The age of an igneous rock that has solidified from magma can be found by analyzing the amount of 40K and 40Ar. Potassium-40 emits a positron to produce argon-40 and the half-life of 40K is 1.25 * 109 years. What is the age of a rock whose 40Ar/40K ratio is 1.42?

Verified step by step guidance

1. Determine the decay equation for Potassium-40 (40K) converting into Argon-40 (40Ar). The decay equation is based on the first-order kinetics: \( N = N_0 e^{-kt} \), where \( N \) is the amount of 40K remaining, \( N_0 \) is the initial amount of 40K, \( k \) is the decay constant, and \( t \) is the time elapsed.
2. Calculate the decay constant (k) using the half-life formula: \( k = \frac{\ln(2)}{t_{1/2}} \), where \( t_{1/2} \) is the half-life of 40K, which is 1.25 * 10^9 years.
3. Set up the ratio of 40Ar to 40K, which is given as 1.42. Since the amount of 40Ar produced is equal to the initial amount of 40K minus the remaining amount of 40K, the ratio can be expressed as \( \frac{N_0 - N}{N} = 1.42 \).
4. Solve the equation \( \frac{N_0 - N}{N} = 1.42 \) for \( N_0 \) in terms of \( N \) and substitute into the decay equation to find \( t \).
5. Solve for \( t \), the age of the rock, using the decay equation with the calculated decay constant and the relationship derived from the 40Ar/40K ratio.

Key Concepts

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. In this context, potassium-40 (40K) decays into argon-40 (40Ar) through positron emission. Understanding this process is crucial for determining the age of rocks, as the ratio of parent isotope (40K) to daughter isotope (40Ar) provides insight into the time elapsed since the rock solidified.

Half-Life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. For potassium-40, the half-life is approximately 1.25 billion years. This concept is essential for calculating the age of the rock, as it allows us to relate the measured ratio of 40Ar to 40K to the time that has passed since the rock formed.

Isotope Ratio Dating

Isotope ratio dating is a method used to determine the age of materials by measuring the relative abundances of isotopes. In this case, the ratio of argon-40 to potassium-40 (40Ar/40K) is used to calculate the age of the igneous rock. By applying the known half-life of 40K, one can derive the age of the rock based on the observed isotope ratio.