Question

For the reaction \( A + B \to C \), the rate law is found to be \( \text{rate} = k[A]^2[B] \). If the concentration of \( A \) is doubled and \( B \) is halved, by what factor does the rate change?

• A. \( 2 \)
• B. \( \frac{1}{2} \)
• C. \( 4 \)
• D. \( 1 \)

Hint:

When concentrations change, apply exponent in rate law carefully to find new rate.

Answer 1

The Correct Option is A

To determine the effect of changing concentrations on the rate of reaction for the given rate law \( \text{rate} = k[A]^2[B] \), we need to analyze how the rate is impacted when the concentration of \( A \) is doubled and \( B \) is halved.

Let's denote the initial concentrations as \([A]_0\) and \([B]_0\). The initial rate is:

\(\text{rate}_0 = k[A]_0^2[B]_0\)

After doubling \([A]\) and halving \([B]\), the new concentrations are \(2[A]_0\) and \(\frac{1}{2}[B]_0\). The new rate \(\text{rate}_1\) is:

\(\text{rate}_1 = k(2[A]_0)^2\left(\frac{1}{2}[B]_0\right)\)

Calculating \(\text{rate}_1\):

\(\text{rate}_1 = k \cdot 4[A]_0^2 \cdot \frac{1}{2}[B]_0 = 2k[A]_0^2[B]_0\)

Now, find the factor by which the rate changes:

\(\frac{\text{rate}_1}{\text{rate}_0} = \frac{2k[A]_0^2[B]_0}{k[A]_0^2[B]_0} = 2\)

Thus, the rate increases by a factor of \( 2 \).

Therefore, the correct answer is: \( 2 \)

Answer 2

Given the reaction \( A + B \to C \) with the rate law \( \text{rate} = k[A]^2[B] \), we need to determine the factor by which the rate changes when the concentration of \( A \) is doubled and \( B \) is halved.

Let's break it down step-by-step:

• Initial rate: \( \text{rate}_1 = k[A]^2[B] \).
• New concentrations: \([A]' = 2[A]\) and \([B]' = \frac{1}{2}[B]\).

Substitute into the rate law:

• New rate: \( \text{rate}_2 = k([A]')^2([B]') = k(2[A])^2\left(\frac{1}{2}[B]\right) \).
• Calculate: \( \text{rate}_2 = k(4[A]^2)\left(\frac{1}{2}[B]\right) = k\left(4 \cdot \frac{1}{2}\right)[A]^2[B] = 2k[A]^2[B] \).

Thus, the new rate compared to the initial rate is:

\[ \frac{\text{rate}_2}{\text{rate}_1} = \frac{2k[A]^2[B]}{k[A]^2[B]} = 2 \]

Therefore, the rate increases by a factor of \( 2 \).