 5: Use Given Information to Solve for Time

We are told that the Shroud of Turin contains 91% of its original carbon-14, so \( C(t) = 0.91 C_0 \). Substitute this into the decay formula:\[ 0.91 C_0 = C_0 e^{-kt}. \]Simplify by cancelling \( C_0 \) from both sides to get:\[ 0.91 = e^{-kt}. \]

Step 6: Solve for t

To find \( t \), take the natural logarithm of both sides:\[ \ln(0.91) = -kt. \]Solve for \( t \):\[ t = \frac{\ln(0.91)}{-k}. \]Substitute \( k \approx 1.2097 \times 10^{-4} \):\[ t \approx \frac{\ln(0.91)}{-1.2097 \times 10^{-4}} \approx 764.61 \text{ years}. \]

Step 7: Conclusion

The Shroud of Turin is approximately 765 years old according to the carbon-14 data.

Key Concepts

Carbon-14

Carbon-14 is a unique radioactive isotope of carbon that is found in all living organisms. It gets replenished while the organism is alive through various natural processes. Once the organism dies, the Carbon-14 starts to decay into the more stable Carbon-12. This process occurs at a known decay rate, making Carbon-14 ideal for dating ancient biological samples.

• Carbon-14 is continuously absorbed by living organisms.
• Upon death, the absorption stops and the decay process begins.
• The decay transforms Carbon-14 into Carbon-12.

The predictable decay of Carbon-14 allows scientists to estimate the time that has passed since the organism died. This technique is widely used in archaeology and geology, known as Carbon Dating. Understanding how Carbon-14 decays over time involves evaluating its exponential decay behavior, which is where differential equations come into play.

Differential Equations

Differential equations are mathematical tools used to model the rate of change of quantities. In the case of radioactive decay, differential equations describe how the quantity of a substance, like Carbon-14, decreases over time.
The equation used is:

• \( C'(t) = -kC(t) \)

- Here, \( C(t) \) is the amount of Carbon-14 present at time \( t \).
- \( k \) is the decay constant, indicating the rate at which Carbon-14 decays.
This equation tells us that the rate of decay of Carbon-14 is proportional to the amount of Carbon-14 present at any given time. The decay process follows an exponential pattern described by the equation:

• \( C(t) = C_0 e^{-kt} \)

- \( C_0 \) represents the initial amount of Carbon-14.
Solving these equations and using known values, like the half-life, allows us to calculate how much time has passed since the death of an organism.

Half-Life

Half-life is a crucial concept in understanding radioactive decay. It is defined as the time it takes for half of the radioactive substance to decay. For Carbon-14, this is notably 5730 years.

• Half-life is constant and specific for each radioactive substance.
• For Carbon-14, after 5730 years, only half of an initial amount remains.

Using the relationship between half-life and the decay constant, we can derive the decay constant with the formula:

• \( T_{1/2} = \frac{\ln(2)}{k} \)

With Carbon-14's half-life known, calculating \( k \) gives us a decay constant of approximately \( 1.2097 \times 10^{-4} \text{ year}^{-1} \).
This decay constant is a vital piece of information that, combined with the natural logarithm method, allows researchers to determine the age of a sample by observing how much Carbon-14 remains compared to its initial quantity. This technique was pivotal in dating the Shroud of Turin and determining its age.