Question

Rate law for the reaction \(\text{aA} + \text{bB} \rightarrow \text{cC} + \text{dD}\) is \(r = k[A][B]\), the rate of reaction doubles if

• A. Concentration of both A and B are doubled.
• B. Concentration of A is doubled and concentration of B is kept constant.
• C. Concentration of B is doubled and concentration of A is halved.
• D. Concentration of A is kept constant and concentration of B is halved.

Hint:

In rate law \(r = k[A][B]\), doubling one reactant doubles the rate, while doubling both makes the rate four times.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants.
For the given reaction, the rate law is \(r = k[A]^1[B]^1\), which means the reaction is first-order with respect to A and first-order with respect to B.
The overall order of the reaction is \(1 + 1 = 2\).
Step 2: Key Formula or Approach:
Substitute the modified concentrations from each option into the given rate law \(r = k[A][B]\) and calculate the new rate \(r'\) to see which one equals \(2r\).
Step 3: Detailed Explanation:
Let the initial rate be \(r_1 = k[A][B]\).
We want to find the condition that makes the new rate, \(r_2\), equal to \(2 \times r_1\).
Let's test each option by plugging the modified concentrations into the rate law:
(A) Double both [A] and [B]:
New concentrations: \([A]' = 2[A]\), \([B]' = 2[B]\).
\(r_2 = k(2[A])(2[B]) = 4k[A][B] = 4r_1\). (The rate quadruples).
(B) Double [A], keep [B] constant:
New concentrations: \([A]' = 2[A]\), \([B]' = [B]\).
\(r_2 = k(2[A])([B]) = 2k[A][B] = 2r_1\). (The rate doubles, matching the requirement).
(C) Double [B], halve [A]:
New concentrations: \([A]' = 0.5[A]\), \([B]' = 2[B]\).
\(r_2 = k(0.5[A])(2[B]) = (0.5 \times 2)k[A][B] = 1k[A][B] = r_1\). (The rate remains unchanged).
(D) Keep [A] constant, halve [B]:
New concentrations: \([A]' = [A]\), \([B]' = 0.5[B]\).
\(r_2 = k[A](0.5[B]) = 0.5k[A][B] = 0.5r_1\). (The rate is halved).
Step 4: Final Answer:
The only scenario where the rate exactly doubles is option (B).