Understanding Reaction Order from Rate Changes

The order of a chemical reaction describes how the rate of the reaction depends on the concentration of the reactants. For a simple reaction involving a single reactant A, the rate law is often expressed as:

$\text{Rate} = k[\text{A}]^n$

where:

• $\text{Rate}$ is the reaction rate.
• $k$ is the rate constant.
• $[\text{A}]$ is the concentration of reactant A.
• $n$ is the order of the reaction with respect to reactant A.

In this problem, we are given two different reaction rates at two different concentrations of the reactant. We can use this information to determine the value of $n$, the order of the reaction.

Setting up the Equations

Let $R_1$ be the rate at concentration $C_1$, and $R_2$ be the rate at concentration $C_2$. According to the rate law:

$R_1 = k[C_1]^n$

$R_2 = k[C_2]^n$

We are given:

• $R_1 = 2.48 \times 10^{-3} \text{ mol L}^{-1} \text{ s}^{-1}$ when $C_1 = 0.6 \text{ M}$
• $R_2 = 4.96 \times 10^{-3} \text{ mol L}^{-1} \text{ s}^{-1}$ when $C_2 = 2.4 \text{ M}$

Substituting these values into the rate law equations:

Equation 1: $2.48 \times 10^{-3} = k[0.6]^n$

Equation 2: $4.96 \times 10^{-3} = k[2.4]^n$

Calculating the Reaction Order

To find the order $n$, we can divide Equation 2 by Equation 1. This cancels out the rate constant $k$:

$\frac{R_2}{R_1} = \frac{k[C_2]^n}{k[C_1]^n}$

$\frac{4.96 \times 10^{-3}}{2.48 \times 10^{-3}} = \frac{[2.4]^n}{[0.6]^n}$

The left side simplifies to:

$\frac{4.96 \times 10^{-3}}{2.48 \times 10^{-3}} = \frac{4.96}{2.48} = 2$

The right side can be written as:

$\frac{[2.4]^n}{[0.6]^n} = \left(\frac{2.4}{0.6}\right)^n = (4)^n$

So, we have the equation:

$2 = 4^n$

To solve for $n$, we need to find what power $n$ applied to 4 gives 2. We know that the square root of 4 is 2, and the square root is equivalent to raising to the power of 0.5 (or 1/2).

$\sqrt{4} = 4^{1/2} = 2$

Comparing this to $2 = 4^n$, we see that $n = 1/2 = 0.5$.

Therefore, the order of the reaction is 0.5.

Summary of Calculation

Conclusion on Reaction Order

Based on the calculation using the changes in reaction rate with respect to changes in reactant concentration, the order of the reaction is found to be 0.5.

Revision Table: Chemical Kinetics Concepts

Additional Information: Determining Reaction Order

The method used in this problem is a common way to determine the order of a reaction when kinetic data is available at different concentrations. Here are some points about reaction order:

• Reaction orders are typically determined experimentally. They are not necessarily related to the stoichiometric coefficients of the balanced chemical equation (unless it's an elementary reaction).
• Reaction orders can be whole numbers (0, 1, 2, etc.), fractions (like 0.5 in this case), or even negative numbers.
• A zero-order reaction means the rate is independent of the reactant's concentration.
• A first-order reaction means the rate is directly proportional to the reactant's concentration.
• A second-order reaction means the rate is proportional to the square of the reactant's concentration.
• Comparing initial rates at varying initial concentrations is another experimental method (Method of Initial Rates) used to find reaction orders.