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 \(\left.5.335 \times 10^{-10} / \mathrm{yr} .\right)\)

Step-by-Step Solution

Answer

The rock is approximately 1.25 billion years old.

Step 1: Understand the Decay Equation

The decay equation used for these types of problems is the exponential decay equation: \( N(t) = N_0 e^{- ext{decay rate} \times t} \), where \( N(t) \) is the remaining amount of the original isotope, \( N_0 \) is the original amount, the decay rate is \( \lambda = 5.335 \times 10^{-10}/\text{year} \), and \( t \) is the time.

Step 2: Set Up the Problem

We know \( N(t) = \text{current K}^{40} \) and \( N_0 = \text{current K}^{40} + \text{current Ar}^{40} \). Given \( 0.00047\% \) of K is current, and \( 0.000079\% \) of Ar is current, we must first convert these percentages into comparable numbers.

Step 3: Convert Percentages into Fractions

Since the percentage for K is \( 0.00047\% \), we convert it to a fraction of the whole: \( 0.00047/100 \). Similarly, for Ar, it is \( 0.000079/100 \).

Step 4: Calculate Initial and Remaining Amounts

\( N_0 = 0.00047 + 0.000079 \) because \( N_0 = \text{current K}^{40} + \text{current Ar}^{40} \). \( N(t) = 0.00047 \). Calculate keeping in mind that \( N_0 \) is based on initial K amount.

Step 5: Use Decay Formula

Use the relation \( \frac{N(t)}{N_0} = e^{-\lambda t} \). First, calculate the ratio \( \frac{N(t)}{N_0} = \frac{0.00047}{0.00047 + 0.000079} \).

Step 6: Solve for Time \( t \)

Take the natural logarithm of both sides: \( \ln\left(\frac{N(t)}{N_0}\right) = -\lambda t \). Plug in all the known values and solve for \( t \): \[ t = -\frac{1}{\lambda} \ln \left(\frac{N(t)}{N_0} \right) \].

Step 7: Compute the Age of the Rock

Calculate \( t \): \[ t = -\frac{1}{5.335 \times 10^{-10}} \ln \left( \frac{0.00047}{0.00047 + 0.000079} \right) \]. Solve this equation to find the age of the rock.

Key Concepts

Isotope Dating

Isotope dating is a fascinating method used by scientists to determine the age of rocks, minerals, and even archaeological artifacts. This process hinges on the existence of isotopes, which are atoms of the same element with different numbers of neutrons. Over time, certain isotopes decay to form different elements or isotopes. By analyzing the ratio of the remaining isotope to the new one it has decayed into, scientists can calculate how much time has passed since the rock was formed. For example, the isotope Potassium-40 (8094 K^{40}80 ) decays into Argon-40 (8094 Ar^{40}80 ) over a very long period of time. Since 8094 Ar^{40}80 is a noble gas, it doesn't bond with other atoms and gets trapped in the mineral structures, providing a stable record for isotope dating. This makes isotopic analysis a reliable scientific tool for dating. By knowing the decay rate (or half-life), which is specific to each isotope, geologists can create a timeline of Earth's history, offering insights into geological events and the age of different rock layers.

Exponential Decay

Exponential decay is a fundamental concept in understanding radioactive decay, and it describes how the quantity of a substance decreases over time. The process can be simply explained by the exponential decay equation: \( N(t) = N_0 e^{-\lambda t} \)Here, \( N(t) \) represents the amount of substance left at time \( t \), \( N_0 \) is the initial amount, and \( \lambda \) is the decay rate. This equation tells us that the reduction of the substance is not a simple arithmetic decrease but rather decreases exponentially, faster at first and then slowing down over time.

• The decay rate, \( \lambda \), is unique for each substance, determined by its atomic structure.
• As time progresses, the exponential decay model explains why only small amounts of the original isotope remain, which is crucial for understanding dating techniques like isotope dating.

Overall, the exponential decay equation is essential in simplifying complex radioactive behaviors to enable clearer, calculable predictions of isotope dating.

Potassium-Argon Dating

Potassium-argon dating is a specialized isotope dating method used primarily to date ancient geological samples, especially volcanic rocks. It exploits the naturally occurring radioactive decay of Potassium-40 into Argon-40. This method is particularly useful because Potassium-40 is present in many common minerals and, after decay, the Argon-40 produced becomes trapped within the mineral structure. Here's how the process works:

• When volcanic rocks form, they start without argon-40, as it gets expelled during the cooling process at volcanic heat.
• Over time, as Potassium-40 within the rock decays to Argon-40, the amount of argon gradually increases.
• By measuring the ratio of Potassium-40 to Argon-40 within the sample and knowing the decay rate, scientists can determine when the rock last cooled from a molten state.
• This technique is effective for dating rocks that are millions of years old, which helps in mapping out Earth's geological history.

It is crucial for understanding geological time scales and often adds invaluable data for reconnecting with our planet's past.