Problem 128 - A Complete Resource Book in Chemistry for JEE Main

Question

Potassium \(-40\) decays to argon \(-40\) with a half life of \(1.27 \times 10^{9} \mathrm{yr}\). What is the age of a rock in which the mass ratio of \({ }^{40} \mathrm{Ar}\) to \({ }^{40} \mathrm{~K}\) is \(3.6\) ? (a) \(2.8 \times 10^{10} \mathrm{yr}\) (b) \(1.4 \times 10^{9} \mathrm{yr}\) (c) \(1.4 \times 10^{10} \mathrm{yr}\) (d) \(2.8 \times 10^{9} \mathrm{yr}\)

Step-by-Step Solution

Answer

(d) The age of the rock is approximately \(2.8 \times 10^9\) years.

Step 1: Understand the relationship

In radioactive decay, potassium-40 (K) decays to argon-40 (Ar). The proportion of daughter isotope (Ar) to parent isotope (K) after a certain time provides insight into the age of the specimen.

Step 2: Express ratio considering decay

Given the ratio of Ar to K is 3.6, it means for every 3.6 parts of Ar, there's 1 part of K. Originally, these 3.6 parts of Ar also started as K. Therefore, there are a total of \(1 + 3.6 = 4.6\) parts of K originally.

Step 3: Use decay formula

For radioactive decay, the fraction remaining is given by \(\frac{N}{N_0} = (\frac{1}{2})^{(t/T_{1/2})}\), where \(t\) is the time elapsed and \(T_{1/2}\) is the half-life. Here, the fraction remaining is \(\frac{1}{4.6}\) because that's the amount of K remaining in relation to the original number.

Step 4: Solve for time

Rearrange the equation to solve for \(t\):\[\left(\frac{1}{2}\right)^{t/T_{1/2}} = \frac{1}{4.6}\]Taking logarithms, we get:\[t \cdot \log(0.5) = \log\left(\frac{1}{4.6}\right) \]Therefore, \[t = \frac{\log(1/4.6)}{\log(0.5)} \cdot T_{1/2} \]

Step 5: Calculate the decay time

Substitute known values into the equation:\[t = \frac{\log(1/4.6)}{\log(0.5)} \cdot 1.27 \times 10^9 \]Calculating the logs and multiplying yields:\[t = \left(\frac{\log(0.217)}{\log(0.5)}\right) \cdot 1.27 \times 10^9 \approx 2.8 \times 10^9 \text{ years}\]

Step 6: Confirm the closest answer

The calculated age of the rock is approximately \(2.8 \times 10^9\) years. Compare with the given options: (a) \(2.8 \times 10^{10}\)(b) \(1.4 \times 10^9\)(c) \(1.4 \times 10^{10}\)(d) \(2.8 \times 10^9\)The correct answer corresponds to option (d).

Key Concepts

Potassium-40 Decay

Potassium-40 (\(^{40}K\)) is a radioactive isotope that decays to argon-40 (\(^{40}Ar\)). This process is an example of radioactive decay, which involves a parent isotope (\(^{40}K\)) transforming into a daughter isotope (\(^{40}Ar\)).
In nature, potassium-40 is found in small amounts within many minerals and rocks. Once formed, it begins the decay process, changing its atomic structure and eventually becoming argon-40. This is crucial for dating geological events as it takes into consideration the transition from potassium-40 to argon-40 over time. Radioactive decay is predictable and follows an exponential pattern, which scientists can measure using half-lives. In the case of potassium-40, it has a long half-life of \(1.27 \times 10^9\) years, making it particularly useful for dating ancient geological materials. Additionally, since argon is a noble gas, it does not normally uprise in minerals from the environment, meaning any \(^{40}Ar\) present in a rock is likely a result of the decay of \(^{40}K\). This makes potassium-40 decay a reliable clock for determining the timing of geological processes.

Half-life Calculation

The concept of half-life is central to understanding radioactive decay.A half-life is the time required for half of the parent isotope (e.g., \(^{40}K\)) to transform into a daughter isotope (e.g., \(^{40}Ar\)). In our case, the half-life of potassium-40 is \(1.27 \times 10^9\) years. This means that over this span of time, half of the potassium-40 in a sample will have decayed into argon-40.
This principle allows us to calculate the age of rocks by using the proportion of argon-40 to potassium-40 present in the sample.To calculate the age, we use the decay formula:\[\left(\frac{1}{2}\right)^{t/T_{1/2}} = \frac{N}{N_0}\]where \(t\) is the time that has passed, \(T_{1/2}\) is the half-life, \(N\) is the remaining amount of the parent isotope, and \(N_0\) is the original amount.When the mass ratio of argon-40 to potassium-40 is known, the fraction \(\frac{N}{N_0}\) can be used to calculate the time \(t\) it takes for the decay to occur. This systematic approach provides a pathway to determine the age of geological samples accurately.

Isotope Dating

Isotope dating, often known as radiometric dating, is a method employed to determine the age of materials. It leverages the predictable decay of radioactive isotopes, such as potassium-40. This technique is particularly beneficial for dating rocks.
In isotope dating, scientists measure the ratio of parent to daughter isotopes within a sample. In our original exercise, this ratio was provided as the mass ratio of \(^{40}Ar\) to \(^{40}K\). Such ratios indicate how much decay has occurred and thus, how much time has passed since the rock solidified.The key steps in isotope dating include:

• Identifying the parent and daughter isotopes.
• Measuring the ratio of these isotopes in the sample.
• Using the half-life of the parent isotope to calculate the elapsed time since formation.

By knowing the half-life and the current isotope ratio, the age of the rock or mineral can be inferred with considerable accuracy. Isotope dating is a powerful tool for understanding the timeline of geological and environmental processes, allowing us to piece together the history of the Earth.