Experimental Data:

How to Calculate the Overall Order of the Reaction A + B → C Using the Initial Rate Method?

Correct Answer: 3

Detailed Solution Using the Initial Rate Method

This chemical kinetics problem asks us to determine the overall order of the reaction from experimentally measured initial concentrations and reaction rates. The most effective method is the method of initial rates, in which two experiments are compared at a time while the concentration of one reactant is kept constant.

For the reaction:

A + B → C

we cannot determine the rate law merely from the coefficients in the balanced chemical equation. Unless the reaction is explicitly stated to be an elementary reaction, the reaction orders must be obtained from experimental data.

Let the reaction be of order m with respect to A and order n with respect to B. The general rate law can therefore be written as:

Rate = k[A]^m[B]ⁿ

Our task is to determine the values of m and n. The overall order will then be calculated by adding these two individual reaction orders.

Step 1: Determine the Order of Reaction with Respect to A

To determine the order with respect to A, we must compare two experiments in which the concentration of B remains constant while the concentration of A changes. Experiments 1 and 2 satisfy this requirement.

In Experiment 1:

[A]₁ = 0.4 mol L⁻¹

[B]₁ = 0.3 mol L⁻¹

Rate₁ = 0.078 mol L⁻¹ s⁻¹

In Experiment 2:

[A]₂ = 0.8 mol L⁻¹

[B]₂ = 0.3 mol L⁻¹

Rate₂ = 0.312 mol L⁻¹ s⁻¹

The concentration of B remains unchanged at 0.3 mol L⁻¹, while the concentration of A increases from 0.4 to 0.8 mol L⁻¹. Therefore, the concentration of A is doubled.

[A]₂ ÷ [A]₁ = 0.8 ÷ 0.4 = 2

At the same time, the reaction rate changes from 0.078 to 0.312 mol L⁻¹ s⁻¹.

Rate₂ ÷ Rate₁ = 0.312 ÷ 0.078 = 4

Thus, doubling the concentration of A causes the rate to become four times greater.

Mathematical Calculation of the Order with Respect to A

Using the general rate law:

Rate = k[A]^m[B]ⁿ

For Experiments 1 and 2:

Rate₂ ÷ Rate₁ = ([A]₂ ÷ [A]₁)^m

The concentration of B does not appear in the simplified ratio because it remains constant in both experiments.

Substituting the experimental values:

0.312 ÷ 0.078 = (0.8 ÷ 0.4)^m

4 = 2^m

Since:

4 = 2²

Therefore:

m = 2

Hence, the reaction is second order with respect to A.

Step 2: Determine the Order of Reaction with Respect to B

To determine the order with respect to B, we need to compare two experiments in which the concentration of A remains constant while the concentration of B changes. Experiments 1 and 3 provide the required comparison.

In Experiment 1:

[A]₁ = 0.4 mol L⁻¹

[B]₁ = 0.3 mol L⁻¹

Rate₁ = 0.078 mol L⁻¹ s⁻¹

In Experiment 3:

[A]₃ = 0.4 mol L⁻¹

[B]₃ = 0.6 mol L⁻¹

Rate₃ = 0.156 mol L⁻¹ s⁻¹

The concentration of A remains unchanged at 0.4 mol L⁻¹. The concentration of B increases from 0.3 to 0.6 mol L⁻¹, which means that B is doubled.

[B]₃ ÷ [B]₁ = 0.6 ÷ 0.3 = 2

The reaction rate changes from 0.078 to 0.156 mol L⁻¹ s⁻¹.

Rate₃ ÷ Rate₁ = 0.156 ÷ 0.078 = 2

Therefore, doubling the concentration of B causes the reaction rate to double.

Mathematical Calculation of the Order with Respect to B

For Experiments 1 and 3:

Rate₃ ÷ Rate₁ = ([B]₃ ÷ [B]₁)ⁿ

Substituting the experimental values:

0.156 ÷ 0.078 = (0.6 ÷ 0.3)ⁿ

2 = 2ⁿ

Therefore:

n = 1

Hence, the reaction is first order with respect to B.

