Problem 61 - Chemistry Structure and Dynamics

Question

The activity of the \({ }^{14} \mathrm{C}\) in living tissue is 15.3 disintegrations per minute per gram of carbon. The limit for reliable determination of \({ }^{14} \mathrm{C}\) ages is 0.10 disintegration per minute per gram of carbon. Calculate the maximum age of a sample that can be dated accurately by radiocarbon dating if the half-life for the decay of \({ }^{14} \mathrm{C}\) is 5730 years.

Step-by-Step Solution

Answer

The maximum age for reliable dating is approximately 41609 years.

Step 1: Understand the Problem

We need to determine the maximum age of a sample that can be reliably dated using carbon-14 dating, given the activity levels and the half-life of carbon-14. We are given:- The initial activity (activity in living organisms) of \({}^{14} \text{C} = 15.3\text{ dpm/g} \text{C}\).- The minimum activity level still reliable is \(0.10\text{ dpm/g} \text{C}\).- The half-life of \({}^{14} \text{C} = 5730 \text{ years}\).Our goal is to find the time at which the activity decays from 15.3 to 0.10 dpm/g C.

Step 2: Use the Decay Formula

The activity of a radioactive isotope decreases over time according to the first-order decay formula:\[ A(t) = A_0 \times \left( \frac{1}{2} \right)^{t/T_{1/2}} \]where:- \(A(t)\) is the activity at time \(t\),- \(A_0\) is the initial activity,- \(T_{1/2}\) is the half-life of the isotope.Substitute the given values into the formula to set up an equation:\[ 0.10 = 15.3 \times \left( \frac{1}{2} \right)^{t/5730} \]

Step 3: Solve for Time (t)

First, divide both sides by 15.3:\[ \frac{0.10}{15.3} = \left( \frac{1}{2} \right)^{t/5730} \]Calculating the left side gives:\[ \frac{0.10}{15.3} \approx 0.00654 \]Next, solve the equation by taking the logarithm of both sides:\[ \log_{10}(0.00654) = \left( \frac{t}{5730} \right) \log_{10}\left(\frac{1}{2}\right) \]Solving for \(t\):\[ t = \frac{\log_{10}(0.00654)}{\log_{10}(0.5)} \times 5730 \]

Step 4: Calculate the Result

Start by calculating the logarithms:\[ \log_{10}(0.00654) \approx -2.184 \]\[ \log_{10}(0.5) \approx -0.3010 \]Substitute these into the equation to find \(t\):\[ t \approx \frac{-2.184}{-0.3010} \times 5730 \]Calculate the quotient:\[ t \approx 7.257 \times 5730 \]Finally, calculate the product:\[ t \approx 41609 \text{ years} \]

Step 5: Final Confirmation

The result for the maximum age a sample can be reliably dated using carbon-14 dating, with the given data, is approximately 41609 years. This considers the half-life of carbon-14 and the given activity levels.

Key Concepts

Carbon-14 Decay

Carbon-14 decay is a fascinating natural process. It helps scientists understand the ages of ancient artifacts, fossils, and geological features. This decay refers to how carbon-14 ( ^{14}C) breaks down over time inside an organic material.

When something was alive—like a plant or animal—it absorbed carbon from its environment. When it stops living, its carbon-14 content begins to decrease. This decay occurs because carbon-14 is a radioactive isotope, which means it is unstable and loses energy. It does so by emitting radiation and ultimately transforming into nitrogen-14 over a very long period.

During this process, the carbon-14 atom undergoes a change in its nucleus, leading to the loss of a part of itself, known as a decay event. Scientists measure this activity over time to calculate how long it has been since the organism died.

• It provides us with a powerful clock for understanding historical events.
• This principle underlies the radiocarbon dating technique.

Half-Life Calculation

The concept of half-life is crucial for understanding radioactive decay, especially for carbon-14. The half-life of an isotope is the time it takes for half of a given sample to decay.

For carbon-14, this duration is approximately 5730 years. This means if you started with a certain amount of carbon-14, in 5730 years, only half of it would be left. Another 5730 years later, you'd be left with half of that amount again, or a quarter of the original.

Understanding half-life helps in calculating the ages of samples.

• It allows scientists to predict how much carbon-14 remains in a sample at any given time.
• Using the half-life formula, they can set up equations to find the time elapsed since the organism was alive.

The half-life calculation involves the decay formula:\[ A(t) = A_0 \times \left( \frac{1}{2} \right)^{t/T_{1/2}} \]Where:

• \(A(t)\) is the remaining activity at time \(t\).
• \(A_0\) is the initial activity.
• \(T_{1/2}\) is the isotope's half-life.

Radioactive Isotope

A radioactive isotope, also known as a radioisotope, is an isotope of an element that has an unstable nucleus. This instability causes the nucleus to release energy by emitting radiation in the form of particles or electromagnetic waves.

One well-known example is carbon-14, which is utilized extensively in radiocarbon dating. What makes an isotope radioactive is that its nucleus has either too many or too few neutrons compared to protons, making it energetically unstable.

Radioactive isotopes have:

• The ability to decay over time into a different element or isotope, releasing radiation in the process.
• Specific decay rates which help scientists determine the age of objects containing these isotopes.

This characteristic is particularly useful in scientific fields like archaeology and geology where dating ancient objects accurately is essential. Understanding radioactive isotopes, their properties, and behaviors is key to fully appreciating the methods used in radiocarbon dating and other nuclear sciences.