Question

The radioactive decay rates of naturally occurring radioactive elements can t used to determine the age of very old materials. For example, \({ }_{6}^{14} \mathrm{C}\) is radioactiv and emits a low-energy electron with a half-life of about 5730 years. Throug a balance of natural processes, the ratio of \({ }^{14} \mathrm{C} /{ }^{12} \mathrm{C}\) is constant in living orgar isms. However, in dead organisms or material, this ratio decreases as the decays. Since the radioactive decay is known to be a first order reaction, the ag of the material can be estimated by measuring the decrease in the \({ }^{14} \mathrm{C}^{12} \mathrm{C}\) ratio Suppose a piece of ancient wool is found in which the ratio has been found decrease by \(20 \%\). What is the age of the wool?

Short answer

The age of the wool is approximately 1,435 years.

Step-by-step solution

Understand the Problem

We are tasked with finding the age of ancient wool using the radioactive decay of carbon-14 ( ). We know the ratio of to has decreased by 20%.

Apply the First Order Decay Formula

Since radioactive decay is a first order reaction, we use the equation: , where is the remaining quantity, is the initial quantity, is the decay constant, and is time. We calculate using the concept that 20% has decayed, .

Calculate the Decay Constant

The decay constant, , can be calculated using the half-life ( ) with the formula: , where . We know .

Solve for Time

Now we rearrange the first order decay formula to solve for time, with being the initial percentage (100%) and being the remaining percentage (80%). Substitute the values into the formula and solve for .

Key concepts

Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by releasing radiation. This occurs as the nucleus changes to a more stable form. It's a random process but governed by statistical probability. Each type of radioactive atom decays with a certain likelihood per unit time. This makes it predictable over large quantities.

• Naturally occurring radioactive elements decay at fixed rates.
• This decay involves the emission of particles such as electrons, also known as beta particles.
• The rate of decay is typically specific to the element in question, like carbon-14 in our example.

The decay process helps in carbon-14 dating, a valuable method to determine the age of archaeological finds and geological samples. By measuring the relative amount of carbon-14 isotopes in a sample, scientists can determine how long it has been since the sample stopped exchanging carbon with its environment.

First Order Reactions

A first order reaction is one where the rate of reaction depends linearly on the concentration of one reactant. For radioactive decay like that of carbon-14, it follows a first order kinetic pattern. Meaning, the rate at which it decays only depends on the amount of _C14_ present at any time.

• The formula for first order decay is: \[ N_t = N_0 e^{-kt} \] where \(N_t\) is the remaining quantity at time \(t\), \(N_0\) is the initial quantity, \(k\) is the decay constant, and \(t\) is the time elapsed.
• Understanding this helps in determining how long it takes for a substance to reduce to a certain level.
• It's a straightforward model but can simplify the complex reality of decay processes.

First order reactions are exponential processes, meaning the rate at which the quantity decreases is proportional to its current value. This unique property allows us to easily calculate the age of objects through their carbon-14 decay.

Half-Life Measurement

The concept of half-life represents the time required for half of the radioactive atoms in a sample to decay. It's a key factor in radioactive dating.

• For carbon-14, the half-life is approximately 5730 years.
• This means that every 5730 years, half of a given quantity of carbon-14 will decay, simplifying the dating process.
• It provides a fixed timetable of decay, allowing scientists to backtrack and determine the age of organic materials.

Calculating the half-life involves the formula: \[ t_{1/2} = \frac{0.693}{k} \]where \(t_{1/2}\) is the half-life and \(k\) is the decay constant. By understanding a sample's current state and tracing back through the half-life timeline, we can determine how many years have elapsed since an organism or object ceased taking in carbon-14, initiating its radioactive countdown. This mechanism, built on the reliability of half-life, is central to techniques like carbon-14 dating, widely used by archaeologists and scientists today.