Step 3: Write the Complete Rate Law

We have now determined both individual reaction orders:

Order with respect to A = 2

Order with respect to B = 1

The general rate law was:

Rate = k[A]^m[B]ⁿ

Substituting m = 2 and n = 1 gives:

Rate = k[A]²[B]

This is the experimentally determined rate law for the given reaction.

Step 4: Calculate the Overall Order of the Reaction

The overall order of a reaction is equal to the sum of the powers of all reactant concentrations appearing in the experimentally determined rate law.

For the rate law:

Rate = k[A]²[B]

the overall order is:

Overall order = 2 + 1

Overall order = 3

Therefore, the reaction is a third-order reaction overall.

Verification Using Experiments 1 and 4

The result can be verified by comparing Experiments 1 and 4. From Experiment 1 to Experiment 4, the concentration of A doubles from 0.4 to 0.8 mol L⁻¹, while the concentration of B also doubles from 0.3 to 0.6 mol L⁻¹.

Since the reaction is second order with respect to A, doubling A should increase the rate by:

2² = 4 times

Since the reaction is first order with respect to B, doubling B should increase the rate by:

2¹ = 2 times

The combined increase in the rate should therefore be:

4 × 2 = 8 times

Now compare the experimental rates:

Rate₄ ÷ Rate₁ = 0.624 ÷ 0.078

= 8

The experimentally observed rate increases exactly eightfold, confirming that the rate law Rate = k[A]²[B] is correct.

Understanding the Initial Rate Method

The initial rate method, also called the method of initial rates, is an experimental technique used to determine reaction orders. The initial rate is measured immediately after reactants are mixed, when their concentrations are accurately known and the effect of product accumulation is minimal.

To determine the order with respect to one reactant, two experiments are selected in which all other reactant concentrations remain constant. The change in the selected reactant concentration is then compared with the corresponding change in reaction rate.

If doubling a reactant concentration produces no change in rate, the reaction is zero order with respect to that reactant. If doubling the concentration doubles the rate, the reaction is first order. If doubling the concentration increases the rate fourfold, the reaction is second order. In the present question, these relationships allow the orders of A and B to be identified directly from the experimental data.

Why Is the Reaction Second Order with Respect to A?

Experiments 1 and 2 clearly show the dependence of the reaction rate on the concentration of A. The concentration of B remains constant, while the concentration of A doubles. The rate increases by a factor of four.

For a second-order dependence:

Rate ∝ [A]²

Therefore, if [A] doubles:

Rate factor = 2² = 4

This exactly matches the experimental observation. Hence, the reaction is second order with respect to A.

Why Is the Reaction First Order with Respect to B?

Experiments 1 and 3 show the dependence of the rate on B. The concentration of A remains constant, while the concentration of B doubles. The rate also doubles.

For a first-order dependence:

Rate ∝ [B]

Therefore, if [B] doubles:

Rate factor = 2¹ = 2

This agrees exactly with the experimental data. Hence, the reaction is first order with respect to B.

Why the Stoichiometric Equation Cannot Directly Give the Reaction Order

The chemical equation is written as:

A + B → C

It may appear that the reaction should be first order in A and first order in B because the stoichiometric coefficients of A and B are both one. However, reaction orders are generally experimental quantities and cannot be determined directly from the balanced equation unless the reaction is known to occur in a single elementary step.

The experimental data show that the reaction is second order in A and first order in B. Therefore, the correct rate law is:

Rate = k[A]²[B]

and not simply Rate = k[A][B].

Final Answer

The correct answer is 3. Comparing Experiments 1 and 2, doubling the concentration of A increases the rate fourfold, so the reaction is second order with respect to A. Comparing Experiments 1 and 3, doubling the concentration of B doubles the rate, so the reaction is first order with respect to B. Therefore, the rate law is Rate = k[A]²[B], and the overall order is 2 + 1 = 3.