Question

For a reaction $A \xrightarrow{K_1} B \xrightarrow{K_2} C$ If the rate of formation of B is set to be zero then the concentration of B is given by:

• A. $K_1K_2[A]$
• B. $(K_1 - K_2)[A]$
• C. $(K_1 + K_2)[A]$
• D. $(K_1/K_2)[A]$

Answer 1

The Correct Option is D

To solve this problem, we need to analyze the reaction sequence and understand the condition provided, which is that the rate of formation of B is set to be zero.

The reaction sequence given is:

\(A \xrightarrow{K_1} B \xrightarrow{K_2} C\)

Here, \(K_1\) is the rate constant for the conversion of \(A\) to \(B\), and \(K_2\) is the rate constant for the conversion of \(B\) to \(C\).

The rate of formation of \(B\) can be expressed as:

\(\frac{d[B]}{dt} = K_1[A] - K_2[B]\)

According to the problem, the rate of formation of \(B\) is set to be zero, i.e.,

\(K_1[A] - K_2[B] = 0\)

Rearranging the terms, we get:

\(K_1[A] = K_2[B]\)

Solving for \([B]\), we find:

\([B] = \frac{K_1}{K_2}[A]\)

Thus, the concentration of \(B\) when the rate of formation is zero is given by:

\(\frac{K_1}{K_2}[A]\)

Therefore, the correct answer is \(\frac{K_1}{K_2}[A]\).

Explanation of Incorrect Options:

• \(K_1K_2[A]\): This would imply multiplying the rate constants and the concentration, which is not correct under the given condition.
• \((K_1 - K_2)[A]\): This suggests subtracting the rate constants, which is not the correct interpretation of the rate condition.
• \((K_1 + K_2)[A]\): This implies adding the rate constants, which is not relevant to solving for the concentration of \(B\) under the specific condition given.

This reasoning helps us conclude that \(\frac{K_1}{K_2}[A]\) is the correct and logical choice based on the condition provided in the problem.

Answer 2

The rate of formation of B is:
\[ \frac{d[\text{B}]}{dt} = k_1[\text{A}] - k_2[\text{B}]. \]
For the rate of formation of B to be zero:
\[ \frac{d[\text{B}]}{dt} = 0. \]
Substitute:
\[ k_1[\text{A}] - k_2[\text{B}] = 0. \]
Rearrange to find [B]:
\[ k_1[\text{A}] = k_2[\text{B}] \implies [\text{B}] = \frac{k_1}{k_2}[\text{A}]. \]
Thus, the concentration of B is:
\[ [\text{B}] = \left(\frac{k_1}{k_2}\right)[\text{A}]. \]