Problem 21 - General Chemistry: Principles and Modern Applications

Question

In the first-order reaction \(A \longrightarrow\) products, it is found that \(99 \%\) of the original amount of reactant \(A\) decomposes in 137 min. What is the half-life, \(t_{1 / 2}\), of this decomposition reaction?

Step-by-Step Solution

Answer

The half-life for this first order decomposition reaction is \(t_{1/2} = \frac{0.693 * 137}{ln(100)}\) minutes

Step 1: Find Rate Constant k

First, calculate the rate constant, k, using the equation for first order reactions \(k = \frac{ln[N_0 / N]}{t}\). Here, [N_0] is the initial concentration and [N] the final concentration. Since it is stated that 99% of A has decomposed, the remaining is 1%. So, [N] = 0.01[N_0]. This simplifies to \(k = \frac{ln(100)}{137}\)

Step 2: Calculate Half Life

Substitute k into the formula for half-life of a first-order reaction \(t_{1/2} = \frac{0.693}{k}\). After substituting you will get \(t_{1/2} = \frac{0.693}{\frac{ln(100)}{137}}\)

Step 3: Simplify the expression

Finally, simplify the expression to find \(t_{1/2}\). The final result after simplification will be the half life time of the reaction.

Key Concepts

Understanding the Rate Constant in First-Order Reactions

In the context of first-order reactions, the rate constant, often represented by the symbol \(k\), is a crucial parameter. It can be understood as a measure of how quickly the reactants are transformed into products. The rate constant is derived from the rate law specific to first-order reactions. This relationship is given by the equation:
\[ k = \frac{ln[N_0 / N]}{t} \]
Where:

• \([N_0]\) represents the initial concentration of the reactant
• \([N]\) is the concentration of the reactant at time \(t\)
• \(t\) is the time elapsed

In practical terms, once you know how much of the reactant remains at a certain time, you can calculate \(k\). For example, if only 1% of the original reactant remains, as it occurs in the problem, you can set \([N]\) to 0.01\([N_0]\), and solve for \(k\).

Mastering Half-Life Calculation for First-Order Reactions

The half-life of a reaction, symbolized as \(t_{1/2}\), is the time it takes for half of the original amount of reactant to decompose. In first-order reactions, the half-life is independent of the initial concentration, making it a convenient characteristic value to calculate. The formula to find the half-life in a first-order reaction is:
\[ t_{1/2} = \frac{0.693}{k} \]
This equation shows that \(t_{1/2}\) is inversely proportional to the rate constant \(k\). Thus, a larger \(k\) indicates a quicker reaction and a shorter \(t_{1/2}\).
After calculating \(k\) using the specific conditions provided in the problem (in this case, when 99% decomposition occurs), you can apply this formula for \(t_{1/2}\). This straightforward approach provides insight into how quickly the reaction progresses over time.

The Nature of Decomposition Reactions

Decomposition reactions are fascinating chemical processes where a single compound breaks down into two or more simpler substances. These reactions are commonly found in various scientific and industrial fields. In the case of a first-order decomposition reaction, like the one described, the reaction rate depends solely on the concentration of the decaying substance.
The reaction of compound \(A\) decomposing can be depicted simply as \(A \longrightarrow \text{products}\).
This illustrates that as \(A\) gets consumed, its decrease in concentration causes the reaction to continue at a predictable pace determined by the rate constant \(k\).
Such reactions are pivotal because they allow scientists and engineers to predict how fast a substance will disappear under certain conditions, which is invaluable in waste management, pollution control, and even in everyday household chores.