Problem 63 - Chemistry Matter and Change

Question

Consider the generic chemical reaction: \(A+B \rightarrow A B\) . Based on experimental data, the reaction is second order in Reactant A. If the concentration of \(\mathrm{A}\) is halved, and all other conditions remain unchanged, how does the reaction rate change?

Step-by-Step Solution

Answer

The reaction rate becomes one quarter of its original value.

Step 1: Define Rate Law

The rate law for a reaction that is second order in reactant A can be expressed as: \( \text{Rate} = k[A]^2[B]^m \), where \( k \) is the rate constant and \( m \) is the order of reaction with respect to B.

Step 2: Understand the Effect of Halving Reactant A

If the concentration of A is halved, the new concentration \([A]'\) is \([A]/2\). Substitute this into the rate law: \(\text{New Rate} = k\left(\frac{[A]}{2}\right)^2[B]^m\).

Step 3: Simplify the New Rate Expression

Simplify the expression for the new rate: \(\text{New Rate} = k\left(\frac{[A]}{2}\right)^2[B]^m = k\frac{[A]^2}{4}[B]^m = \frac{1}{4}k[A]^2[B]^m\).

Step 4: Compare Initial Rate to New Rate

The new rate of reaction is \(\frac{1}{4}\) of the initial rate. This indicates that the rate of reaction becomes one quarter of its original value when the concentration of A is halved.

Key Concepts

Rate Law

When exploring chemical reactions, the **rate law** is fundamental in understanding how the reaction progresses. The rate law is a mathematical expression that links the concentration of reactants to the speed at which a reaction occurs (known as the reaction rate). It is typically written as:

• Rate = k[A]ⁿ[B]ᵐ

Here, **k** is the rate constant, while **n** and **m** represent the reaction order for reactants A and B, respectively. In a **second order reaction** (which this exercise focuses on), the rate depends on the square of one reactant's concentration or a combination of the first orders of two reactants. The rate law not only tells us how concentrations affect rate but also helps in determining reaction mechanisms.
Knowing the rate law is essential, especially in industrial chemistry and research, as it aids in controlling reaction speeds for desired outcomes. Thus, understanding it can be incredibly powerful when predicting how fast a product is formed under given conditions.

Concentration of Reactants

The **concentration of reactants** plays a pivotal role in determining the speed of chemical reactions. This is because reaction rates are directly influenced by how frequently reactant molecules collide. The higher the concentration, generally, the more collisions occur per unit of time, leading to an increase in the reaction rate.
In this exercise, when the concentration of A is halved, it impacts the reaction significantly since the reaction is second order in relation to this reactant. This means that the rate is proportional to the square of

• [A]², so any change in [A] will have a squared effect on the reaction rate.

Thus, if [A] is halved, the rate becomes a quarter of its initial value. Monitoring and adjusting reactant concentrations is, therefore, a crucial strategy in experimental chemistry to manage and optimize how fast reactions proceed.

Second Order Reaction

A **second order reaction** often involves reactants whose concentrations are squared in the rate law, indicating that these concentrations have a pronounced effect on reaction speed. In the context of this exercise, we know that A affects the rate more than B does due to its order of 2, which can be expressed as:

• Rate = k[A]²[B]ᵐ

The double impact of A's concentration makes any changes in this reactant significant. Halving the concentration of A reduces the reaction rate by a factor of four, as illustrated by the rate equation: if [A] is halved to [A]/2, substituting this into the rate law gives (1/2)² = 1/4. This highlights that a small change in concentration can drastically halt or speed up a reaction.
Understanding this can explain reaction behavior more comprehensively, not only providing insights into chemical kinetics but also aiding in controlled reaction engineering in pharmaceuticals, polymers, and more.