Problem 41 - Chemistry: The Central Science

Question

Potassium- 40 decays to argon- 40 with a half-life of \(1.27 \times 10^{9}\) yr. What is the age of a rock in which the mass ratio of \({ }^{40} \mathrm{Ar}\) to \({ }^{40} \mathrm{~K}\) is \(4.2\) ?

Step-by-Step Solution

Answer

The age of the rock is approximately \(1.74 \times 10^9\) years.

Step 1: Recall the radioactive decay formula

The radioactive decay formula is given by: \[ N_t = N_0 \cdot (0.5)^\frac{t}{t_{1/2}} \] Where: - \(N_t\) is the amount of the substance remaining after time \(t\), - \(N_0\) is the initial amount of the substance, - \(t_{1/2}\) is the half-life of the substance, and - \(t\) is the time elapsed.

Step 2: Write the formula in terms of the mass ratios

We have the mass ratio of Ar-40 to K-40 in the rock, which can be represented as: \[ \frac{N_\text{Ar}}{N_\text{K}} = 4.2 \] Since Potassium-40 decays to Argon-40, we know that the initial amount of Argon-40 in the rock was 0. Let's denote the initial amount of Potassium-40 as \(N_0\). Therefore, after time \(t\), we'll have \(N_0 - N_t\) as the mass of Argon-40. So our mass ratio equation now looks like: \[ \frac{N_0 - N_t}{N_t} = 4.2 \]

Step 3: Substitute the radioactive decay formula into the mass ratio equation

We know that \(N_t = N_0 \cdot (0.5)^\frac{t}{t_{1/2}} \), so we can substitute this into our mass ratio equation: \[ \frac{N_0 - N_0 \cdot (0.5)^\frac{t}{t_{1/2}}}{N_0 \cdot (0.5)^\frac{t}{t_{1/2}}} = 4.2 \]

Step 4: Simplify and solve for \(t\)

Notice that \(N_0\) appears in both the numerator and the denominator, so it can be canceled out: \[ \frac{1 - (0.5)^\frac{t}{t_{1/2}}}{(0.5)^\frac{t}{t_{1/2}}} = 4.2 \] Now we need to solve for \(t\). We can begin by isolating the exponential term: \[1 - (0.5)^\frac{t}{t_{1/2}} = 4.2 \cdot (0.5)^\frac{t}{t_{1/2}} \] \[5.2 \cdot (0.5)^\frac{t}{t_{1/2}} = 1\] Now we can take the natural logarithm of both sides: \[ \frac{t}{t_{1/2}} \cdot \ln(0.5) = \ln{\frac{1}{5.2}} \] Now, we can solve for \(t\): \[ t = \frac{\ln{\frac{1}{5.2}}}{\ln(0.5)} \cdot t_{1/2} \] We are given that the half-life of Potassium-40, \(t_{1/2} = 1.27 \times 10^9\) years, so: \[ t = \frac{\ln{\frac{1}{5.2}}}{\ln(0.5)} \cdot (1.27 \times 10^9) \text{ years} \] Calculate the value of \(t\): \[ t \approx 1.74 \times 10^9 \text{ years} \] Therefore, the age of the rock is approximately \(1.74 \times 10^9\) years.

Key Concepts

Half-life

The concept of half-life is integral to understanding radioactive decay. In simple terms, a half-life is the time it takes for half of a given quantity of a radioactive substance to decay. This is a constant rate of decay, characteristic of each radioactive isotope.

• For Potassium-40, the half-life is approximately \(1.27 imes 10^9\) years.
• This means that if you start with 100 grams of Potassium-40, only 50 grams would remain after 1.27 billion years.

Understanding half-life helps scientists to not only calculate the age of rocks but also track environmental changes and understand fundamental principles of nuclear physics.
Knowing the half-life gives us a mathematical foundation to solve decay problems, like determining how much time has passed given a ratio of parent isotope to daughter product isotope.

Potassium-40

Potassium-40 is a naturally occurring isotope comprised of the element potassium. It is radioactive, meaning it undergoes decay over time, and this characteristic is pivotal in scientific studies such as geochronology and archaeology.

• Potassium-40 decays to Argon-40 and Calcium-40.
• Due to its long half-life, it is particularly useful in dating geological formations and archaeological artifacts.

Understanding Potassium-40's decay process allows scientists to perform radiometric dating, providing crucial insights into Earth's history and the evolution of life.
The ability of Potassium-40 to transform into Argon-40 over millions or billions of years is what allows scientists to use it as a dating tool for ancient rocks and minerals.

Argon-40

Argon-40 is the stable daughter isotope that results from the radioactive decay of Potassium-40. When Potassium-40 decays, it transforms into Argon-40 within a rock or mineral.

• Because Argon is a noble gas, it doesn't bond with other elements, making it easy to isolate and measure.
• This property ensures that Argon-40 remains trapped within mineral lattices until analyzed.

In the context of radiometric dating, the presence of Argon-40 is used to calculate the age of a rock sample.
By measuring the ratio of Argon-40 to Potassium-40 in a sample, geologists can determine how long the Potassium-40 in a rock has been decaying, which in turn tells them the age of the rock.

Radiometric Dating

Radiometric dating is a scientific technique used to date materials by comparing the abundance of a radioactive isotope within the material to the abundance of its decay products.

• The method leverages the constant rate of decay of radioactive isotopes, such as Potassium-40.
• It provides highly accurate ages for rocks, minerals, and even archaeological objects.

Radiometric dating, including Potassium-Argon dating, is one of the most reliable methods for age determination. By using ratios of parent isotopes to daughter products, scientists can deduce the time that has elapsed since a rock or mineral cooled to a closed system. Whether it's the moon's rocks, ancient fossils, or Earth's earliest rock formations, radiometric dating offers a window into our planet's past.