Question

The half-life for the radioactive decay of \(\mathrm{C}-14\) is 5730 years and is independent of the initial concentration. How long does it take for \(25 \%\) of the \(\mathrm{C}-14\) atoms in a sample of \(\mathrm{C}-14\) to decay? If a sample of C-14 initially contains 1.5 mmol of C-14, how many millimoles are left after 2255 years?

Short answer

It takes 11,460 years for 25% of the C-14 atoms to decay (two half-lives). After 2255 years, approximately 1.32 mmol of C-14 remains in the sample.

Step-by-step solution

Understanding the problem

We need to determine the time it takes for 25% of the C-14 atoms in a sample to decay. The half-life of C-14 is 5730 years. Also, we need to calculate how many millimoles remain after 2255 years if we start with 1.5 mmol.

Calculating the decay time for 25%

Since the half-life is the time for half the sample to decay, after one half-life (5730 years), 50% would remain. Another half-life would mean half of the remaining 50% decays, leaving 25%. Therefore, it takes two half-lives for 25% to decay. Time taken is 2 multiplied by 5730 years.

Applying the decay formula

The remaining amount of C-14 can be found using the exponential decay formula: \( N(t) = N_0 e^{-kt} \), where \( N_0 \) is the initial quantity, \( N(t) \) is the quantity after time t, k is the decay constant, and t is the time. We need to find k using the half-life formula: \( t_{1/2} = \frac{\ln(2)}{k} \).

Calculating the decay constant (k)

Using the half-life formula: \( 5730 = \frac{\ln(2)}{k} \) to solve for k, we get \( k = \frac{\ln(2)}{5730} \).

Applying the exponential decay model to 2255 years

Now we use the decay constant to find how much C-14 remains after 2255 years: \( N(2255) = 1.5 \, mmol \cdot e^{-(\frac{\ln(2)}{5730}) \cdot 2255} \).

Calculate the remaining mmol after 2255 years

Plug in the values into the exponential decay formula to get the final amount of C-14 remaining after 2255 years.

Key concepts

Half-life

Half-life is a term used in the realm of radioactive decay to describe the amount of time it takes for half of a radioactive substance to lose its activity by decay. It is an important concept because it is a constant value for a given substance, no matter the amount of the substance present.

For carbon-14 (¹⁴C), which is used in archaeological dating, the half-life is approximately 5730 years. That means if you start with a certain amount of ¹⁴C, after 5730 years, only half of that ¹⁴C would remain due to the radioactive decay process. After another 5730 years, only a quarter would be left, and so on. This predictable decline allows scientists to estimate the age of organic materials such as wood, cloth, or bone.

Exponential decay formula

The exponential decay formula provides a mathematical expression for the process of decay over time. It is frequently expressed as:

\[ N(t) = N_0 e^{-kt} \],

where

• ¹⁴C is represented as \( N(t) \),
• \( N_0 \) is the initial quantity of ¹⁴C at time zero,
• e is the base of the natural logarithm,
• \( k \) is the decay constant unique to the substance,
• and \( t \) is the time elapsed.

This formula can be used to calculate the amount of radioactive material remaining after a given period or, conversely, to determine the time elapsed given an initial quantity and the remaining quantity. It is the foundational equation for understanding radioactive decay in numerous applications, from medical imaging to carbon-14 dating.

Carbon-14 dating

Carbon-14 dating, also known as radiocarbon dating, is a method used to determine the age of an object containing organic material by measuring the amount of carbon-14 it contains. Living organisms constantly exchange carbon with their environment until death, at which point they stop absorbing carbon-14. From that moment, the ¹⁴C atoms begin to decay at a predictable rate, given by its half-life.

By measuring how much carbon-14 remains in a sample and using the half-life and the exponential decay formula, scientists can calculate when the organism died—this is the essence of carbon-14 dating. The accuracy and limitations of this dating method depend on various factors, including the precision in measuring the remaining ¹⁴C and in knowing its initial quantity, which is commonly assumed based on a constant ratio of carbon-14 in the atmosphere.

Decay constant

The decay constant, symbolized as \( k \) in the exponential decay formula, is a probability rate that describes the likelihood of decay of a single atom of a radioactive substance in a given time period. It is related to the half-life of the substance via the formula:

\[ t_{1/2} = \frac{\ln(2)}{k} \],

where \( t_{1/2} \) represents the half-life. The decay constant is a fundamental property of each radioactive isotope, and it is crucial in the calculations of radioactive decay, as it enables the determination of the amount of a substance that will remain after a given time has passed. In the context of carbon-14 dating, once the decay constant for ¹⁴C is known, it can be used in the exponential decay formula to calculate the age of the carbonaceous material with fairly high precision.