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: \({ }_{19}^{40} \mathrm{~K}+{ }_{-1}^{0} \mathrm{e} \longrightarrow{ }_{18}^{40} \mathrm{Ar}(10.7 \%) \quad t_{1 / 2}=1.27 \times 10^{9}\) years \({ }_{19}^{40} \mathrm{~K} \longrightarrow{ }_{20}^{40} \mathrm{Ca}+{ }_{-1}^{0} \mathrm{e}(89.3 \%)\) 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 \({ }^{40} \mathrm{Ar}\) escaped from the sample?

Short answer

a. Argon-40/Potassium-40 ratios are used to date materials instead of Calcium-40/Potassium-40 ratios because Argon-40 is a noble gas that does not bond with other elements, making it easier to distinguish and measure. Calcium-40 is a common element that bonds with other elements, making it harder to determine the decay ratio accurately. b. Assumptions made using this technique include: initially, no Argon-40 in the sample; Argon-40 increases solely due to Potassium-40 decay; no Argon-40 loss or addition since sample formation; and constant Potassium-40 decay rates. c. The age of the sedimentary rock with an Argon-40/Potassium-40 ratio of 0.95 is approximately 401 million years. d. If some Argon-40 escaped from the sample, the measured age of the rock would be less than the real age, as the lower Argon-40-to-Potassium-40 ratio would lead to a younger estimated age.

Step-by-step solution

a. Why Argon-40/Potassium-40 ratios are used to date materials instead of Calcium-40/Potassium-40 ratios

Argon-40 is a noble gas, and it does not bond with other elements, allowing it to accumulate after the decay of Potassium-40. This allows the Argon-40 to be easily distinguished and measured compared to the materials surrounding it. Calcium-40, on the other hand, is a common element in the Earth's crust and can easily bond with other elements. This makes it harder to determine which calcium ions resulted from Potassium-40 decay, and therefore the Calcium-40/Potassium-40 ratio would not provide accurate dating information.

b. Assumptions made using this technique

Some important assumptions made while using the Argon-40/Potassium-40 dating technique are: 1. The initial amount of Argon-40 was zero when the rock sample formed. 2. The amount of Argon-40 in the sample has only increased due to the decay of Potassium-40. 3. No Argon-40 has been lost or added to the rock sample since its formation. 4. The decay rates of Potassium-40 remained constant over time.

c. Calculating the age of the rock

We are given the Argon-40/Potassium-40 ratio as 0.95. Let's denote the amount of Argon-40 as A and the amount of Potassium-40 as K, so A/K = 0.95. To calculate the age of the rock, we will use the decay equation \( t = \frac{1}{\lambda} * \ln(1 + \frac{A_{40}}{K_{40}}) \), where \(t\) is the age of the rock, and \(\lambda\) is the decay constant related to half-life by the equation: \(\lambda = \frac{\ln(2)}{t_{1/2}}\). We have half-life \(t_{1/2} = 1.27 \times 10^9\) years. Now, we will calculate the decay constant, \(\lambda\). \(\lambda = \frac{\ln(2)}{1.27 \times 10^9} \approx 5.46 \times 10^{-10}\mathrm{~} \text{year}^{-1}\) Now we can calculate the age of the rock using the Argon-40/Potassium-40 ratio: \(t = \frac{1}{5.46 \times 10^{-10}} * \ln(1 + 0.95) = 4.01 \times 10^8\) years. The age of the rock is approximately 401 million years.

d. Comparing the measured age of a rock with its actual age if some Argon-40 escaped from the sample

If some Argon-40 escaped from the sample, the measured Argon-40/Potassium-40 ratio would be smaller than the actual ratio. As a result, the calculated age of the rock using the above equation would be less than the real age. The lower Argon-40-to-Potassium-40 ratio would lead to a younger estimated age, as it implies less Potassium-40 decayed into Argon-40 than what actually occurred.

Key concepts

Potassium-Argon Dating

Potassium-Argon dating is a method used to determine the age of geological samples, particularly rocks. The technique relies on the radioactive decay of Potassium-40 ( ^{40} K) to Argon-40 ( ^{40} Ar). Potassium-40 is a naturally occurring isotope of potassium that decays to stable Argon-40, a noble gas. Since Argon-40 does not bond with other elements, it accumulates within the rock matrix, making it measurable regardless of the surrounding conditions. This accumulation is key for determining the age of rocks.

The Potassium-Argon dating method is particularly useful for dating volcanic rocks and can date materials that are over several millions of years old. Due to its long half-life— approximately 1.27 billion years—Potassium-40 is ideal for studying ancient geological events and formations. Additionally, because Argon is a gas, it can migrate, allowing for assumptions such as initially having no Argon-40 in the system, thus starting from zero in the isotopic ratio at a rock's formation.

Decay Processes

Decay processes involved in Potassium-Argon dating are crucial for understanding how isotopes change over time. Potassium-40 decays via two processes: it can transform into Argon-40 through electron capture (10.7% probability) or decay into Calcium-40 ( ^{40} Ca) through beta decay (89.3% probability). These processes involve the transformation of isotopes and are governed by the principles of nuclear physics.

- **Electron Capture**: In this process, an electron is absorbed by the nucleus, transforming a proton into a neutron, resulting in the formation of Argon-40. - **Beta Decay**: This process involves the emission of an electron from a neutron turning it into a proton, producing Calcium-40.
The choice of using Argon-40 for dating is due to its non-reactive nature. Unlike Calcium-40, Argon-40's gaseous form means it remains trapped within minerals, providing a clear signal for dating. Understanding these decay processes is vital for interpreting Potassium-Argon dating results.

Half-life Calculation

Half-life is a fundamental concept in radiometric dating. It represents the time it takes for half of the radioactive isotope in a sample to decay. For Potassium-40, the half-life is 1.27 billion years, making it suitable for dating geological events spanning significant timescales. This extended half-life allows scientists to measure the age of older samples, providing insight into Earth's history.

When calculating age using Potassium-Argon dating, the decay constant ( λ ) is required. This is related to the half-life by the equation:\[λ = \frac{\ln(2)}{t_{1/2}}\]This constant allows for determining the amount of time that has passed since the formation of a rock. Using the decay equation ( t = \frac{1}{λ} \cdot \ln(1 + \frac{A_{40}}{K_{40}}) ), where A_{40}/K_{40} is the measured ratio of Argon-40 to Potassium-40, we can calculate the age of the rock. This method assumes that the system was closed, meaning no loss or gain of isotopes has occurred since the rock formed.

Geologic Age Estimation

Geologic age estimation using Potassium-Argon dating involves interpreting isotope ratios to determine the timescales of geological formations. By measuring the ( ^{40} Ar/^{40} K ) ratio, scientists can estimate how long a rock has existed since it ceased to accumulate additional Potassium-40 or lost Argon-40.

For reliable age estimation, several assumptions are made:

• Initial Argon-40 was zero, meaning any Argon-40 present formed through decay.
• No Argon-40 has been lost or added since the rock was formed.
• The decay constant has remained constant over time.

If some Argon-40 has escaped due to geological processes, the apparent age may appear younger than the actual age. This is because the ratios would indicate less decay, underscoring the importance of understanding geological context when using dating methods. Accurate geologic age estimation helps reconstruct Earth's history and the timing of past events.