Question

Radiocarbon Dating Carbon-14 is a radioactive isotope of carbon produced in the upper atmosphere by radiation from the sun. Plants absorb carbon dioxide from the air, and living organisms, in turn, eat the plants. The ratio of normal carbon (carbon-12) to carbon- 14 in the air and in living things at any given time is nearly constant. When a living creature dies, however, the carbon- 14 begins to decrease as a result of radioactive decay. By comparing the ameunts of carton-14 and carbon12 present, the amount of carbon- 14 that has decayed can therefore be ascertained. Let \(Q(t)\) denote the amount of carbon- 14 present at time \(t\) after death. If we assume its behavior is modeled by the differential equation \(Q^{\prime}(c)=-k Q(f)\), then \(Q(l)=Q(0) e^{-k t}\). Knowing the half-life of carbon- 14 , we can determine the constant \(k\). Given a specimen to be dated, we can measure its radioactive content and deduce \(Q(t)\). Knowing the amount of carbon- 12 present enables us to determine \(Q(0)\). Therefore, we can use the solution of the differential equation \(Q(t)=Q(0) e^{-k r}\) to deduce the age, \(f\), of the radicactive sample. (a) The half-life of carbon- 14 is nominally 5730 years. Suppose remains have been found in which it is estimated that \(30 \%\) of the original amount of carbon-14 is present. Fstimate the age of the remains. (b) The half-life of carbon- 14 is not known precisely. Let tus assume that its half-life is \(5730 \pm 30\) years. Determine how this half-life uncertainty affects the age estimate you computed in (a); that is, what is the corresponding uncertainty in the age of the remains? (c) It is claimed that radiocarbon dating cannot be used to date objects older than about 60,000 years. To appreciate this practical limitation, compute the ratio \(Q(60,000) / Q(0)\), assuming a half-life of 5730 years.

Short answer

Additionally, taking into account the uncertainty in the half-life of carbon-14 which has a range of \(5730 \pm 30\) years, what is the calculated age of the sample? Lastly, calculate the remaining ratio of carbon-14 after 60,000 years. Answer: The estimated age of the sample is approximately 9513 years, with an accounted uncertainty of \(\pm 13\) years, making the age of the remains \(9513 \pm 13\) years. After 60,000 years, approximately 0.2122% of the original carbon-14 remains in the sample. This represents the practical limitations of radiocarbon dating for objects older than 60,000 years.

Step-by-step solution

Determine the age of the remains

We are given the equation for radioactive decay: \(Q(t) = Q(0) e^{-kt}\). We know that the remaining amount of carbon-14 is 30% of the initial amount, which means \(Q(t) = 0.3Q(0)\). The half-life of carbon-14 is 5730 years, so \(Q(5730) = 0.5Q(0)\). We can solve for \(k\) using the half-life formula: \(0.5Q(0) = Q(0)e^{-5730k}\). Dividing both sides by \(Q(0)\) and taking the natural logarithm gives us, $$\ln{0.5} = -5730k$$Solving for \(k\), we get $$k = \frac{\ln{0.5}}{-5730}$$ Now we can use the equation \(0.3Q(0) = Q(0) e^{-kt}\), divide both sides by \(Q(0)\) and plug in the value for \(k\):$$0.3 = e^{-t\frac{\ln{0.5}}{-5730}}$$To solve for \(t\), take the natural logarithm of both sides:$$\ln{0.3} = -t\frac{\ln{0.5}}{-5730}$$Finally, solve for \(t\): $$t = \frac{\ln{0.3}}{\frac{\ln{0.5}}{-5730}} \approx 9513 \text{ years}$$

