Problem 88 - A Complete Resource Book in Chemistry for JEE Main

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

Step-by-Step Solution

Answer

The age of the rock sample is \(9.0 \times 10^{9}\) years, corresponding to option (d).

Step 1: Understand the Problem

We have a rock sample that initially had no lead. Over time, the uranium present undergoes radioactive decay to form lead. The current molar ratio of uranium to lead is 1:3. This suggests that for every 1 mole of uranium, there are 3 moles of lead. We need to find the age of the rock using the half-life of uranium-238.

Step 2: Analyze the Molar Ratios

Since the molar ratio of uranium to lead is 1:3, it means for every 1 mole of uranium remaining, 3 moles of lead have been formed. Initially, the moles of uranium (U) can be expressed as (1+3)x, where x is the moles of uranium remaining, because each lead atom formed corresponds to a uranium atom that decayed.

Step 3: Calculate the Fraction of Uranium Remaining

We have x moles of uranium remaining out of the initial (1+3)x moles. Thus, the fraction of uranium left is \( \frac{x}{4x} = \frac{1}{4} \). This means that \( \frac{3}{4} \) of the uranium has decayed into lead.

Step 4: Utilizing the First Order Decay Formula

For first-order reactions, the fraction remaining can be expressed as \( \frac{1}{4} = e^{-kt} \). Using the half-life \( t_{1/2} = 4.5 \times 10^{9} \) years, the decay constant \( k \) is given by \( k = \frac{0.693}{t_{1/2}} \). Substituting \( t \) and \( k \) into the first-order decay formula allows us to solve for \( t \).

Step 5: Calculate the Decay Constant

Using the relation \( k = \frac{0.693}{t_{1/2}} \), we find the decay constant \( k = \frac{0.693}{4.5 \times 10^{9}} \approx 1.54 \times 10^{-10} \text{ year}^{-1} \).

Step 6: Solve for Time (Age of the Rock)

Substitute \( k \) and the fraction \( \frac{1}{4} \) into the equation from Step 4: \( \frac{1}{4} = e^{-kt} \). Thus, \( e^{-kt} = 0.25 \). Taking natural logarithms on both sides yields \( -kt = \ln(0.25) \). Solving for \( t \), we find \( t = \frac{-\ln(0.25)}{1.54 \times 10^{-10}} \approx 9.0 \times 10^{9} \text{ years} \).

Step 7: Choose the Correct Answer

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

Key Concepts

Uranium-238

Uranium-238 is an isotope of uranium, which is a heavy metal with significant importance in nuclear chemistry. Known for its stability among radioactive isotopes, uranium-238 naturally occurs in the environment. It undergoes a process known as radioactive decay, where it transforms into different elements over time, eventually becoming lead. This decay chain involves the emission of particles from the uranium nucleus.

Uranium-238 is particularly notable for its abundance, making up about 99.3% of natural uranium. Its slow decay rate contributes to its extensive half-life. This long half-life is a crucial factor in geological dating methods since uranium-238's decay is predictable over vast periods.

• Commonly used in dating rocks and understanding the age of the Earth.
• Provides insight into natural radioactive processes.

The element plays a pivotal role in nuclear reactors and, although not directly usable for energy production, it breeds plutonium-239, another vital nuclear fuel.

Half-Life

The concept of half-life is central to understanding radioactive decay. Half-life refers to the time it takes for half of a sample of a radioactive substance to decay into a different element. For uranium-238, the half-life is an incredibly long 4.5 billion years. This lengthy duration means it decays very slowly, making it perfect for dating rocks that are millions to billions of years old.

A half-life allows scientists and geologists to track the age of minerals and rocks since it gives a measurable timeframe over which uranium decays into lead. This transformation offers a natural clock,

• This concept helps in quantifying the age of geological samples.
• Used to estimate the remaining life of nuclear materials.

Understanding the half-life of uranium-238 is pivotal for applying the principles of radioactive dating effectively.

First-Order Reaction

Radioactive decay, like that of uranium-238, is an example of a first-order reaction. In chemical kinetics, a first-order reaction depends linearly on the concentration of one reactant. In this scenario, the rate of decay is directly proportional to the quantity of uranium remaining at any time.

The formula for a first-order reaction can be expressed as:\[N = N_{0}e^{-kt}\]Where:

• \(N\) is the amount of substance remaining.
• \(N_{0}\) is the initial amount.
• \(k\) is the decay constant.
• \(t\) is time.

This equation helps determine how much of a radioactive isotopic sample remains after a given time. By knowing the decay constant (calculated from the half-life) and using the equation, we can solve for unknowns such as the time passed or amount remaining.

Molar Ratios

Molar ratios are used to describe the relative quantities of substances involved in a chemical reaction. In the context of uranium to lead in a rock sample, the molar ratio tells us how much of the decay product is present compared to the original amount of uranium.

In radioactive dating, given a molar ratio of 1:3 (uranium to lead), we can deduce that for every 1 mole of uranium left, 3 moles of lead have formed. This is because the transformation is a 1:1 process, indicating that as 1 mole of uranium decays, 1 mole of lead is produced.

• Essential in stoichiometry to calculate the progression of reactions.
• Helps identify the initial and remaining amounts of substances in decay scenarios.

Understanding molar ratios allows us to track and quantify the changes in the sample over time, thus assisting in age determination of geological samples.