The \(\beta^{-}\) -activity of a sample of \(\mathrm{CO}_{2}\) prepared from a contemporary wood gave a count rate of \(25.5\) counts per minute (c.p.m.). The same mass of \(\mathrm{CO}_{2}\) from an ancient wooden statue gave a count rate of \(20.5 \mathrm{cpm} .\), in the same counter condition. Calculate its age to the nearest 50 year taking \(t_{1 / 2}\) for \({ }^{14} \mathrm{C}\) as 5770 year. What would be the expected count rate of an identical mass of \(\mathrm{CO}_{2}\) from a sample which is 4000 year old?

The concept of half-life is pivotal to understanding radioactive decay and is widely used in fields such as geology, archaeology, and environmental science. The half-life of a radioactive isotope refers to the amount of time required for half of the isotope present in a sample to decay. For many students, the calculation of half-life can appear challenging, but by breaking it down, the concept can be made quite accessible.

Let's consider a practical example where a student is asked to calculate the age of an ancient wooden artifact based on the carbon-14 dating method. To start, the student must understand that the half-life of carbon-14 is 5770 years. This means that every 5770 years, the amount of carbon-14 in a given sample will be reduced to half of its original quantity due to radioactive decay. Knowing this constant allows for the utilization of the exponential decay law to determine the age of the sample, as shown in the textbook solution.

The exponential decay law describes the process of gradual decrease in the quantity of a radioactive substance over time. It's essential for students to grasp that this decrease is not linear, but rather, it follows an exponential pattern. This law is mathematically expressed by the equation \(N(t) = N_0 e^{-\frac{ln(2)t}{t_{1/2}}}\), where \(N(t)\) stands for the number of particles at time \(t\), \(N_0\) represents the initial number of particles, \(t_{1/2}\) is the half-life, and \(e\) is the base of the natural logarithm.

When students tackle radioactive decay problems, they often apply this formula to determine the remaining quantity of a substance after a certain period. By rearranging this formula, you can find the time passed for a given change in quantity, aiding in calculations such as dating or determining remaining activity of radioactive materials. The textbook problem provides a scenario where the rearrangement of the decay law is used to find the age of an ancient artifact, an application known as carbon-14 dating.

Carbon-14 dating, or radiocarbon dating, is a method used to determine the age of carbon-containing materials up to about 60,000 years old. The technique is based on the decay rate of the radioactive isotope carbon-14. Carbon-14 is continually being formed in the atmosphere and incorporated into biological systems, allowing scientists to use it as a dating tool.

In our textbook problem, the difference in count rates between contemporary and ancient wood samples gives students a real-world example of how carbon-14 decay can be observed and measured. Using the provided count rates and the exponential decay law, students can calculate the time that has elapsed since the wood was cut, effectively dating the ancient statue. Improvements in understanding can be made by ensuring that students comprehend how the natural logarithm in the decay equation is used to solve for the age of the wood and how this relates to the observable count rate. Furthermore, students should be encouraged to explore why carbon-14 dating is suitable only for organic materials and how it fits into the broader context of radiometric dating methods.