Problem 75 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The decomposition of hydrogen peroxide in aqueous solution is a first-order reaction: Time in min \(0 \quad 10\) Volume \((V\), in \(\mathrm{ml}) \quad 25.0 \quad 20.0\) where \(V\) is the number of \(\mathrm{ml}\) of potassium permanganate required to decompose a definite volume of hydrogen peroxide solution, at the given time. From the following data, the rate constant (in \(\min ^{-1}\) ) of reaction is \((\ln 5=1.6, \ln 2=0.7)\) (a) \(0.09\) (b) \(0.02\) (c) \(0.2\) (d) \(0.16\)

Step-by-Step Solution

Answer

The rate constant of the reaction is \(0.2\ \min^{-1}\).

Step 1: Understanding the Concept

First-order reactions have a rate constant k that is determined by the equation \[\ln\left(\frac{[A]_0}{[A]}\right) = kt\], where \([A]_0\) is the initial concentration, \([A]\) is the concentration at time t, and k is the rate constant.

Step 2: Setting Up the Equation

For the given first-order reaction, we can set up the equation using the volumes of potassium permanganate as a proxy for concentration, since volume is directly proportional to concentration for a given solution. Substitute \(V_0 = 25.0\ ml\) as the initial volume, \(V = 20.0\ ml\) as the volume at time \(t = 10\ min\), and solve for k.

Step 3: Substitute the Given Values

Substitute the given values into the equation to yield \[\ln\left(\frac{25.0}{20.0}\right) = k \cdot 10\]. Simplify the equation by reducing the fraction and using the given values for natural logarithms.

Step 4: Calculate the Rate Constant

Divide both sides by 10 to isolate k and make use of the given natural logarithms: \[\ln(1.25) = k \cdot 10\] \[\ln(5) - \ln(4) = k \cdot 10\] Since \(\ln(4) = 2\ln(2)\), we can further simplify the equation to find the value of k.

Step 5: Solve for k

Using the properties of logarithms and known values, we obtain \[1.6 - 2\cdot 0.7 = k \cdot 10\], and solving for k gives us the rate constant of the reaction.

Key Concepts

Chemical Kinetics

Chemical kinetics is the branch of physical chemistry that studies the rates of chemical reactions, as well as the mechanisms by which they occur.
This field is pivotal for understanding how reactions take place and for predicting the speed at which they progress. The rate at which a reactant is consumed or a product is formed over a specific time period is fundamental to kinetics. By measuring these changes, chemists can determine the rate law, identify the reaction order, and calculate the rate constant for the reaction.

In the context of the decomposition of hydrogen peroxide, a classic first-order kinetics process, the rate at which the reaction occurs is proportional to the concentration of hydrogen peroxide. Therefore, finding the rate constant provides insight into the efficiency of the reaction under specific conditions.

Reaction Rate

The reaction rate tells us the speed at which reactants are converted into products in a chemical reaction. It can be determined by measuring the change in concentration of reactants or products over time.
For a first-order reaction, the rate is directly proportional to the concentration of the single reactant. This relationship can be represented mathematically as rate = k[A], where 'k' is the rate constant and '[A]' is the concentration of the reactant. In the exercise provided, the reaction rate can be deduced by investigating the change in the amount of potassium permanganate over time.

Understanding this concept allows students to connect how the decomposition of hydrogen peroxide, measured via potassium permanganate titration, relates directly to the key principles of reaction rates in chemical kinetics.

Natural Logarithm in Kinetics

The natural logarithm is a crucial mathematical concept used in chemical kinetics. It helps express the relationship between the concentration of reactants and time in a manner that is particularly useful for first-order reactions.
The equation \(\ln\left(\frac{[A]_0}{[A]}\right) = kt\) showcases how the natural log of the ratio of initial concentration to the concentration at any given time (t) is equal to the product of the rate constant (k) and time.

By using the properties of logarithms, complex reaction dynamics become simpler to analyze. For instance, the volume of potassium permanganate used in a titration can be connected to the concentration of hydrogen peroxide, and thus, the natural logarithm helps us to determine the reaction's rate constant through a clear, linear relationship.

Potassium Permanganate Titration

Potassium permanganate titration is a type of redox titration used to determine the concentration of oxidizable substances - in this case, hydrogen peroxide.
The volume of potassium permanganate needed to react with a certain volume of hydrogen peroxide decreases over time as the peroxide decomposes. This change can be measured to follow the reaction's progress. The substance's initial concentration can be found by analyzing the volume of the potassium permanganate solution used in the titration at the starting point of the reaction.

Potassium permanganate titration is not just a tool to indicate the endpoint of a reaction but also serves as a means to gather quantitative data necessary for calculating reaction rates and constants in chemical kinetics. This is beautifully demonstrated in measuring the decomposition of hydrogen peroxide, where the titration provides a direct method to follow the reaction kinetics.