Question

The rate law for the reaction is \( r = k[A]^m[B]^n \). If the concentration of \( A \) is doubled and that of \( B \) is halved, the new rate is \( r_2 \); then what is the ratio of \( r_2/r_1 \)?

• A. \( a - b \)
• B. \( \frac{1}{2(a+b)} \)
• C. \( a + b \)
• D. \( 2(a-b) \)

Hint:

When changing the concentration of reactants, use the rate law to calculate the change in rate by applying the exponents corresponding to the reactants.

Answer 1

The Correct Option is D

Step 1: Understanding the rate law.
The rate law for the reaction is given as \( r = k[A]^m[B]^n \). If the concentration of \( A \) is doubled and that of \( B \) is halved, the new rate \( r_2 \) can be written as: \[ r_2 = k(2[A])^m \left(\frac{1}{2}[B]\right)^n = 2^m \times \frac{1}{2^n} \times r_1 \] This gives the ratio \( \frac{r_2}{r_1} = 2^{m-n} \). Thus, the ratio is \( 2(a-b) \), where \( a = m \) and \( b = n \).
Step 2: Analyzing the options.
(A) \( a - b \): Incorrect. This is not the correct ratio.
(B) \( \frac{1}{2(a+b) \):} Incorrect. This is not the correct expression for the ratio.
(C) \( a + b \): Incorrect. This does not give the correct ratio.
(D) \( 2(a-b) \): Correct — This is the correct ratio, where \( a = m \) and \( b = n \).
Step 3: Conclusion.
The correct answer is (D) \( 2(a-b) \), as this represents the correct ratio of rates in the given rate law.