The experimental data are:

Experiment 1: [A] = 0.2 mol L⁻¹, [B] = 0.2 mol L⁻¹, Initial rate = 0.3 mol L⁻¹ s⁻¹
Experiment 2: [A] = 0.4 mol L⁻¹, [B] = 0.2 mol L⁻¹, Initial rate = 0.6 mol L⁻¹ s⁻¹
Experiment 3: [A] = 0.4 mol L⁻¹, [B] = 0.4 mol L⁻¹, Initial rate = 2.4 mol L⁻¹ s⁻¹

The order of the reaction with respect to A is _____.

How to Determine the Order of Reaction with Respect to A Using Initial Rate Data

Correct Answer: 1

The order of reaction with respect to A is determined by comparing two experiments in which the concentration of A changes while the concentration of B remains constant. In the given data, experiments 1 and 2 satisfy this condition. The concentration of B remains fixed at 0.2 mol L⁻¹, while the concentration of A increases from 0.2 to 0.4 mol L⁻¹. At the same time, the initial reaction rate increases from 0.3 to 0.6 mol L⁻¹ s⁻¹.

Thus, doubling the concentration of A while keeping B constant causes the reaction rate to double. This means that the rate is directly proportional to the first power of the concentration of A. Therefore, the reaction is first order with respect to A, and the required answer is 1.

Understanding the Method of Initial Rates

What Is the Initial Rate Method?

The method of initial rates is an experimental technique used to determine the order of a chemical reaction with respect to individual reactants. The initial rate of a reaction is measured for different starting concentrations of the reactants. By comparing experiments in which only one reactant concentration changes, the effect of that reactant on the reaction rate can be determined.

This method is particularly useful because the stoichiometric equation alone does not generally reveal the experimentally observed rate law. For the reaction A + B → C, the coefficients of A and B in the balanced equation do not automatically mean that the reaction is first order in both reactants. The actual powers of the concentrations in the rate equation must be determined from experimental data.

Step-by-Step Solution for Finding the Order with Respect to A

Step 1: Write the General Rate Law

For the reaction:

A + B → C

the general rate law can be written as:

Rate = k[A]^m[B]ⁿ

Here, k is the rate constant, m is the order of reaction with respect to A, and n is the order of reaction with respect to B. The question asks only for the value of m.

Therefore, we need to select two experiments in which [B] remains unchanged while [A] changes. This allows the effect of A on the rate to be isolated.

Step 2: Select the Correct Experiments

Experiments 1 and 2 are the correct pair for determining the order with respect to A because the concentration of B is identical in both experiments.

For experiment 1:

[A]₁ = 0.2 mol L⁻¹
[B]₁ = 0.2 mol L⁻¹
Rate₁ = 0.3 mol L⁻¹ s⁻¹

For experiment 2:

[A]₂ = 0.4 mol L⁻¹
[B]₂ = 0.2 mol L⁻¹
Rate₂ = 0.6 mol L⁻¹ s⁻¹

Since [B] remains constant at 0.2 mol L⁻¹, any change in the reaction rate between these two experiments is caused by the change in [A].

Step 3: Compare the Change in Concentration of A

The concentration of A changes from 0.2 mol L⁻¹ to 0.4 mol L⁻¹. Therefore:

[A]₂ / [A]₁ = 0.4 / 0.2 = 2

Thus, the concentration of A is doubled.

Step 4: Compare the Change in Initial Rate

The initial reaction rate changes from 0.3 mol L⁻¹ s⁻¹ to 0.6 mol L⁻¹ s⁻¹. Therefore:

Rate₂ / Rate₁ = 0.6 / 0.3 = 2

Thus, the reaction rate is also doubled.

The experimental observation is therefore:

Doubling [A] → Doubling the reaction rate

This behavior indicates a first-order dependence on A.

Step 5: Calculate the Order Mathematically

The rate equations for experiments 1 and 2 can be written as:

Rate₁ = k[A]₁^m[B]₁ⁿ

and:

Rate₂ = k[A]₂^m[B]₂ⁿ

Dividing the second equation by the first equation gives:

Rate₂ / Rate₁ = ([A]₂ / [A]₁)^m × ([B]₂ / [B]₁)ⁿ

Since [B]₂ = [B]₁, the B term becomes 1 and can be removed:

0.6 / 0.3 = (0.4 / 0.2)^m

Therefore:

2 = 2^m

Comparing the powers of 2:

m = 1

Hence, the order of the reaction with respect to A is 1.

Why Experiments 1 and 2 Must Be Compared

To determine the order of reaction with respect to one reactant, all other reactant concentrations must remain constant. Experiments 1 and 2 are ideal for finding the order with respect to A because only [A] changes, while [B] remains fixed.

Comparing experiments 1 and 3 directly would not allow the order with respect to A to be determined because both [A] and [B] change simultaneously. The rate changes from 0.3 to 2.4 mol L⁻¹ s⁻¹, but this change is caused by the combined effects of both reactants.

Therefore, selecting the correct pair of experiments is the most important step in applying the initial rate method accurately.

What Does First Order with Respect to A Mean?

A reaction that is first order with respect to A has a rate proportional to the first power of [A]. The dependence can be written as:

Rate ∝ [A]¹

This means that if the concentration of A is doubled while all other conditions remain unchanged, the reaction rate doubles. If [A] is tripled, the rate becomes three times greater. If [A] is reduced to one-half, the contribution of A to the rate also becomes one-half.

The experimental data show exactly this behavior. Increasing [A] by a factor of 2 increases the rate by the same factor of 2. Therefore, the exponent of [A] in the rate equation is 1.

Further Analysis of the Complete Experimental Data

Although the question asks only for the order with respect to A, the third experiment helps us understand the complete rate behavior. Comparing experiments 2 and 3, the concentration of A remains constant at 0.4 mol L⁻¹, while the concentration of B doubles from 0.2 to 0.4 mol L⁻¹.

During this change, the rate increases from 0.6 to 2.4 mol L⁻¹ s⁻¹, which is a fourfold increase:

2.4 / 0.6 = 4

Since doubling [B] causes the rate to increase by a factor of 4:

2ⁿ = 4 = 2²

Therefore, n = 2, meaning the reaction is second order with respect to B. The complete experimentally determined rate law is:

Rate = k[A][B]²

The overall order of the reaction would therefore be 1 + 2 = 3. However, the question specifically asks only for the order with respect to A, so the required answer remains 1.

Complete Calculation in One Expression

The order with respect to A can be calculated directly from experiments 1 and 2:

Rate₂ / Rate₁ = ([A]₂ / [A]₁)^m

0.6 / 0.3 = (0.4 / 0.2)^m

2 = 2^m

m = 1

Final Answer

To determine the order of the reaction with respect to A, experiments 1 and 2 are compared because the concentration of B remains constant at 0.2 mol L⁻¹. The concentration of A doubles from 0.2 to 0.4 mol L⁻¹, and the initial rate also doubles from 0.3 to 0.6 mol L⁻¹ s⁻¹.

Since doubling the concentration of A causes the rate to double, the rate is directly proportional to [A]¹. Therefore, the reaction is first order with respect to A.

Correct Answer: 1