Reversible Reaction Networks

Introduction

Many network arrows are reversible. A product may convert back to reactant, a surface species may adsorb and desorb, and an intermediate can exchange among several forms. Net rate is the difference between forward and reverse fluxes. At equilibrium those fluxes can both be substantial even though the net amount of each species is unchanged. A kinetic model must respect the equilibrium constants implied by reaction thermodynamics.

Core explanation

For elementary A ⇌ B under ideal conditions, forward flux is kf[A] and reverse flux is kr[B], giving net v = kf[A] − kr[B]. At equilibrium v = 0, so [B]/[A] = kf/kr for a concentration convention with compatible standard states. In nonideal systems use activities; the equilibrium constant is formally dimensionless when activities are referenced to standard states. The ratio changes with temperature according to thermodynamics, not arbitrary curve fitting.

For A + B ⇌ C, v = kf[A][B] − kr[C] in a dilute ideal model. The units of kf and kr differ, so their raw numerical ratio has units unless standard-state concentration factors are included. A dimensionless equilibrium constant is defined with activities: K = aC/(aA aB). This detail matters when combining reactions of different molecularity. Writing “K = kf/kr” without stating concentration or activity conventions can hide a dimensional inconsistency.

In a multistep closed network at thermodynamic equilibrium, each elementary forward and reverse pair satisfies detailed balance. If A ⇌ B, B ⇌ C and C ⇌ A form a cycle, the equilibrium ratios around the loop must multiply to one. A model with ratios that multiply to two would predict a preferred circulation around a closed cycle even without external energy, violating thermodynamic consistency. Driven chemical or photochemical systems can sustain nonzero cycles only because energy or material enters through an explicit boundary.

At nonequilibrium steady state, an intermediate's net accumulation can be zero while individual net step fluxes are nonzero. For example, A is continuously fed, B forms and converts to P, and P is removed. The B pool may remain constant, but forward and reverse steps need not balance individually. Distinguish this flux balance from detailed balance at equilibrium. An ACS study of network thermodynamics discusses the role of network structure and energy constraints.

Step-by-step reasoning

1. Write both directions for each reversible elementary step. 2. Define rate laws with activities or stated concentration approximations. 3. Compute net flux as forward minus reverse. 4. At equilibrium, impose detailed balance and verify thermodynamic ratios. 5. For driven operation, identify the source and sink that sustain net flux.

Visual explanation

Draw A ⇌ B with two arrows of equal width at equilibrium, both labelled 5 units/min, and a net arrow labelled zero. Draw a second state with forward width 8 and reverse width 3, giving net 5 toward B. For a three-step cycle, place equilibrium ratio labels around its edges and show their product must be one in a closed equilibrium model.

Real-world analogy

People can enter and leave a room at equal rates while occupancy stays constant. That resembles equal forward and reverse chemical fluxes, but chemical equilibrium additionally requires thermodynamic consistency across every microscopic route. A room with people continuously entering from one door and leaving another resembles a driven steady state instead.

Real-world example

A ligand binds reversibly to a metal catalyst. At high ligand concentration, the bound form dominates and may slow substrate access; at low concentration, dissociation frees a site. A model using only ligand association would predict indefinite accumulation of bound metal and fail to explain recovery when ligand concentration falls. Including dissociation and its equilibrium relation allows the model to reproduce both directions.

Why?

Why do forward and reverse reactions continue at equilibrium? Molecules still collide and transform through thermal motion. Their average fluxes cancel, giving no net macroscopic change. Equilibrium is dynamic at the molecular level, not a frozen state.

Common misconception

“At equilibrium all rates are zero” is false. “Any set of reverse constants can be fitted independently” can violate thermodynamics. “Constant intermediate concentration proves detailed balance” confuses steady state with equilibrium. “A closed cycle can drive itself forever without fuel” ignores the need for an external free-energy source.

Worked example

For A ⇌ B, take kf = 0.4 min⁻¹, kr = 0.2 min⁻¹, [A] = 0.5 M and [B] = 0.5 M. Forward flux is 0.20 M/min and reverse flux 0.10 M/min, so net flux is 0.10 M/min toward B. At equilibrium for the same total 1.0 M in this ideal one-to-one system, [B]/[A] = 0.4/0.2 = 2. Thus [A] = 1/3 M and [B] = 2/3 M. Forward flux is about 0.133 M/min and reverse flux about 0.133 M/min: both continue while net flux is zero. The ratio is dimensionless here because both constants have the same units.

Quick check

1. Are forward and reverse elementary fluxes zero at thermodynamic equilibrium? Answer: No. They are equal, so their net flux is zero.

Exam focus

Calculate forward, reverse and net flux for a reversible step. Derive a simple equilibrium ratio with appropriate standard-state caution. Distinguish detailed balance from an open-system steady state and explain why closed cycles must satisfy thermodynamic consistency.

Advanced insight

Microscopic reversibility constrains models even when some steps are too fast to measure separately. Thermodynamic measurements can therefore reduce the number of independent kinetic parameters. For an externally driven cycle, a nonzero net circulation can be consistent only when the overall coupled process consumes a free-energy source, such as fuel, applied voltage or photons.

Summary

Reversible networks require both forward and reverse fluxes. Equilibrium imposes detailed balance and consistent cycle ratios, while driven steady states can maintain nonzero net flux through explicit energy and material inputs.

Practice questions

1. If forward flux is 7 and reverse flux 4 mmol/L/min, what is net flux? Answer: 3 mmol/L/min in the written forward direction. 2. If kf = kr for ideal A ⇌ B, what is [B]/[A] at equilibrium? Answer: One, under the same standard-state convention. 3. Can a steady B concentration coexist with net A → B → P flux? Answer: Yes. B production can equal its consumption while material flows through it. 4. What must support a sustained net cycle in a nonequilibrium network? Answer: An explicit supply of free energy or material, with corresponding output or dissipation.