Problem 78 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Consider the following consecutive firstorder reaction: $$ \mathrm{A} \stackrel{K_{1}}{\longrightarrow} \mathrm{B} \stackrel{K_{2}}{\longrightarrow} \mathrm{C} $$ If \(K_{1}=0.01 \mathrm{~min}^{-1}\) and \(K_{1}: K_{2}=1: 2\), after what time from the start of reaction, the concentration of ' B' will be maximum? \((\ln 2=0.7)\) (a) \(70 \mathrm{~min}\) (b) \(140 \mathrm{~min}\) (c) \(35 \mathrm{~min}\) (d) \(700 \mathrm{~min}\)

Step-by-Step Solution

Answer

An error seems to have occurred as the calculated time is not reflected in the provided options. The calculated time for maximum concentration of B is roughly 23.333 min, which is not listed.

Step 1: Understanding the Reaction Series

For the given consecutive reactions A → B → C with rate constants K1 and K2, the concentration of B will firstly increase as it is formed from A and then decrease as it is converted to C.

Step 2: Calculating the Rate Constants

Given that K1 = 0.01 min^{-1} and the ratio K1:K2 = 1:2, we can determine that K2 = 2 * K1, which means K2 = 0.02 min^{-1}.

Step 3: Determining the Maximum Concentration of B

The concentration of B is at a maximum when the rate of formation from A equals the rate of consumption into C. This means that the rate K1[A] = K2[B].

Step 4: Finding the Time for Maximum Concentration of B

We use the formula t_max = 1/(K1 + K2) * ln(K2/K1). Substituting K1 = 0.01 and K2 = 0.02 and using the provided ln(2)=0.7 we find that t_max = 1/(0.01 + 0.02) * ln(0.02/0.01) = 1/0.03 * ln(2) = 1/0.03 * 0.7 ≈ 23.333 min, which is not one of the provided options. The options may be incorrect or there might be a mistake in the given information or interpretation.

Key Concepts

Chemical Kinetics

Chemical kinetics is the branch of physical chemistry that deals with understanding the rates of chemical reactions. The central goal of kinetics is to provide a detailed description of how changes in conditions affect the speed of reactions. It involves studying various factors such as reactant concentrations, temperature, and catalyst presence, which can influence how quickly reactants turn into products.

For instance, in a first-order reaction, the rate at which reactants are transformed into products is directly proportional to the concentration of the reactants. This principle is crucial because it enables chemists to predict how long a reaction will take under specified conditions, and thus, they can control and optimize various industrial and laboratory processes.

Reaction Rate Constants

Reaction rate constants are numerical values that represent the speed of a chemical reaction. In the context of first-order reactions, the rate constant (usually denoted as K) has units of time^-1, such as min^-1 or s^-1, reflecting the reaction's time dependency. The rate equation for a first-order reaction is typically expressed as: rate = K[Reactant].

The value of the rate constant provides insight into how quickly a chemical reaction proceeds. A higher rate constant means a faster reaction. Importantly, rate constants are not constant across different conditions but can vary with changes in temperature, pressure, and the usage of catalysts. It’s important to note that rate constants provide vital information for designing chemical processes and understanding reaction mechanisms.

Consecutive Reactions

Consecutive reactions involve multiple steps where the product of one reaction becomes the reactant for the next. This sequence of reactions is common in complex chemical processes and pharmacokinetics, where drugs go through a series of metabolic steps.

In the context of the exercise, the sequence A → B → C consists of two first-order reactions, each governed by their own rate constants K1 and K2. Understanding the dynamics of these consecutive reactions is important, as it determines the concentration of intermediate B over time. Specifically, the intermediate B reaches its maximum concentration when the rate of its formation from A equals the rate of its conversion to C. This principle is vital in chemical manufacturing, to ensure maximum yield of desired products, and in pharmacology, to control drug availability in the body.