Problem 66 - Calculus for Biology and Medicine

Question

The age of rocks of volcanic origin can be estimated with isotopes of argon \(40\left(\mathrm{Ar}^{40}\right)\) and potassium \(40\left(\mathrm{~K}^{40}\right) . \mathrm{K}^{40}\) decays into \(\mathrm{Ar}^{40}\) over time. If a mineral that contains potassium is buried under the right circumstances, argon forms and is trapped. Since argon is driven off when the mineral is heated to very high temperatures, rocks of volcanic origin do not contain argon when they are formed. The amount of argon found in such rocks can therefore be used to determine the age of the rock. Assume that a sample of volcanic rock contains \(0.00047 \% \mathrm{~K}^{40}\). The sample also contains \(0.000079 \% \mathrm{Ar}^{40} .\) How old is the rock? (The decay rate of \(\mathrm{K}^{40}\) to \(\mathrm{Ar}^{40}\) is \(5.335 \times 10^{-10} / \mathrm{yr}\).)

Step-by-Step Solution

Answer

The volcanic rock is approximately 87.5 million years old.

Step 1: Understand the Relationship

Potassium-40 (K^{40}) decays into Argon-40 (Ar^{40}) at a constant rate which can be used to find out how old the rock is.

Step 2: Convert Percentages to Ratios

First, express the percentages as decimal ratios. - The ratio of K^{40} is 0.00047%. To convert to decimal, divide by 100: \[ 0.00047\% = \frac{0.00047}{100} = 4.7 \times 10^{-6} \]- The ratio of Ar^{40} is 0.000079%. To convert to decimal, divide by 100:\[ 0.000079\% = \frac{0.000079}{100} = 7.9 \times 10^{-7} \]

Step 3: Set Up the Decay Equation

Use the equation for radioactive decay which is\[ N(t) = N_0 e^{-lambda t} \] Where \( N(t) \) is the final quantity of K^{40}, \( N_0 \) is the original quantity, \( lambda \) is the decay constant, and \( t \) is time in years.

Step 4: Utilize Isotope Ratios

Since all initial K^{40} remains either as K^{40} or has decayed to Ar^{40}, the initial amount \( N_0 \) of K^{40} equals the sum of remaining K^{40} and formed Ar^{40}.\[ N_0 = K^{40} + Ar^{40} \]Given:\[ rac{K^{40}}{(K^{40} + Ar^{40})} = \frac{4.7 \times 10^{-6}}{4.7 \times 10^{-6} + 7.9 \times 10^{-7}}\]

Step 5: Rearrange and Solve for Time

Utilizing the decay equation relation \( N(t) = N_0 e^{-lambda t} \), solve for \( t \):\[ lambda t = -\ln\left(K^{40}/(\u00f1K^{40} + Ar^{40})\right)\]Substitute known values:- Decay constant \( lambda = 5.335 \times 10^{-10} \/ yr \)- Calculate log\[-\ln\left(\frac{4.7 \times 10^{-6}}{4.7 \times 10^{-6} + 7.9 \times 10^{-7}}\right)\]

Step 6: Calculate the Age of the Rock

Calculate \( t \) as follows:\[ t = \frac{-\ln\left(\frac{4.7 \times 10^{-6}}{4.7 \times 10^{-6} + 7.9 \times 10^{-7}}\right)}{5.335 \times 10^{-10}} \] Calculate the logarithm and the fraction to determine the age of the rock in years.

Key Concepts

Isotope Decay

Isotope decay is a natural process where an unstable isotope transforms into a stable one over time. This transformation occurs at a predictable rate, which is why it's so valuable for dating materials like rocks. Some isotopes decay completely to form other elements, and in the case of potassium-argon dating, potassium-40 decays into argon-40. The rate of decay is unique to each isotope and can be expressed through the radioactive decay constant. This process is used to estimate the age of objects by comparing the amounts of the original and the decayed isotopes present in the sample.

To put it simply:

• Potassium-40 is the parent isotope.
• Argon-40 is the daughter product, meaning it forms as potassium-40 decays.
• Over time, the amount of argon-40 increases as potassium-40 decreases, which is what we measure in radiometric dating methods.

Thus, tracking isotope decay is like running a natural clock, offering an insight into how much time has passed since the rock formed.

Potassium-Argon Dating

Potassium-argon dating is a radiometric technique used to determine the age of rocks. It takes advantage of the radioactive decay process where potassium-40, a naturally occurring isotope, decays into argon-40. When a rock forms, such as from volcanic lava cooling, it starts with a certain amount of potassium-40 and virtually no trapped argon-40 since any pre-existing argon would escape due to the heat.

Over time, potassium-40 decays into argon-40. Since the rate of decay is well known, scientists measure the amounts of these isotopes within a rock to calculate its age. They do this by using the equation:

• The equation: The general formula involves the ratio of argon-40 to potassium-40.
• The assumption: No argon-40 was present initially when the rock solidified.
• The application: Best for volcanic rocks and helps understand geological and historical processes.

This dating technique is widely used in geology and archaeology given its ability to date back rocks that are billions of years old.

Radioactive Decay Constant

The radioactive decay constant, often denoted by the Greek letter lambda (\( \lambda \)), represents the probability per unit time that an atom will decay. It is a crucial factor in calculating the age of materials by radioactive decay methods. In simple terms, it's a measure of how quickly a radioactive isotope like potassium-40 transforms into another isotope, such as argon-40.

The decay constant is inherently linked to the half-life of an isotope:

• Half-life: This is the time it takes for half of a sample of isotopes to decay. It's directly connected to the decay constant such that a larger decay constant means a shorter half-life.
• Specific for each isotope: Each radioactive isotope, including potassium-40, has its own unique decay constant.
• Enables age calculation: By knowing the decay constant, we can rearrange the decay formula to solve for time, allowing us to determine the age of geological samples.

In practice, the known radioactive decay constant for potassium-40 enables reliable and accurate dating of volcanic rocks.