Batch reactor kinetics – first-order irreversible, isothermal, variable volume For an isothermal batch reactor with a first-order irreversible reaction, which relation between conversion (X_A) and time (t) is applicable (even if volume changes with conversion)?

Introduction / Context:
Relating conversion to time is fundamental for reactor design and performance prediction. A common doubt is whether changing volume during reaction alters the classic first-order exponential behavior in batch systems.

Given Data / Assumptions:

• Isothermal batch reactor.
• Elementary first-order irreversible reaction: r_A = −k C_A.
• Volume may vary with conversion due to stoichiometry (ε_A ≠ 0).

Concept / Approach:
For a batch system, the total moles of A, N_A, satisfy dN_A/dt = r_A V = −k C_A V. Because C_A = N_A/V, the product C_A V equals N_A. Hence dN_A/dt = −k N_A regardless of how V changes with time. This decouples the kinetics from the volume change for first-order reactions in batch mode and leads to an exponential decay of N_A and the familiar logarithmic conversion relation.

Step-by-Step Solution:

Verification / Alternative check:
Check limiting behavior: as t → 0, X_A → 0; as t grows, X_A approaches 1 asymptotically, consistent with first-order kinetics.

Why Other Options Are Wrong:

• X_A = k t: Linear in time is characteristic of zero-order, not first-order.
• ε_A * ln(1 − X_A) = k t: ε_A affects concentration–conversion relationships in flow reactors, not the batch first-order N_A decay law.

Common Pitfalls:
Assuming variable volume changes the exponential decay of N_A for first order in batch; it does not, because C_A V = N_A cancels V(t).

Final Answer:
-ln(1 - X_A) = k t