Question

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

Short answer

The age of the rock is approximately \(9.18 \times 10^8 \,\mathrm{yr}\).

Step-by-step solution

Write down the given information

We are given the following information: - Mass ratio of \(^{40}Ar\) to \(^{40}K\) is \(4.2\). - Half-life of potassium-40 (\(t_{1/2}\)) is \(1.27 \times 10^9 \mathrm{yr}\).

Calculate the decay constant

The decay constant (\(λ\)) is related to the half-life by the following formula: \[t_{1/2} = \frac{\ln{2}}{λ}\] We need to solve for \(λ\): \[λ = \frac{\ln{2}}{t_{1/2}}\] Now, we can plug in the given value for the half-life: \[λ = \frac{\ln{2}}{1.27 \times 10^9 \mathrm{yr}}\] Calculating this, we get: \[λ \approx 5.45 \times 10^{-10}\,\mathrm{yr^{-1}}\]

Calculate the age of the rock using the decay formula

The decay formula relates the current amount of parent isotopes (\(N\)), the original amount of parent isotopes (\(N_0\)), the decay constant (\(λ\)), and the age of the sample (\(t\)): \[N = N_0 e^{-λt}\] We are given the mass ratio of \(^{40}Ar\) to \(^{40}K\), so we can express the current amount of parent isotopes (\(N\)) as: \[N = \frac{1}{1+4.2}\] To find the age of the rock, we need to solve for \(t\) in the decay formula. We can rewrite the decay formula as: \[t = \frac{\ln{\frac{N_0}{N}}}{λ}\] Now, we can plug in the values for \(N\), \(λ\), and \(N_0\): \[t = \frac{\ln{\frac{1}{\frac{1}{1+4.2}}}}{5.45 \times 10^{-10}\,\mathrm{yr^{-1}}}\] Calculating this, we get: \[t \approx 9.18 \times 10^8 \,\mathrm{yr}\]

State the age of the rock

The age of the rock is approximately \(9.18 \times 10^8 \,\mathrm{yr}\).

Key concepts

Radioactive Decay

When we speak of radioactive decay, we refer to the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs naturally in unstable isotopes and can take the form of emitting alpha particles, beta particles, or gamma rays. Radioactive decay is a random process at the level of single atoms, meaning that it is impossible to predict when a particular atom will decay. However, for a large number of atoms, the decay rate can be expressed statistically and is found to be exponential.

In essence, as time progresses, we can observe that the amount of the original radioactive substance, often called the parent isotope, decreases, while the amount of the decay product, known as the daughter isotope, increases. This process continues until a stable isotope is formed or the sequence of decay has concluded.

Half-Life

The half-life of a radioactive isotope is defined as the time it takes for half of the atoms in a given sample to decay. It is a key concept in understanding radioactive decay because it is not dependent on how much material is present and remains constant over time for a specific isotope.

Knowing the half-life is crucial for various applications, such as radioactive dating. For example, if the half-life of a substance is 1,000 years, after 1,000 years, only half of the original amount of the substance remains. In another 1,000 years, half of that remaining amount will have decayed, leaving a quarter of the original amount, and so on. This concept of half-life provides a way to measure the age of substances and objects based on the principles of radioactive decay.

Decay Constant

The decay constant, denoted by the symbol \(λ\), represents the probability per unit time that a given atom will decay. It is a fundamental property of the radioactive isotope and is directly related to the half-life. To find the decay constant of an isotope, you can use the formula \[λ = \frac{\ln{2}}{t_{1/2}}\], where \(t_{1/2}\) is the half-life.

The decay constant is central to calculating the time it takes for a particular amount of radioactive substance to decay as it factors into equations that model the behavior of radioactive decay. In our potassium-argon dating example, the decay constant was used to calculate the age of a rock based on the remaining amount of potassium-40 and the produced argon-40.

Isotopic Dating Methods

Isotopic dating methods are used to determine the age of materials based on the radioactive decay of isotopes within them. Potassium-argon dating, highlighted in our exercise, is one such method. It relies on the decay of potassium-40 (\(^{40}K\)) to argon-40 (\(^{40}Ar\)), with the ratio of daughter and parent isotopes serving as a 'clock' to measure the time since the rock solidified.

The basic principle of isotopic dating is simple: measure how much of the parent isotope remains and how much daughter isotope has been produced. From that data, and knowing the half-life of the isotope, one can calculate the time that has elapsed since the parent isotope started to decay, which usually corresponds to the age of the rock. Thanks to these methods, geologists can construct detailed chronologies of Earth's history and the timing of major events such as volcanic eruptions or the formation of mineral deposits.