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:
- $K_1K_2[A]$
- $(K_1 - K_2)[A]$
- $(K_1 + K_2)[A]$
- $(K_1/K_2)[A]$
The Correct Option is D
Approach Solution - 1
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.
Approach Solution -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:
\[ [\text{B}] = \left(\frac{k_1}{k_2}\right)[\text{A}]. \]
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