Question

The mass ratios of \(^{40} \mathrm{Ar}\) to \(^{40} \mathrm{K}\) also can be used to date geologic materials. Potassium-40 decays by two processes: $$\begin{aligned} &_{19}^{40} \mathrm{K}+_{-1}^{0} \mathrm{e} \longrightarrow_{18}^{40} \mathrm{Ar}(10.7 \%) \quad t_{1 / 2}=1.27 \times 10^{9} \text { years }\\\ &_{19}^{40} \mathrm{K} \longrightarrow_{20}^{40} \mathrm{Ca}+_{-1}^{0} \mathrm{e}(89.3 \%) \end{aligned}$$a. Why are \(^{40} \mathrm{Ar} /^{40} \mathrm{K}\) ratios used to date materials rather than \(^{40} \mathrm{Ca} /^{40} \mathrm{K}\) ratios? b. What assumptions must be made using this technique? c. A sedimentary rock has an \(^{40} \mathrm{Ar} /^{40} \mathrm{K}\) ratio of 0.95. Calculate the age of the rock. d. How will the measured age of a rock compare to the actual age if some \(^{40}\) Ar escaped from the sample?

Short answer

The Ar-40/K-40 ratio is used to date materials because the radioactive decay of Potassium-40 into Argon-40 has a known half-life, while using the Ca-40/K-40 ratio is not useful since the isotopes have different decay processes. Assumptions in this technique include no initial Ar-40 when the rock formed, a constant decay rate, and a closed system for the rock. Given an Ar-40/K-40 ratio of 0.95, the age of the rock is approximately 1.18 × 10^9 years. If Ar-40 escapes from the sample, the measured age will be underestimated compared to the actual age.

Step-by-step solution

Explain why Ar-40/K-40 ratios are used to date materials

The Ar-40/K-40 ratio is used to date materials because Potassium-40 decays into Ar-40, and this radioactive decay process has a known half-life. This means that the ratio of Ar-40 to K-40 can be used to determine the age of a sample, since it can be correlated to the time passed from when the decay of K-40 started. By contrast, using the Ca-40/K-40 ratio is not useful because both isotopes have different decay processes, and we cannot link the ratio's change directly to the passage of time.

List the assumptions made using Ar-40/K-40 dating technique

There are several assumptions made when using the Ar-40/K-40 dating technique: 1. No initial Ar-40 was present when the rock formed. 2. The rate of decay of K-40 to Ar-40 has remained constant over time. 3. The rock has remained a closed system, meaning that no Ar-40 or K-40 has been added to or removed from the sample since its formation.

Calculate the age of the rock with an Ar-40/K-40 ratio of 0.95

To calculate the age of the rock, we will use the following formula derived from radioactive decay equations: \(t = \frac{1}{\lambda} \ln(1 + \frac{Ar}{K})\), where \(t\) is the age of the rock, \(\lambda\) is the decay constant of K-40, and \(Ar\) and \(K\) represent the amounts of Ar-40 and K-40 respectively. Using the given half-life \(t_{1/2} = 1.27 \times 10^9\) years, we can calculate the decay constant \(\lambda = \frac{\ln(2)}{t_{1/2}}\). The ratio of Ar-40/K-40 is given as 0.95. Consequently, the age of the rock is: \(t = \frac{1}{\lambda} \ln(1 + 0.95) = \frac{1.27 \times 10^9}{\ln(2)} \ln(1.95)\). After solving for \(t\), we find that the age of the rock is approximately 1.18 × 10^9 years.

Discuss the impact of Ar-40 escaping from the sample on the measured age

If some Ar-40 escapes from the sample, the measured ratio of Ar-40/K-40 will be lower than it should be for the actual age of the rock. This will result in an age calculation that is underestimated. As a consequence, the measured age will be less than the actual age of the rock, since the true ratio of Ar-40/K-40 would be higher if no Ar-40 had escaped.

Key concepts

Radioactive Decay

Understanding radioactive decay is fundamental when studying isotopic dating methods like potassium-argon dating. Radioactive decay is a natural process that occurs at a stable rate, in which unstable isotopes transform into other isotopes or elements by emitting particles or radiation. In the case of potassium-40 ((^{40})K), it decays over time to argon-40 ((^{40})Ar) and calcium-40 ((^{40})Ca), through electron capture and beta decay, respectively.

The stability of the decay rate allows scientists to use the byproducts of decay, in this case, (^{40})Ar, as a 'clock' to measure the age of geological materials. When evaluating the age of a rock, one must consider that over time, (^{40})K decays into (^{40})Ar at a predictable rate, giving us a snapshot of when the rock was formed.

Half-life

The half-life of a radioactive isotope is the time it takes for half of the original isotope to decay into its daughter products. In the context of potassium-argon dating, we focus on the half-life of (^{40})K, which is approximately 1.25 billion years. This means that after 1.25 billion years, half of the (^{40})K in a sample would have decayed to (^{40})Ar. Knowing the half-life allows us to calculate the age of a rock by measuring the ratio of (^{40})Ar to (^{40})K.

It's like a cosmic stopwatch that ticks away as the potassium isotopes transform, this ticking rate, or half-life, remains constant, which is pivotal for making accurate age estimations of rocks and minerals.

Geochronology

Geochronology is the science of determining the age of rocks, fossils, and sediments using signatures inherent in the rocks themselves. The discipline involves the integration of stratigraphy (the study of rock layers) with chronology to establish a time scale for geological events. By using methods like potassium-argon dating, geochronologists can pin down the time during which a rock layer was formed.

For instance, when potassium-40 decays to argon-40 within a rock, it effectively starts the geological clock. As geochronologists examine these isotopic ratios, they unravel the historical timeline of Earth's crust, allowing us to understand not just the age of rocks but the history of our planet's surface as well.

Isotopic Dating Methods

Isotopic dating methods involve analyzing the concentration of radioactive isotopes and their decay products to establish the age of material. Each method varies depending on the specific isotopes involved. Potassium-argon dating is one such method, specifically designed to date volcanic and igneous rocks that contain appreciable amounts of potassium mineral.

As the unstable (^{40})K isotope decays to (^{40})Ar, it generates a measurable ratio which provides a basis for the age calculations. This technique is highly advantageous for dating rocks that are beyond the range of radiocarbon dating, which has an upper limit of around 50,000 years. Potassium-argon dating extends our capacity to date rocks up to billions of years old.

Closed System Assumption

A critical component in accurate isotopic dating is the closed system assumption. This principle posits that the rocks being dated have not been affected by external factors that could change the natural decay of isotopes within them. It assumes, for the duration of the rock's existence, no isotopes have entered or left the system, allowing for an undisturbed decay process.

In the context of potassium-argon dating, this means that after the rock has cooled and solidified, neither (^{40})K nor (^{40})Ar has been added or removed from the sample. If this assumption is satisfied, a direct correlation between the measured isotopic ratio and the elapsed time since the rock's formation is established. However, if the system has been open, perhaps through erosion or other geological events, this can result in inaccuracies when dating the sample.