For identical feed composition, flow rate, and conversion, and for all positive reaction orders, how does the required reactor volume of a mixed flow reactor (CSTR) compare to that of a plug flow reactor (PFR)?

Introduction / Context:
Choosing between a continuous stirred-tank reactor (CSTR) and a plug flow reactor (PFR) has major implications for reactor size and operating cost. For positive-order reactions, concentration decreases with conversion, altering rate profiles across the reactor volume. Understanding the CSTR vs. PFR volume requirement is a foundational design insight.

Given Data / Assumptions:

• Isothermal operation with identical inlet composition and flow rate.
• Same target conversion in both reactors.
• Positive overall reaction order so that rate increases with concentration.

Concept / Approach:
In a CSTR, the exit concentration equals the reactor concentration due to perfect mixing, so the entire volume operates at the lowest driving concentration (for a given conversion). In contrast, a PFR experiences high concentration at the inlet, where the reaction is fastest, and progressively lower concentration toward the outlet, using the kinetics more efficiently. Hence, for positive orders, V_CSTR > V_PFR for the same conversion.

Step-by-Step Solution:
Define rate law r = k * C^n, with n > 0.CSTR design: V = F_A0 * (X / r_exit) with r_exit based on low exit concentration.PFR design: V = ∫ (F_A0 / r) dX benefiting from higher rates at higher concentrations near inlet.Thus, V_CSTR / V_PFR > 1 for n > 0.

Verification / Alternative check:
Graphical design using 1/−r_A vs. X shows a rectangle (CSTR) larger than the area under the curve (PFR) to reach the same X when 1/−r_A increases with X (positive order).

Why Other Options Are Wrong:
Equal to 1 or less than 1: contradict positive-order behavior.Equal to reaction order: no such direct equality exists.Indeterminate: kinetics sign (positive order) is sufficient to establish inequality.

Common Pitfalls:
Assuming mixing always helps; it helps for zero/negative orders, not positive.Ignoring nonisothermal effects that can change rate profiles.

Final Answer:
Greater than 1