Problem 66 - Chemistry The Molecular Nature of Matter

Question

Radon- 220 is radioactive, and decays into polonium- 216 by emitting an alpha particle. This is a first-order process with a rate constant of \(0.0125 \mathrm{~s}^{-1}\). When the concentration of radon- 220 is \(1.0 \times 10^{-9} \mathrm{~mol} \mathrm{~L}^{-1}\), what is the rate of the reaction?

Step-by-Step Solution

Answer

The rate of the reaction is \(1.25 \times 10^{-11}\mathrm{~mol} \mathrm{~L}^{-1} \mathrm{s}^{-1}\).

Step 1: Understand the Rate Law for First-Order Reactions

For a first-order reaction, the rate law is given by the equation: \( rate = k[\text{A}] \), where \( k \) is the rate constant and \([\text{A}]\) is the concentration of the reactant. In this case, \( k = 0.0125\mathrm{~s}^{-1} \) and the concentration of Radon-220 is \( 1.0 \times 10^{-9}\mathrm{~mol} \mathrm{~L}^{-1} \).

Step 2: Plug in the Values into the Rate Law Equation

Substitute the given values into the rate law equation: \( rate = 0.0125\mathrm{~s}^{-1} \times 1.0 \times 10^{-9}\mathrm{~mol} \mathrm{~L}^{-1} \).

Step 3: Calculate the Rate of the Reaction

Perform the multiplication to find the rate of the reaction: \( rate = 0.0125\mathrm{~s}^{-1} \times 1.0 \times 10^{-9}\mathrm{~mol} \mathrm{~L}^{-1} = 1.25 \times 10^{-11}\mathrm{~mol} \mathrm{~L}^{-1} \mathrm{s}^{-1} \).

Key Concepts

Rate Law

Understanding the rate law is crucial for studying the speed of chemical reactions. It is a mathematical equation that relates the rate of a reaction to the concentration of its reactants. For first-order reactions, the rate law has a simple form:
\[ rate = k[\text{A}] \]
In this equation, \( k \) represents the rate constant, a unique value for each reaction that depends on factors like temperature, and \( [\text{A}] \) symbolizes the concentration of the reactant. Focusing on our example with Radon-220, by knowing the rate constant \( k \) and the initial concentration of Radon-220, we can easily apply the formula to calculate the reaction rate. Through this process, it becomes transparent how the concentration directly influences the rate at which the reaction proceeds. This equation is a powerful tool because once the rate constant is determined, it allows us to predict the reaction rate at any given concentration of reactants.

Radioactive Decay

Radioactive decay is a spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. This phenomenon is a natural occurrence found in elements such as Uranium, Carbon-14, and as mentioned in our example, Radon-220. The decay of these unstable isotopes follows a first-order reaction, where each nucleus has a fixed probability of decaying per unit time, hence the use of the rate law.
In our exercise, Radon-220 decays by emitting an alpha particle, transforming into Polonium-216. The rate constant (\( 0.0125 \text{s}^{-1} \)) tells us how quickly this decay process occurs. Radioactive decay is not only crucial for the field of nuclear chemistry but also for applications such as radiometric dating, cancer treatment, and understanding the behavior of naturally occurring and man-made radionuclides.

Half-Life

The concept of half-life is intimately connected with radioactive decay and chemical kinetics. Half-life, often represented by the symbol \( t_{1/2} \), is the time required for half the amount of a reactant to be consumed or for half of a radioactive substance to decay. For a first-order reaction, the half-life is given by the equation:
\[ t_{1/2} = \frac{\ln(2)}{k} \]
It is a constant value that does not depend on the initial concentration of reactant. This property makes it particularly useful since knowing the half-life and rate constant allows you to understand the longevity and decay profile of a substance. For many real-life applications, such as pharmaceuticals and environmental monitoring, knowing the half-life is crucial for safety and effectiveness analysis.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed of chemical reactions and the factors that influence this rate. It encompasses not just the rate law but a broader set of principles that dictate how and why chemical reactions occur as they do. Factors such as temperature, catalysts, surface area, and concentration all play integral roles in reaction kinetics. Understanding kinetics is essential for controlling processes like synthesizing new materials, degrading pollutants, and even digesting food. Kinetics also helps researchers develop better catalysts that increase reaction efficiency, which is critical in industries ranging from pharmaceuticals to energy production. The study of chemical kinetics bridges the gap between knowing a reaction can occur and understanding how it can be harnessed practically.