Problem 27 - Calculus with Concepts in Calculus

Question

Potassium \(\left(\mathrm{K}^{40}\right)\) decays into two substances, calcium \(\left(\mathrm{Ca}^{40}\right)\) and argon \(\left(\mathrm{Ar}^{40}\right) .\) By measuring the amounts of potassium and argon it is possible to determine the age of an object from which they are extracted. The formula, which is more complicated than the one in Exercise 25 , is $$ t=(1.885) 10^{9} \ln \left(9.068 \frac{D(t)}{P(t)}+1\right) $$ where \(D(t)\) is the amount of argon and \(P(t)\) the amount of potassium at time \(t\) (in years). a. A sample of basalt taken from lava in the Grand Canyon contained \(1.95 \times 10^{-12}\) moles per gram of argon and \(2.885 \times 10^{-8}\) moles per gram of potassium. \(^{*}\) Determine the age of the basalt. b. A rock from the lunar plains collected during the Apollo 16 mission was determined by the potassiumargon method to be approximately \(4.19\) billion years old. Assuming that the rock contained no argon when it was formed, determine the present ratio of the amount of argon to the amount of potassium in the lunar rock.

Step-by-Step Solution

Answer

(a) 1.155 million years. (b) 0.797.

Step 1: Identify Given Values

For part (a): - Amount of Argon, \(D(t) = 1.95 \times 10^{-12}\) moles per gram.- Amount of Potassium, \(P(t) = 2.885 \times 10^{-8}\) moles per gram.For part (b), the age of the rock \(t = 4.19\) billion years (\(4.19 \times 10^9\) years).

Step 2: Use Age Formula for (a)

Insert the given values into the formula to find the age:\[t = (1.885 \times 10^9) \ln \left(9.068 \frac{1.95 \times 10^{-12}}{2.885 \times 10^{-8}} + 1\right)\]

Step 3: Simplify the Expression for (a)

First, calculate the ratio in the logarithm:\[\frac{1.95 \times 10^{-12}}{2.885 \times 10^{-8}} = 6.7597 \times 10^{-5}\]Then, calculate:\[9.068 \times 6.7597 \times 10^{-5} = 0.0006129\]

Step 4: Solve the Logarithm for (a)

Add 1 to the result and find the natural logarithm:\[\ln(0.0006129 + 1) = \ln(1.0006129)\]Calculate \(\ln(1.0006129) \approx 0.0006127\).

Step 5: Calculate Age for (a)

Multiply by \(1.885 \times 10^9\):\[t = (1.885 \times 10^9) \times 0.0006127 \approx 1.155 \times 10^6\, \text{years}\]So, the age of the basalt is approximately 1.155 million years.

Step 6: Understand the Relationship for (b)

For part (b), you need the ratio \(\frac{D(t)}{P(t)}\) after \(t = 4.19 \times 10^9\) years, using the decay formula:\[t = (1.885 \times 10^9) \ln \left(9.068 \frac{D(t)}{P(t)} + 1\right)\]

Step 7: Rearrange Formula for (b)

Rearrange the formula to solve for \(\frac{D(t)}{P(t)}\):\[\ln \left(9.068 \frac{D(t)}{P(t)} + 1\right) = \frac{t}{1.885 \times 10^9}\]

Step 8: Calculate Ratio for (b)

Substitute \(t = 4.19 \times 10^9\) to find:\[\frac{t}{1.885 \times 10^9} \approx 2.223\]Then,\[e^{2.223} = 9.068 \frac{D(t)}{P(t)} + 1\]\[e^{2.223} - 1 = 9.068 \frac{D(t)}{P(t)}\]Finally, solve to get:\[\frac{D(t)}{P(t)} = \frac{e^{2.223} - 1}{9.068} \approx 0.797\]

Step 9: Conclusion: Answer Parts (a) and (b)

(a) The age of the basalt from the Grand Canyon is approximately 1.155 million years. (b) The current ratio of argon to potassium in the lunar rock is approximately 0.797.

Key Concepts

Potassium-Argon Dating

Potassium-Argon dating is a radiometric technique used to determine the age of minerals and rocks. It leverages the decay of potassium-40 ( K^{40} ) to argon-40 ( Ar^{40} ). This process is key for geologists to estimate geological or historical events. Unlike other dating methods, potassium-argon dating is advantageous for dating geological samples that are billions of years old due to its long half-life of approximately 1.25 billion years.

The method requires measuring the amount of Ar^{40} and comparing it to the remaining K^{40} in a sample. This decay process gives a viable estimate of the time elapsed since the rock or mineral last cooled and, thus, provides an indirect indication of its age. As argon gas can escape from molten rock, only after solidification does the argon remain trapped, marking the "zero" point in the dating process.

• Ideal for dating relatively old geological formations.
• Useful in geological studies and archaeology.
• Depends on precise measurements of parent and daughter isotopes.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. This concept is crucial in understanding radiometric dating, as radioactive isotopes decay exponentially over time. The decay can be expressed as:\[ N(t) = N_0 e^{-kt} \]where \(N(t)\) is the remaining quantity at time \(t\), \(N_0\) is the initial amount, \(k\) is the decay constant, and \(e\) is the base of the natural logarithm. This equation illustrates how the amount of a radioactive substance diminishes over a specific time frame.

The exponential decay model is significant in potassium-argon dating as it helps calculate the age of a sample based on the proportions of K^{40} and Ar^{40}. It allows scientists to back-calculate to determine how much parent isotope was originally present, which is crucial in pinpointing the time elapsed since the rock crystallized.

• Decay processes are characterized by their half-lives.
• Critical for accurately aging geological and archaeological samples.
• Provides a mathematical basis for understanding time-dependent decay.

Natural Logarithm

The natural logarithm (\ln) is a mathematical function represented by the symbol \(\ln(x)\), and it is the inverse of the exponential function. It is useful in many scientific calculations, particularly those involving exponential decay, such as in radioisotope dating.

In the context of potassium-argon dating, the natural logarithm is instrumental in determining the age equation:\[t = (1.885 \, \times \, 10^9) \ln \left(9.068 \frac{D(t)}{P(t)} + 1\right)\]In this equation, the \(\ln\left(\right)\) part allows for transforming the exponential decay relationship into a linear one, making it easier to calculate the elapsed time. Essentially, the \ln function simplifies complex calculations involving exponential decay into more manageable steps.

• Converts multiplicative relationships into additive ones.
• Simplifies equations significantly when dealing with exponential models.
• Helps determine unknown time periods in radioactive dating.

Geological Age Estimation

Geological age estimation is the process of determining the age of geological materials like rocks and minerals. This is a fundamental aspect of geology as it helps to chart the timeline of earth’s history, evolutionary milestones, and tectonic events.

To estimate geological age using potassium-argon dating, geologists measure the ratio of argon to potassium within a sample. These ratios, combined with known decay rates and logarithmic calculations, provide a reliable estimation of when the rock was last molten. For example, in a lunar rock sample analyzed using this method, a known ratio of argon to potassium allowed scientists to deduce an age of 4.19 billion years.

This process is vital for:

• Understanding earth's evolutionary history.
• Mapping past climatic events and environmental changes.
• Providing context for archaeological finds and historical geology.

Such age estimations are crucial not just in geology but also in paleontology, archaeology, and climatology.