Problem 92 - Chemistry The Molecular Science

Question

All radioactive decays are first order. Why is this so?

Step-by-Step Solution

Answer

Radioactive decay is first order because the decay rate is proportional to the number of undecayed nuclei, fitting first-order kinetics.

Step 1: Understanding First-Order Reactions

First-order reactions are characterized by the rate of reaction being directly proportional to the concentration of a single reactant. In mathematical terms, this is expressed in the rate law as \( r = k[A] \), where \( r \) is the reaction rate, \( k \) is the rate constant, and \( [A] \) is the concentration of the reactant.

Step 2: Understanding Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This process depends solely on the radioactive substance itself and not on its concentration, environment, or the presence of other substances.

Step 3: Analyzing the Relationship Between Decay and First-Order Kinetics

Radioactive decay fits the model of a first-order reaction because the rate of decay is directly proportional to the number of undecayed nuclei present at any given time. This aligns with the first-order rate law \( r = k[N] \), where \( [N] \) is the number of radioactive nuclei.

Step 4: Expression of Radioactive Decay Mathematically

The mathematical expression for radioactive decay is \( N(t) = N_0 e^{-kt} \), where \( N(t) \) is the amount of substance remaining at time \( t \), \( N_0 \) is the initial amount of substance, \( k \) is the decay constant, and \( t \) is the time elapsed. This equation is derived from the first-order kinetics model.

Step 5: Conclusion and Reasoning

Radioactive decay follows first-order kinetics because it is an exponential decay process where the rate of decay is a constant proportion of the remaining substance at any time, independent of the concentration, thus fitting the definition of first-order reactions.

Key Concepts

First-Order Reactions

First-order reactions are essential in understanding many chemical processes, especially radioactive decay. These reactions are unique because their rate is directly proportional to the concentration of one reactant. This can be expressed with the rate law equation:

• \( r = k[A] \)
• \( r \): reaction rate
• \( k \): rate constant
• \( [A] \): concentration of the reactant

In simpler terms, this means the rate at which the reaction occurs is reliant on just one factor—the concentration of the reactant. This relationship helps predict how quickly a reaction will proceed under different conditions.

Rate Law

The rate law provides a precise mathematical way to describe how a reaction rate depends on the concentration of reactants. It is significant because it quantitatively tells us about the reaction's speed and mechanism. In the case of first-order reactions, the rate law is:\[ r = k[A] \]Here are the key components:

• \( r \) - the speed of the reaction
• \( k \) - the specific rate constant for the reaction, unique to each reaction and conditions like temperature
• \( [A] \) - the concentration of the reactant

By understanding the rate law, scientists can manipulate reaction conditions to control how fast or slow a reaction takes place.

Exponential Decay

Exponential decay is a pattern where quantities decrease rapidly at first and then slow down significantly as time goes on. This concept is central to radioactive decay, where unstable nuclei lose energy.The mathematical representation of exponential decay in radioactive materials is:\[ N(t) = N_0 e^{-kt} \]Where:

• \( N(t) \): number of radioactive atoms remaining after time \( t \)
• \( N_0 \): initial number of atoms
• \( e \): base of natural logarithms
• \( k \): decay constant
• \( t \): time

This equation shows that over equal time intervals, the percentage of the original radioactive material decreases by a consistent amount, showcasing the nature of exponential functions.

Decay Constant

The decay constant, denoted by \( k \), is a pivotal element in understanding the rate of radioactive decay. It is a measure of the probability per unit time that a given atom will decay. The larger the decay constant, the quicker the rate of decay.In the exponential decay equation \( N(t) = N_0 e^{-kt} \), \( k \) governs how rapidly the number of undecayed atoms decreases over time. The decay constant links the unique properties of a substance to its half-life, which is the time required for half of the radioactive atoms in a sample to decay. This relationship is given by:\[ t_{1/2} = \frac{0.693}{k} \]Where \( t_{1/2} \) is the half-life. Understanding \( k \) not only reveals how quickly a substance decays but also helps in making practical decisions in fields such as nuclear medicine, carbon dating, and nuclear energy.