Question

Potassium- 40 has a half-life of \(1.26 \times 10^{9} \mathrm{yr}\). decaying to calcium \(\cdot 40\) and argon- 40 in a ratio of \(8.54\) to 1 . If a rock sample contained no argon when it formed a solid but now contains one argon- 40 atom for every potassium-40 atom, how old is the rock?

Short answer

The rock is \(1.26 \times 10^{9}\) years old.

Step-by-step solution

Identify given values and formulate the half-life formula

Firstly, it's important to understand the half-life concept, which is the time required for half of a quantity of a radioactive substance to decay. We know the half-life (\(t_{0.5}\)) of Potassium-40 is \(1.26 \times 10^{9} \, \mathrm{yr}\), and a 1:1 ratio of Potassium-40 to Argon-40 atoms is now present in the rock. The generic formula for half-life is \(N = N_{0} \cdot \left( \frac{1}{2} \right)^{\frac{t}{t_{0.5}}}\) where \(N\) is the final amount, \(N_{0}\) is the initial amount, \(t\) is the time elapsed, and \(t_{0.5}\) is the half-life.

Substitute the known values within the formula

Let's start with equal number of Potassium-40 and Argon-40 atoms today. That means half the potassium has decayed because half remains, and we have 1 remaining Potassium-40 atom for every 1 Argon-40 atom. Hence, we can substitute \(N_0 = 1\), \(N = 0.5\) along with given \(t_{0.5} = 1.26 \times 10^{9} yr\) in the half-life formula to find the elapsed time \(t\). This gives us \(0.5 = 1 \cdot \left( \frac{1}{2} \right)^{\frac{t}{1.26 \times 10^{9}}}\).

Solve for t using logarithms

Rearranging gives \(\left( \frac{1}{2} \right)^{\frac{t}{1.26 \times 10^{9}}}\) = 0.5. To simplify, we'll need to apply logarithms to solve for \(t\). Taking log base 0.5 on both sides provides \(\frac{t}{1.26 \times 10^{9}}\) = log_{0.5} 0.5. Considering \(\log base \, a \, a = 1\), the right-hand-side becomes 1. Solving for \(t\), \(t = 1.26 \times 10^{9} yr\).

Key concepts

half-life

Half-life is a fundamental concept in radioactive decay, crucial for understanding how long it takes for half of the radioactive isotopes in a sample to decay. It signifies stability or the lack thereof for some elements.
In the context of Potassium-40, its half-life is extremely long, approximately \(1.26 \times 10^9\) years. This extended duration means that only half of the original Potassium-40 atoms will have decayed over that vast period.

Understanding half-life also explains why Potassium-40 is a prevalent choice for isotopic dating. As radioactive isotopes decay exponentially, the equation used to quantify this process is:
\[ N = N_0 \cdot \left( \frac{1}{2} \right)^{\frac{t}{t_{0.5}}} \]
Where:

• \(N\) is the number of remaining radioactive atoms.
• \(N_0\) stands for the initial quantity of the atoms.
• \(t_{0.5}\) corresponds to the half-life.
• \(t\) is the elapsed time.

This formula helps calculate how long a given sample has been undergoing decay by rearranging and solving for \(t\). It's particularly useful for geological studies and researchers analyzing the age of rock samples.

isotopic dating

Isotopic dating is an essential method used to determine the age of objects, like rocks or archaeological artifacts, through measuring radioisotopes.
It exploits the law of radioactive decay to assess the time elapsed since a particular isotope started decaying.

One popular technique is called Potassium-Argon dating, specifically targeting Potassium-40 (\(K-40\)) decay to Argon-40 (\(Ar-40\)).
This approach is widely used because:

• Potassium is abundant in many materials, including minerals and rocks.
• The decay product, Argon, is a noble gas and doesn’t resume reacting under natural conditions, remaining in the sample.
  

When rocks form, they typically start with just Potassium-40, but over billions of years, this changes.
Rock samples, initially devoid of Argon-40, develop more as time passes, presenting an isotope ratio over time. Calculating the ratio of Potassium to Argon allows scientists to estimate precisely when a rock solidified, offering insights into Earth's history.

potassium-40

Potassium-40 (\(K-40\)) is a naturally occurring radioactive isotope of potassium. It is one of the few isotopes that occur with any appreciable natural abundance on Earth.
This isotope has an exceptionally prolonged half-life of about \(1.26 \times 10^9\) years, making it useful for dating ancient geological formations.

Potassium-40 decays in two primary ways: a hefty 89% to calcium-40 through beta decay, and the remaining 11% to Argon-40 through electron capture or positron emission.

• Beta decay to calcium-40 is not directly exploitable for dating.
• The decay to Argon-40, however, is a powerful tool for scientists. Its half-life and formation of an inert gas retain Argon in place within the rock.
  

This decay property provides a natural clockwork mechanism, helping determine the age of the rocks that contain Potassium. Therefore, even in a state of pure rock-solid condition, Argon presence increases over time, marking a vital indicator for isotopic dating studies.