Rate Law for the Reaction A + B ⟶ C + D

The rate law for a chemical reaction expresses the relationship between the rate of the reaction and the concentrations of the reactants. In this case, we need to determine the rate law for the reaction A + B ⟶ C + D based on the given information.

Step 1: Comparing Concentration Changes and Rate Changes

According to the given information, doubling the concentration of both reactants (A and B) increases the rate by 8 times. This implies that the rate is directly proportional to the concentration of both A and B, raised to a power, let's call it 'n'.

We can express this relationship using the following equation:

Rate ∝ [A]^m [B]^n

Where [A] and [B] are the concentrations of reactants A and B, respectively, and m and n are the respective reaction orders for A and B.

Step 2: Doubling the Concentration of B

The second piece of information states that doubling the initial concentration of reactant B simply doubles the reaction rate. This suggests that the rate is directly proportional to the concentration of B, raised to the power of 1. Therefore, the reaction order for B is 1.

Rate ∝ [A]^m [B]^1

Step 3: Doubling the Concentration of Both Reactants

When both reactants A and B are doubled in concentration, the rate increases by 8 times. This can be expressed as:

Rate (2[A])^m (2[B])^1 = 8 * Rate [A]^m [B]^1

Simplifying the equation, we get:

2^m * 2 = 8

2^(m+1) = 8

2^(m+1) = 2^3

m + 1 = 3

m = 2

Step 4: Final Rate Law Expression

Now that we have determined the reaction order for A and B, we can write the rate law expression for the given reaction:

Rate = k [A]^2 [B]

Where k is the rate constant.

Summary:

The rate law for the reaction A + B ⟶ C + D is:

Rate = k [A]^2 [B]

This means that the rate of the reaction is directly proportional to the square of the concentration of A and the concentration of B, and the proportionality constant is represented by k.

(A)² (B)¹