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: $$_{19}^{40} \mathrm{K}+_{-1}^{0} \mathrm{e} \longrightarrow_{\mathrm{i} 8}^{40} \mathrm{Ar}(10.7 \%)$$ $$_{19}^{40} \mathrm{K} \longrightarrow_{20}^{40} \mathrm{Ca}+_{-1}^{0} \mathrm{e}(89.3 \%)$$ $$t_{1 / 2}=1.27 \times 10^{9}$$ 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 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 Argon-40 to Potassium-40 ratio is used for dating geological materials because Argon is a noble gas that can be easily trapped and measured in the rock, making the ratio more reliable than Calcium-40 to Potassium-40 ratio. The assumptions involved in this technique include no initial Argon-40 trapped in rock, constant decay rate of Potassium-40, and no loss of Argon-40 since rock formation. Given an Argon-40 to Potassium-40 ratio of 0.95, the age of the rock is approximately \(3.06 \times 10^8 \) years. If some Argon-40 escapes from the rock sample, the measured age will be less than the actual age of the rock.

Step-by-step solution

(Explanation for choosing Argon to Potassium ratio)

In order to date geological materials, we need a stable and reliable end product of decay that can be isolated and measured from the original material to estimate the age. In this case, choosing the Argon-40 to Potassium-40 ratio is preferred because Argon is a noble gas, and it can be easily trapped in the rock as it gets formed. Contrary to this, Calcium, being a reactive solid element, is a part of several minerals and forms various compounds, making it difficult to isolate and measure the Calcium-40 produced from Potassium-40 decay. Therefore, the Argon to Potassium ratio is more reliable and suitable for dating materials.

(Assumptions involved in this technique)

Using the Argon-40/ Potassium-40 ratio in dating materials involves a few major assumptions: 1. No Argon-40 was initially trapped in the rock when it was formed. All Argon-40 found in the rock now has been produced only by the decay of Potassium-40. 2. The decay rate of Potassium-40 remains constant over time. 3. No Argon-40 has been lost from the rock since it was formed until now. If any of these assumptions are not met, the calculation of the age of the rock may result in inaccuracies.

(Calculate the age of the rock)

Given the Argon-40 to Potassium-40 ratio, we can calculate the age of the rock using the following equation: \[ N = N_0e^{\lambda t} \] Where \( N \) is the amount of remaining Potassium-40, \( N_0 \) is the initial amount of Potassium-40, \( \lambda \) is the decay constant, and \( t \) is the age of the rock. Since we are given the ratio, we can write: \[ \frac{\mathrm{Ar}^{40}}{\mathrm{K}^{40}} = \frac{N_0 - N}{N} = \frac{0.95}{1} \] Now, we can find N using this relation: \[ N = \frac{N_0}{1.95} \] We can now apply the decay equation to find the age of the rock: \[ \frac{1}{1.95} = e^{\lambda t} \] The decay constant (\( \lambda \)) can be calculated using the half-life given (\( t_{1/2} = 1.27 \times 10^9 \)): \[ \lambda = \frac{ln(2)}{t_{1/2}} = \frac{ln(2)}{1.27*10^9} \] Now, we can solve the equation for \( t \): \[ t = \frac{ln(1.95)}{\lambda} = \frac{ln(1.95)}{\frac{ln(2)}{1.27*10^9}} \] Calculating the value for \(t\): \[ t \approx 3.06 \times 10^8 \] The age of the rock is approximately \( 3.06 \times 10^8 \) years.

(Measured vs. Actual age with Argon-40 escape)

If some Argon-40 has escaped from the rock sample, the measured Argon-40 to Potassium-40 ratio will be lower than the actual ratio. As a result, the calculated age will be less than the actual age of the rock, since a lower ratio suggests that less Potassium-40 has decayed into Argon-40. The actual age of the rock will always be more than the measured age if some Argon-40 has escaped from the sample.

Key concepts

Potassium-40 decay

The process of Potassium-40 decay is important in the field of geologic dating. Potassium-40 (\(^{40} \mathrm{K}\)) is a radioactive isotope that decays through different paths. One decay pathway results in the formation of Argon-40 (\(^{40} \mathrm{Ar}\)), while another leads to Calcium-40 (\(^{40} \mathrm{Ca}\)).

The decay to Argon-40 occurs when Potassium-40 captures an electron and transforms into an Argon atom, which is stable. This process happens in about 10.7% of the decay events. The majority, about 89.3%, results in the production of Calcium-40. Since Calcium is reactive and forms various compounds in nature, it isn't as useful for dating purposes as Argon, which is a noble gas and remains isolated in rock formations, making it easier to measure.

Argon-40 dating

Argon-40 dating is a technique used to estimate the age of geological materials. Given its stable nature, Argon-40 provides a reliable measure because it accumulates over time as Potassium-40 decays.

The technique involves measuring the Argon-40 and Potassium-40 ratio in a rock. The presence of Argon-40 implies that Potassium-40 has decayed, and this decay can be quantified to determine how long it has been occurring. Since no Argon gas escapes easily from the rock, under ideal circumstances, its concentration provides an accurate age of the rock.

Assumptions critical to this method include the initial absence of Argon-40 in the rock at formation and the lack of net Argon loss over time. If these conditions hold, scientists can reliably trace the history encapsulated within the rock.

half-life calculations

Understanding half-life is crucial for calculating the age of geological materials through isotope dating. The half-life is the period it takes for half of a given quantity of a radioactive isotope to decay into another element.

Potassium-40 has a half-life of about 1.27 billion years, which makes it suitable for dating very old geological materials. The decay constant (\( \lambda \)) related to this half-life can be calculated using the formula:\[\lambda = \frac{\ln(2)}{t_{1/2}}\]where \(t_{1/2}\) is the half-life.With the decay constant, we can calculate the elapsed time (\( t \)) since the rock formed using the formula:\[N = N_0 e^{-\lambda t}\]Here, \(N\) is the current amount of Potassium-40, and \(N_0\) is its initial quantity. Solving these equations gives the rock's age, reflecting how Potassium-40 has been converting to Argon-40 over time.

geologic materials dating

Dating geological materials involves analyzing the chemical processes that alter them over time. Through methods like Argon-40 dating, scientists determine the age of rocks by tracing isotopic changes resulting from radioactive decay.

The technique requires that the rock initially contains no Argon-40 and retains all the Argon-40 produced by decay over the millennia. When these criteria are met, the ratio of Argon-40 to Potassium-40 reveals the rock's age with high accuracy.

Any leakage of Argon-40 from the rock post-formation skews this understanding, leading to underestimations of the actual age. This is because escaping Argon reduces the apparent amount produced, thus portraying the rock as younger than it is. Ensuring no Argon loss is crucial for precise dating, and methods are continually refined to account for this and other potential discrepancies in geological dating.