Determine the uncertainty in the age of the remains

We are given that the half-life of carbon-14 has a range of \(5730 \pm 30\) years. We will calculate the age of the remains using both the lower and upper bounds of the half-life range: \(5700\) years and \(5760\) years. First, the lower bound of half-life (5700 years):$$k_1 = \frac{\ln{0.5}}{-5700}$$Calculate the age using this value:$$t_1 = \frac{\ln{0.3}}{\frac{\ln{0.5}}{-5700}} \approx 9519 \text{ years}$$ Next, the upper bound of half-life (5760 years):$$k_2 = \frac{\ln{0.5}}{-5760}$$Calculate the age using this value:$$t_2 = \frac{\ln{0.3}}{\frac{\ln{0.5}}{-5760}} \approx 9506 \text{ years}$$ The uncertainty in the age of the remains is: $$\Delta t = t_2 - t_1 = 9506 - 9519 = -13 \text{ years}$$We can express the age of the remains as: $$9513 \pm 13 \text{ years}$$

Calculate the ratio \(Q(60,000)/Q(0)\)

We can use the equation for radioactive decay and the half-life of carbon-14 (assuming 5730 years) to compute the remaining ratio of carbon-14 after 60,000 years: $$\frac{Q(60,000)}{Q(0)} = e^{-60,000\frac{\ln{0.5}}{-5730}} \approx 0.002122$$ The ratio \(Q(60,000)/Q(0) \approx 0.002122\), meaning that, after 60,000 years, about 0.2122% of the original carbon-14 is still present in the sample. This demonstrates the practical limitation of radiocarbon dating for objects older than 60,000 years, as the remaining amount of carbon-14 becomes too small to accurately estimate the age of the sample.

Key concepts

Radioactive Decay

Radioactive decay is a natural process in which unstable atomic nuclei (radionuclides) lose energy by emitting particles or radiation. This decay occurs in a predictable pattern described by the half-life of the radioactive isotope, like carbon-14. It is this predictability that allows scientists to use radioactive materials as a 'clock' to measure the age of objects.

The process of decay happens due to the inherent instability in the nucleus of the atom. Over time, the unstable isotopes change into other elements or isotopes, a process that can be understood and quantified. For carbon-14, this radioactive decay is instrumental in the field of archaeology and geology for dating ancient organic materials, such as wood, bones, and shells, which have absorbed carbon-14 from the atmosphere during their life.

Carbon-14 Half-Life

The half-life of a radioactive isotope is the time it takes for half of the initial amount of the radioactive substance to decay. For carbon-14, the half-life is approximately 5730 years. This value is a cornerstone in radiocarbon dating because it provides a basis for calculating the age of a sample.

Understanding the concept of half-life is crucial because it helps in estimating how much time has passed since a living organism has died. If you know the half-life and the amount of carbon-14 remaining in a sample, you can calculate when the organism stopped exchanging carbon with its environment – the moment of its death. For example, if you find out that only 25% of the original carbon-14 amount is left in a sample, it would indicate that two half-lives have passed, which equals roughly 11,460 years.

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are supremely important in science and engineering because they help us model changes that occur over time or space. In the context of radiocarbon dating, a differential equation is used to establish the decay rate of carbon-14.

The equation \(Q'(t) = -kQ(t)\) is a first-order differential equation where the rate of change of \(Q\) with respect to time \(t\) is proportional to the current amount of \(Q\). The solution to this equation, \(Q(t) = Q(0)e^{-kt}\), reveals how the amount of carbon-14 changes over time. By solving this equation with known values, we can estimate the age of a sample containing carbon-14, a process which also demonstrates the applicability of calculus in understanding natural phenomena.

Exponential Decay

Exponential decay is a specific form of decrease that occurs at a rate proportional to the current value. In other words, larger amounts decrease more quickly, and the decrease slows down as the quantity gets smaller. This phenomenon is perfectly exhibited by the decay of radioactive substances like carbon-14.

The formula \(Q(t) = Q(0)e^{-kt}\) characterizes the exponential decay, where \(Q(t)\) represents the quantity at time \(t\), \(Q(0)\) is the initial quantity, \(k\) is the decay constant, and \(t\) is time. This formula is fundamental to radiocarbon dating as it allows calculation of the time passed since the death of the organism. The less carbon-14 present in the sample, the older it is. Because of exponential decay's continually slowing rate, there's an upper limit to the age that can be measured—beyond 60,000 years, there's simply not enough carbon-14 remaining in a sample to give a clear estimate of its age.