Question

Hydrogen peroxide, which decomposes in a first-order reaction, has a half-life of 10 hours in air. How long will it take for hydrogen peroxide to decompose to \(10 \%\) of its original concentration?

Short answer

It will take approximately \( 10 \cdot \frac{\ln(10)}{\ln(2)} \) hours or roughly 33 hours for the hydrogen peroxide to decompose to 10% of its original concentration.

Step-by-step solution

Understanding First-Order Reaction

First-order reactions have a characteristic that their rate of reaction is directly proportional to the concentration of the reactant. The half-life of a first-order reaction is constant regardless of the initial concentration. The formula relating time, initial concentration, and remaining concentration for first-order reactions is given by: \( t = \frac{\ln(\frac{[A]_0}{[A]})}{k} \), where \( t \) is the time, \( [A]_0 \) is the initial concentration, \( [A] \) is the remaining concentration, and \( k \) is the rate constant.

Determine the Rate Constant (k)

Since the half-life (\( t_{1/2} \)) is given, and for first-order reactions \( t_{1/2} = \frac{\ln(2)}{k} \), we can rearrange to solve for the rate constant \( k \): \( k = \frac{\ln(2)}{t_{1/2}} \). Plugging in the given half-life of 10 hours, we get: \( k = \frac{\ln(2)}{10} \).

Calculate the Time to Reach 10% Remaining Concentration

We want to find the time \( t \) when the concentration \( [A] \) reaches \( 10\% \) of the original concentration \( [A]_0 \). Let's set \( \frac{[A]}{[A]_0} = 0.10 \). Using the equation from Step 1 and substituting in the value for \( k \), we get: \[ t = \frac{\ln(\frac{[A]_0}{[A]})}{k} = \frac{\ln(\frac{1}{0.10})}{\frac{\ln(2)}{10}} = \frac{\ln(10)}{\frac{\ln(2)}{10}} \].

Perform the Calculation

Now, we can calculate the time \( t \) by evaluating the expression: \( t = \frac{10 \cdot \ln(10)}{\ln(2)} \).

Key concepts

Half-Life of a Reaction

The half-life of a reaction is an important concept in chemical kinetics, representing the time it takes for half of the reactant to decompose or be consumed. For first-order reactions, the half-life is particularly significant because it remains constant regardless of the starting concentration of the reactant. This constancy allows for the straightforward calculation of the reaction rate constant and the estimation of reaction progress over time. Understanding the half-life provides insights into the stability and longevity of substances and is crucial for applications spanning from pharmaceuticals to environmental science. The half-life serves not only as a benchmark for the speed of a reaction but also aids in the design of experiments and the analysis of reaction mechanisms.

Reaction Rate Constant

The reaction rate constant, represented by the symbol k, is a crucial parameter in the field of chemical kinetics. It quantifies the speed of a chemical reaction, correlating the rate of reaction to the concentrations of reactants. In a first-order reaction, the rate constant allows us to connect the reaction rate to the concentration through a simple proportionality relationship. The value of k can be derived from experimental data, such as the half-life, through rearranging the first-order reaction formula. With the rate constant known, predicting the amount of reactant that will be consumed over a given period becomes possible. This constant is essential for chemists and engineers to design chemical processes and understand how different conditions affect the speed of reactions.

First-Order Reaction Kinetics

First-order reaction kinetics describes reactions where the rate is directly proportional to the concentration of a single reactant. These kinetics are characterized by their linear relationship between the natural logarithm of reactant concentration and time, which provides a straightforward method for analyzing reaction rates and predicting reaction behavior. The rate law for a first-order reaction is rate = k[A], where [A] is the concentration of the reactant. This relationship is the foundation for deriving other useful expressions, such as the mathematical formula to calculate the time needed for a reactant to reach a certain concentration, making it indispensable for studies in reaction mechanisms and chemical process optimization.

Chemical Kinetics

Chemical kinetics is the study of reaction rates and the factors affecting them, encompassing the analysis of how quickly reactions occur and the steps involved in their pathways. The field involves examining variables such as temperature, pressure, concentration, and the presence of catalysts, to understand the intricacies of reaction mechanisms. It's a cornerstone of understanding chemical processes and is vital for the development and improvement of industrial reactions, environmental modeling, and much more. By exploring the principles of chemical kinetics, students build a deeper understanding of how and why chemical reactions take place, learning to manipulate conditions to achieve the desired reaction rates.

Concentration and Time Relationship

The relationship between concentration and time is fundamental to the understanding of reaction kinetics, particularly in first-order reactions. The concentration of a reactant decreases exponentially over time, as described by the kinetic equations. This relationship allows for the derivation of a formula that includes the rate constant and enables calculation of the time required for a reactant to reach a certain level of decomposition. It's through this lens that chemists can predict concentrations at various times and simulate reaction progression. Mastery of this concept is essential for fields that depend on precise reaction control, such as the pharmaceutical industry, where the timing of drug degradation can impact efficacy and safety.