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 sample is \(9.0 \times 10^{9}\) years, corresponding to option (d).

Step-by-step solution

Understand the Reaction

The reaction is a first-order radioactive decay of Uranium-238 transforming into Lead through multiple decay reactions. We need to find out the time it has taken to achieve a Uranium to Lead ratio of 1:3.

Use First-Order Decay Equation

The first-order decay equation is given by \( N(t) = N_0 e^{-kt} \), where \( N(t) \) is the amount of substance left after time \( t \), \( N_0 \) is the initial amount, and \( k \) is the decay constant. We will find \( k \) first.

Calculate Decay Constant

The decay constant \( k \) can be found using the half-life equation: \( k = \frac{ln(2)}{t_{1/2}} \). Given \( t_{1/2} = 4.5 \times 10^{9} \) years, calculate \( k \).

Determine Initial and Final Quantities

Let the initial quantity of Uranium be \( N_0 \). After decay, \( N(t) = \frac{N_0}{4} \) because 1 part Uranium to 3 parts Lead signifies 4 parts total with one part Uranium remaining. Therefore, \( N_0 - N(t) = 3 \times N(t) \).

Apply the Formula for Current Quantity

Using the current quantity \( N(t) = \frac{N_0}{4} \), substitute into the first-order decay formula: \( \frac{N_0}{4} = N_0 e^{-kt} \). Simplify to find \( e^{-kt} \).

Solve for Time t

From \( \frac{1}{4} = e^{-kt} \), take the natural log: \( ln\left(\frac{1}{4}\right) = -kt \). Solve for \( t \) using \( k \) calculated earlier. This gives \( t = \frac{-ln(1/4)}{k} \).

Calculate Time

Use the calculated decay constant and natural log value to compute \( t \). You should find \( t = 9.0 \times 10^{9} \) years.

Key concepts

First-Order Reaction

Radioactive decay, like that of uranium-238 to lead, is often classified as a first-order reaction. In simple terms, a first-order reaction is one where the rate of reaction is directly proportional to the amount of the substance that is left. This means if you have more of the substance, it decays faster compared to when you have less of it.
For first-order reactions, the mathematical description follows the form of exponential decay. The amount of a substance at any given time is represented by the equation:

• \[ N(t) = N_0 e^{-kt} \]

Where:

• \( N(t) \) is the remaining amount at time \( t \)
• \( N_0 \) is the original amount
• \( k \) is the decay constant, which we will discuss later on
• \( e \) is the base of the natural logarithm

Understanding this equation helps to see how quickly a substance decreases over time due to radioactive decay.

Half-Life Calculation

In the context of radioactive decay, the half-life is a crucial concept. It is the time it takes for half of the original radioactive sample to decay. For uranium-238, this half-life is known to be approximately 4.5 billion years. This timeframe is incredibly long, which contributes to the use of uranium in dating geological formations.
To calculate the decay constant \( k \), you can use the half-life equation:

• \[ k = \frac{ln(2)}{t_{1/2}} \]

Where:

• \( ln(2) \) is the natural logarithm of 2, approximately 0.693
• \( t_{1/2} \) is the half-life of the substance

So for uranium-238, the decay constant can be computed once you substitute the known values, providing an essential tool for further calculations related to age determination.

Uranium-Lead Dating

Uranium-lead dating is one of the oldest and most reliable methods of radiometric dating. It involves calculating the age of a rock or mineral by examining the ratio of uranium-238 to lead. Over millions and billions of years, uranium-238 decays to lead.
In the problem discussed, a rock initially depleted of lead is found to have a ratio of uranium to lead as 1:3. This means for every unit of uranium, there are three equivalent units of lead as a result of decay. Understanding this ratio allows scientists to determine the age of the rock. By using the first-order decay equation, they calculate how long it took for the original uranium to partially decay into lead. The derived age, in this case, is 9 billion years.

Decay Constant

The decay constant, symbolized as \( k \), is a pivotal factor in the radioactive decay equation. It represents a probability rate at which a single atom might decay. Thus, it helps quantify the speed at which the whole sample undergoes decay.
To compute \( k \), we utilize the half-life of the substance through:

• \[ k = \frac{ln(2)}{t_{1/2}} \]

The constant \( k \) provides a link between the mathematical model of decay and the actual physical process happening at a molecular level. The higher the decay constant, the faster the rate of decay. In contexts such as the calculated age of the rock, knowing \( k \) is essential as it allows for accurate age estimations through formulary calculations.