Question

Lead is the final product formed by a series of changes in which the rate determining stage is the radioactive decay of uranium-238. This radioactive decay is a first order reaction with a half-life of \(4.5 \times 10^{9}\) years. What would be the age of a rock sample originally lead free, in which the molar proportion of uranium to lead is now \(1: 3\) ? (a) \(1.5 \times 10^{9}\) years (b) \(2.25 \times 10^{9}\) years (c) \(4.5 \times 10^{9}\) years (d) \(9.0 \times 10^{9}\) years

Short answer

The age of the rock is \(9.0 \times 10^{9}\) years.

Step-by-step solution

Understanding the Half-life Formula

The half-life formula for a first order reaction is \( t_{1/2} = \frac{0.693}{k} \), where \( t_{1/2} \) is the half-life and \( k \) is the decay constant. Given that the half-life \( t_{1/2} \) is \( 4.5 \times 10^{9} \) years, we first need to find \( k \).

Calculate Decay Constant

Using the half-life formula \( k = \frac{0.693}{t_{1/2}} \), we substitute \( t_{1/2} = 4.5 \times 10^{9} \). So, \( k = \frac{0.693}{4.5 \times 10^{9}} \approx 1.54 \times 10^{-10} \text{ year}^{-1} \).

Understanding the Molar Relationship

The problem states that the molar proportion of uranium to lead is \( 1:3 \). This means for every 4 parts, 1 part is uranium and 3 parts are lead.

Using the First Order Decay Equation

For first-order decay, we use the equation \( N_t = N_0 e^{-kt} \). Here, \( N_t \) is the remaining amount of uranium, and \( N_0 \) is the initial amount of uranium. After decay, \( N_t \) is now 1/4 of \( N_0 \) because 3/4 has decayed to lead. Therefore, \( \frac{N_t}{N_0} = \frac{1}{4} \).

Solve for Time, \( t \)

Using the equation \( \frac{N_t}{N_0} = e^{-kt} \), substitute \( \frac{1}{4} = e^{-1.54 \times 10^{-10}t} \). Taking natural logarithms on both sides gives us \( \ln(\frac{1}{4}) = -1.54 \times 10^{-10}t \), and thus, \( t = \frac{- \ln(\frac{1}{4})}{1.54 \times 10^{-10}} \approx 9 \times 10^{9} \text{ years} \).

Compare with Given Options

The calculated age of the rock is \( 9.0 \times 10^{9} \) years, which matches option (d).

Key concepts

First Order Reaction

Radioactive decay, like the decay of uranium-238, follows the principle of **first-order reaction kinetics**. This means that the rate of decay depends only on the quantity of the substance left. Imagine it as a slow ticking clock, with the hands of the clock moving at a rate proportional to how much remains.

For first-order reactions, the decay rate can be mathematically represented by the equation: \[ r = k[N] \] where \( r \) is the rate of reaction, \( k \) is the decay constant, and \([N]\) is the concentration of the reactant.
When dealing with large numbers of atoms, we can predict how long it will take for a certain portion to decay by using this principle.

The beauty of a first-order reaction lies in its simplicity. No matter how much uranium-238 there is initially, the time it takes for half of it to decay is constant, defined by the half-life specific to the element. This makes it easier for scientists to predict the age of materials, like our rock sample, just by knowing the remaining proportion of the material.

Uranium-238

Uranium-238 is one of the most common isotopes of uranium and is found naturally in the earth's crust. It holds particular significance in geological and archaeological dating due to its long half-life.

Let’s delve into what makes uranium-238 special: - **Long Half-life**: With a half-life of approximately \(4.5 \times 10^9\) years, it decays very slowly, making it ideal for dating ancient rocks.- **Decay Process**: Throughout its decay, uranium-238 goes through a series of transformations, finally turning into lead-206. This is a multi-step process that involves the emission of alpha particles and, in some instances, beta particles.
This decay pathway is a valuable tool for understanding the age of rocks and even the Earth itself. Through the radioactive decay chain, scientists can also learn about the thermal history and evolutionary processes of the Earth's crust. When we measure the remaining uranium versus the produced lead in a rock sample, we gain insights into its age and history.

Half-life Calculation

The concept of half-life is crucial for understanding radioactive decay. It represents the time required for half of a given quantity of a radioactive substance to decay. This makes predicting and calculating the age of samples possible, providing a spotlight into ancient worlds.

Here's how half-life works: - **Consistent Timeframe**: Regardless of the initial amount, half of the substance will decay over each subsequent half-life.- **Mathematical Representation**: This is illustrated through the formula: \[ t_{1/2} = \frac{0.693}{k} \] where \( t_{1/2} \) is the half-life, and \( k \) is the decay constant interpreted from the first order kinetics.
In our exercise, we used the knowledge of uranium-238's half-life to deduce the age of a rock by looking at the uranium to lead ratio. Starting from a known half-life and understanding the proportion of decayed to original material allows you to solve for the sample's age with precision. By rearranging equations and using logarithms, you get the time frame indicating how long it has been since the rock was lead-free. This use of half-life is fundamental in the fields of geology and archaeology.