Question

If the concentration of a reactant is doubled, by what factor will the rate increase if the reaction is second order with respect to that reactant?

Short answer

The rate will increase by a factor of 4.

Step-by-step solution

Understand the concept of reaction order

For a reaction that is second order with respect to a reactant, the rate of the reaction is proportional to the square of the concentration of that reactant. Mathematically, if the rate is denoted by 'r' and the concentration by '[A]', then the rate law for this reaction can be written as r = k[A]^2, where k is the rate constant.

Calculate the rate change when concentration is doubled

If the initial concentration of the reactant is '[A]', the initial rate is r_1 = k[A]^2. When the concentration is doubled to '2[A]', the new rate will be r_2 = k(2[A])^2 = 4k[A]^2. Thus, the rate increases by a factor of 4.

Key concepts

Rate Law

To understand the impact of changes in reactant concentration on the reaction rate, it is essential to begin with the basics of the rate law. The rate law is an equation that relates the rate of a chemical reaction to the concentration of its reactants. It is generally expressed in the form of an equation such as \( r = k[A]^n \) where \( r \) is the reaction rate, \( k \) is the rate constant, \( [A] \) is the concentration of reactant A, and \( n \) is the reaction order with respect to A.

In the case of a second-order reaction, the value of \( n \) is 2, indicating that the rate of reaction is proportional to the square of the concentration of reactant A. For example, with an initial concentration \( [A] \) and rate \( r_1 \), if the concentration is increased to \( 2[A] \) the rate \( r_2 \) would be calculated using \( r_2 = k(2[A])^2 \). This relationship shows the direct impact of changes in concentration to the rate at which the reaction proceeds, giving students a quantitative understanding of the rate law's significance in chemical kinetics.

For students who require further clarification, considering the relationship between rate law and reaction order can be enlightening. Reaction order essentially tells us how the rate is affected by the concentration of one or more reactants, and by determining it, we can predict how the rate will change if conditions are altered.

Rate Constant

The rate constant, symbolized as \( k \) in the rate law equation, holds significant importance in understanding chemical reactions. It is a proportionality constant that relates the reaction rate to the concentrations of reactants. For a specific reaction at a given temperature, the rate constant is fixed and does not change with the concentrations of the reactants.

The value of the rate constant provides insight into the inherent speed of the reaction. A larger rate constant means a faster reaction at a given concentration of reactants. For example, if you have two reactions with the same concentrations and orders, the one with the higher rate constant will proceed at a faster rate. Since the rate constant is also dependent on temperature, studying its variations can give us a better understanding of how reaction conditions, such as temperature, affect the reaction speed.

Another key aspect is that the rate constant has units that vary depending on the overall order of the reaction, which can often be a source of confusion for students. In our specific case of a second-order reaction, the units for \( k \) would be \( M^{-1} s^{-1} \), where \( M \) stands for molarity. This consideration is vital when solving problems or comparing rate constants across different reactions.

Reactant Concentration

The concentration of reactants in a chemical reaction bears a direct correlation with the rate at which the reaction occurs. This correlation is especially critical when examining reactions of different orders. In simple terms, reactant concentration, often indicated by square brackets \( [A] \) or \( [B] \) for reactants A and B respectively, represents the amount of a substance within a particular volume of solution.

For a second-order reaction, increasing the concentration of a reactant leads to a quadratic increase in the rate. Doubling the concentration would consequently quadruple the reaction rate, as demonstrated by the exercise. This quadratic relationship is a key concept for students to grasp when it comes to reactions of this type. Essentially, if you're working with a reaction that is second-order in a particular reactant, you can expect large changes in the rate with even moderate adjustments in that reactant's concentration.

In practice, understanding how concentration affects reaction rate can be useful in several applications, such as controlling the speed of a reaction in industrial processes or predicting the time it will take for a reaction to reach completion. Additionally, it can help students solve kinetic problems accurately and develop a deeper comprehension of the reaction kinetics.