Reaction engineering — constant-volume vs variable-volume batch operation (first order) A first-order irreversible reaction A → B is carried out separately in a constant-volume batch reactor and in a variable-volume batch reactor for the same duration. What does this imply about conversion and concentration in the two cases?

Introduction / Context:
This question tests understanding of first-order kinetics in batch reactors when volume is either constant (typical liquid-phase assumption) or allowed to change (e.g., certain gas-phase systems). It probes whether conversion X and concentration C are affected the same way by volume changes for a first-order irreversible reaction.

Given Data / Assumptions:

• Reaction: A → B, first order in A with rate rA = −k CA.
• Two separate batch operations, same temperature and same reaction time t.
• Case 1: Constant volume V = constant.
• Case 2: Variable volume V = V(t) (e.g., due to gas moles change).

Concept / Approach:
Conversion X is defined on moles of A, X = (NA0 − NA)/NA0. For first-order batch at a fixed k and fixed t, the exponential decay of NA (moles of A) is governed primarily by the integrated rate law. Differences arise in CA = NA/V. Thus, even when NA (and therefore X) matches at a given t, the concentrations will differ if the volumes differ.

Step-by-Step Solution:
For constant V: dCA/dt = −k CA ⇒ CA = CA0 exp(−k t); NA = V CA.For variable V(t): NA still decays based on kinetics, while CA = NA/V(t) differs because V changes.At the same t with the same k, the fraction of A reacted (conversion) can be the same; however, CA values are different as V differs.

Verification / Alternative check:
If volume expands, the same NA corresponds to a smaller CA; if volume contracts, CA is larger. Conversion depends on NA, not directly on V.

Why Other Options Are Wrong:

• (a) Concentrations cannot be identical when volumes differ.
• (c) Conversion need not differ if time and k are the same.
• (d) There is a specific correct relationship (option b).

Common Pitfalls:

• Equating concentration behavior with conversion; they are related but not identical in variable-volume systems.

Final Answer:
Conversion is the same in both cases, but the concentrations are different