Question

Geologists can estimate the age of rocks by their uranium- 238 content. The uranium is incorporated in the rock as it hardens and then decays with first- order kinetics and a half-life of 4.5 billion years. A rock contains \(83.2 \%\) of the amount of uranium- 238 that it contained when it was formed. (The amount that the rock contained when it was formed can be deduced from the presence of the decay products of U-238.) How old is the rock?

Short answer

The age of the rock is \( \log_2(1/0.832) \times 4.5 \) billion years.

Step-by-step solution

Understanding First-Order Kinetics and Half-Life

First-order kinetics implies that the rate of decay of the uranium-238 is proportional to the amount that remains. The half-life is the time required for half of the substance to decay. Uranium-238 has a half-life of 4.5 billion years, which means every 4.5 billion years, the amount of Uranium-238 in the rock will be reduced to half of its previous amount.

Calculating the Number of Half-Lives

Using the half-life formula \(N_t = N_0 \times (1/2)^{t/t_{\frac{1}{2}}}\), where \(N_t\) is the amount of substance at time \(t\), \(N_0\) is the initial amount of substance, \(t\) is the elapsed time, and \(t_{\frac{1}{2}}\) is the half-life. We can rearrange the formula to solve for the number of half-lives as follows: \( (1/2)^{t/t_{\frac{1}{2}}} = N_t/N_0 \), and subsequently \(t/t_{\frac{1}{2}} = \log_2(N_0/N_t)\).

Solve for the Age of the Rock

With the given data that the rock contains \(83.2\%\) of the Uranium-238 it started with, we have \(N_t/N_0 = 0.832\). Taking the logarithm to the base 2, we get \(\log_2(1/0.832)\), and multiplying this value by the half-life of Uranium-238 (4.5 billion years) gives us the age of the rock. Hence, \(t = \log_2(1/0.832) \times 4.5 \) billion years.

Key concepts

First-Order Kinetics

Understanding the decay of radioactive substances is crucial in multiple scientific fields. First-order kinetics is a term used to describe processes where the rate of reaction is directly proportional to the concentration of the reactant that remains. In the context of radioactive decay, it means the rate at which uranium-238 decays is directly linked to how much of it is still present. To put it simply, the more uranium-238 you start with, the faster it will decrease in quantity. However, this decrease occurs at a consistent percentage rate over time rather than a constant amount.

This concept is essential to grasp because it helps us calculate how much time has passed based on the remaining quantity of uranium-238 in a rock. Since the rate of decay is steady as per first-order kinetics, we can use mathematical models to predict how long ago the rock formed, assuming we know the half-life of uranium-238 and the remaining percentage of it in the rock.

Half-Life Calculation

The half-life of a radioactive substance is a key concept in understanding its behavior over time. It is the period it takes for half of the radioactive atoms to decay. For uranium-238, this period is 4.5 billion years. How is this useful? Essentially, by knowing the half-life, you can determine how many of these periods have passed based on how much of the original material is left.

With each passing half-life, the quantity of the remaining uranium-238 gets halved. So, if you were to observe a sample over 9 billion years, you would expect it to have one-quarter of the original uranium-238 remaining. Mathematically speaking, you can calculate the number of half-lives that have elapsed to help estimate the age of rocks, which is what geologists commonly do in their research.

Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. In our case, uranium-238 undergoes radioactive decay until it becomes lead-206, a stable element. This happens through a series of steps that involve emitting different types of particles and radiation over time. The decay rate is consistent and can be represented mathematically, which is indispensable for techniques like uranium-238 dating.

What's important to understand is that the decay of uranium-238 isn't something that occurs instantly. It's a slow process that unfolds over billions of years, allowing geologists to look back in time and estimate the age of the Earth's oldest rocks.

Geological Age Estimation

The accurate estimation of the age of rocks is a cornerstone in the field of geology, providing insights into Earth's history and the formation of its landscapes. Uranium-238 dating is one method geologists use to date ancient rock formations. By measuring the remaining uranium-238 and its decay products, scientists can backtrack to determine when the rock solidified and consequently formed.

The calculation hinges on understanding and applying the principles of radioactive decay and first-order kinetics. Once geologists determine how much uranium-238 remains in the rock compared to what they would expect to find if it were new, they can estimate how much time has elapsed. The process involves complex calculations and careful analysis, but the fundamental idea is that by examining what's left behind in these ancient materials, we can unlock the secrets of our planet's distant past.