Question

For series reactions A $\to$ B $\to$ C, maximum concentration of intermediate B in a PFR occurs when:

• A. Rate of formation of B = Rate of decomposition of B
• B. dC$_B$/dt = 0
• C. k$_1$ = k$_2$
• D. Rate of formation of B = Rate of decomposition of B and dC$_B$/dt = 0

Hint:

At the start of the reactor, \( \frac{dC_B}{dt} \gt 0 \) because the formation of B is faster than its decomposition.
After the peak, \( \frac{dC_B}{dt} \lt 0 \) as B decomposes faster than it is formed.
The peak concentration occurs exactly where these two rates are equal.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the condition under which the concentration of the intermediate component B reaches its maximum value in a Plug Flow Reactor (PFR) for the series reaction network \( A \xrightarrow{k_1} B \xrightarrow{k_2} C \).

Step 2: Key Formula or Approach:
For a plug flow reactor, the change in concentration along the reactor can be modeled using a batch-like time domain, where space-time \( t \) represents the position.
The rate of change of concentration of B is:
\[ \frac{dC_B}{dt} = r_B = k_1 \cdot C_A - k_2 \cdot C_B \]
where \( k_1 \cdot C_A \) is the rate of formation of B, and \( k_2 \cdot C_B \) is the rate of decomposition of B.

Step 3: Detailed Explanation:

• Mathematical Condition for Maximum: From calculus, a continuous function \( C_B(t) \) reaches its maximum value when its first derivative with respect to time is equal to zero:
\[ \frac{dC_B}{dt} = 0 \]

• Physical Condition for Maximum: Substituting this derivative condition into the rate equation gives:
\[ 0 = k_1 \cdot C_A - k_2 \cdot C_B \quad \implies \quad k_1 \cdot C_A = k_2 \cdot C_B \]
This means the rate of formation of B must equal the rate of decomposition of B at the point of maximum concentration.

• Both the mathematical condition (\( dC_B/dt = 0 \)) and the physical rate condition (\( \text{Rate of formation} = \text{Rate of decomposition} \)) are equivalent statements of the same physical state.


Step 4: Final Answer:
The maximum concentration of intermediate B occurs when the rate of formation of B equals its rate of decomposition, which is mathematically represented by \( dC_B/dt = 0 \).