Question

For \(A_{(s)}+B_{(s)} \longrightarrow C_{(s)} ;\) rate \(=k[A]^{1 / 2}[B]^{2}\), if initial concentration of \(A\) and \(B\) are increased by factors 4 and 2 respectively, then the initial rate is changed by the factor: (a) 4 (b) 6 (c) 8 (d) None of these

Short answer

The initial rate is changed by the factor 8.

Step-by-step solution

Write original rate expression

The original rate expression for the reaction is given by rate = k[A]^{1 / 2}[B]^{2}. This represents the rate of the reaction in terms of the concentrations of reactants A and B, where k is the rate constant.

Represent the change in concentration of A and B

The initial concentration of A is increased by a factor of 4, and B by a factor of 2. Representing this change algebraically, the new concentration of A can be written as 4[A] and that of B as 2[B].

Write the new rate expression

The rate expression with the new concentrations will be rate' = k(4[A])^{1 / 2}(2[B])^{2}. Simply this expression using the properties of exponents.

Simplify the new rate expression

Simplify the new rate expression by evaluating the exponents. The new rate is rate' = k(2[A]^{1 / 2})(4[B]^{2}). Further simplification gives rate' = 8k[A]^{1 / 2}[B]^{2}.

Calculate the change in rate

To find the factor by which the initial rate has changed, divide the new rate by the original rate. The rate change factor is (8k[A]^{1 / 2}[B]^{2}) / (k[A]^{1 / 2}[B]^{2}) = 8.

Key concepts

Reaction Rate Expression

Understanding the reaction rate expression is crucial for students delving into the kinetics of chemical reactions. In the context of the given exercise, the rate of reaction for the solid reactants transforming into a solid product is characterized by the equation: rate = k[A]^{1 / 2}[B]^{2}. This mathematical formulation encapsulates the velocity at which reactants A and B convert into product C. The symbols [A] and [B] denote the molar concentrations of reactants A and B, respectively, while the exponentials represent the effect each reactant concentration has on the rate of the reaction. The rate constant 'k' is a coefficient that provides the necessary scaling to translate concentration changes into rate changes, unique to every reaction at a given temperature.

It's vital to recognize that the exponents of the concentrations (also known as reaction orders) are determined empirically and do not necessarily correspond to the stoichiometric coefficients of the reaction equation. The rate expression is derived from experimental observations and can inform us about the reaction mechanism—a topic further explored in the section on Stoichiometry and Reaction Mechanism.

Effect of Concentration on Rate

In chemical kinetics, the concentration of reactants is a pivotal factor influencing the rate at which a reaction proceeds. The relationship between reactant concentration and reaction rate is articulated within the rate law expression. For example, the exercise provided showcases that the rate of the reaction increases when the concentration of reactants A and B is raised. The change in reaction rate is not always directly proportional to changes in concentration; the actual effect is dictated by the reaction orders. In the exercise, reactant A affects the rate to the 1/2 power while reactant B does so squared.

In practical terms, when the initial concentration of A multiplies by a factor of four, the rate-related term involving A, [A]^{1 / 2}, increases by a factor of two. Similarly, doubling the concentration of B results in a fourfold increase in the rate-related term for B, [B]^{2}. By considering these changes, the cumulative effect on the overall rate of reaction can be accurately predicted.

Stoichiometry and Reaction Mechanism

The stoichiometry of a reaction provides a fundamental understanding of the proportions in which reactants combine and products form. However, stoichiometry alone doesn't explain the intricate details of how reactants transform into products—this is where the reaction mechanism steps in. The reaction mechanism gives insight into the step-by-step sequence of elementary reactions that lead to the overall chemical change. It details the breaking and formation of bonds, the intermediate species produced, and the order in which reactants interact.

An intriguing aspect of the reaction mechanism is that it can sometimes reveal why reactants influence the rate of reaction in a non-intuitive manner. For instance, the rate law's exponents often stem from the mechanism rather than the balanced equation's coefficients. In practice, detailed mechanisms are substantiated through experiments and help explain why altering concentrations can dramatically alter the rate, as exemplified by the exercise problem where the exponents indicate a complex relationship between reactant concentration and reaction rate. Understanding these concepts not only aids in solving textbook problems but also prepares students for practical laboratory situations in the real world.