Conservation Laws in a Network

Introduction

A reaction network can fit measured time courses and still be chemically impossible if it creates atoms or charge from nowhere. Conservation laws are hard constraints that should be checked before estimating rate constants. They also reveal conserved pools, such as total catalyst sites or the sum of several forms of one molecular skeleton. In an open reactor, these pools can change through feed and removal even while reactions themselves conserve them.

Core explanation

Every chemically complete reaction conserves each element. For C, H, O and other elements, count atoms in all reactants and products, including water, protons, counterions, gases and surfaces when they participate. Charge must also balance; electron-transfer reactions include electrons as external species or use an explicit electrical boundary. If an overall equation appears unbalanced, the mechanism may be missing a species, not merely a coefficient. An ACS educational analysis of reaction mechanisms connects stoichiometric matrices to mass-conservation tests.

Let N be a stoichiometric matrix. A row vector l assigns each species a count of some conserved item. If lᵀN = 0, every reaction column leaves the weighted sum lᵀc unchanged in a closed, constant-volume system, because d(lᵀc)/dt = lᵀNv = 0. For an element, l contains the number of its atoms in each species. A moiety vector can count a preserved molecular fragment even if it is transferred among forms. A catalyst-site vector can count vacant and occupied sites: θ + θA + θB = 1 if those are all possible states.

Conservation must match the model boundary. In an open flow reactor, incoming and outgoing material changes the pool: d(lᵀc)/dt includes feed and removal. In an electrochemical cell, a charge balance must include current and counterion transport. A model that treats H⁺ as a fixed reservoir is making an approximation about a large buffer or external source. It should not then claim to be a fully closed proton balance.

Conserved quantities help validate data. If the sum of labelled catalyst forms falls during a run, some species may be undetected, deposited or leached. If carbon recovery is 70%, a selectivity based only on detected products may be overstated. A conserved pool can reduce the number of independent ODEs and serve as a diagnostic check for numerical integration. Yet apparent conservation can be broken by real side reactions crossing the chosen boundary, so unexplained residuals deserve investigation.

Step-by-step reasoning

1. List molecular formulas and charges for every tracked species. 2. Check each reaction column separately for every element and charge. 3. Identify weighted combinations of species preserved by all reactions. 4. State whether feed, outflow, light, electrons or surfaces cross the boundary. 5. Compare predicted conserved totals with measurements and investigate gaps.

Visual explanation

Draw a table with reaction columns and rows for carbon, hydrogen, oxygen and charge. Every balanced column totals zero. Beside it draw a catalyst cycle with free site , A and B , connected by arrows, and a horizontal bracket showing total site population remains fixed. A leak arrow from B to dissolved metal makes the site total inside the catalyst boundary fall.

Real-world analogy

Money transferred among three bank accounts preserves the combined balance if no money enters or leaves. A network moiety moves among molecular forms similarly. A salary or withdrawal changes the total because it crosses the system boundary. Atoms are more strictly conserved than money, and chemical charge accounting must include electrons and ions correctly.

Real-world example

A kinetic model predicts that a dissolved organic substrate forms three products. The measured carbon in products plus unreacted substrate accounts for only 85% of incoming carbon. The missing 15% may reside in an unmeasured gas, surface deposit, sampling loss or analytical error. Rather than adjusting rate constants to fit only the observed products, researchers seek the missing carbon pool and update the network.

Why?

Why can a charge-balanced half-reaction require explicit electrons even when atoms already balance? Chemical oxidation and reduction move electrical charge across the electrode boundary. If electrons are omitted, the written species charges may not sum equally. Explicit electron coefficients connect the reaction network to measured current.

Common misconception

“A balanced overall reaction proves every proposed elementary step is balanced” is false. “A conserved pool cannot change in a flow reactor” ignores feed and effluent. “A missing product balance can always be repaired by changing kinetics” is false if species are absent from the model. “Charge is irrelevant to neutral products” ignores ionic intermediates and electron transfer.

Worked example

For reactions A → B and B → C, suppose A, B and C each contain one conserved carbon skeleton. With rows A, B, C, l = (1,1,1), so lᵀN = 0 and [A] + [B] + [C] is constant in a closed batch system. If initial concentrations are 2.0, 0 and 0 mol/L, then a later measurement of [A] = 0.5 and [B] = 0.8 predicts [C] = 0.7 mol/L. If measured C is only 0.4, the 0.3 mol/L gap suggests an omitted species, sampling loss or invalid assumption. In a reactor fed with A, the total can rise or fall because flow terms enter the balance; the reaction columns themselves still conserve the skeleton.

Quick check

1. What does lᵀN = 0 imply for a closed, constant-volume reaction system? Answer: The weighted species total lᵀc does not change due to the encoded reactions.

Exam focus

Check each proposed reaction for atom and charge balance. Write a conserved-moiety expression and explain its left-null-vector form. Distinguish reaction conservation from total concentration changes caused by an open-system boundary.

Advanced insight

Not every left-null relation corresponds to an obvious elemental composition; some arise from network topology and conserved cofactors or catalyst pools. Numerical rank analysis can reveal these hidden relationships, but interpretation must check whether the species definitions and boundary make them physically meaningful.

Summary

Element, charge and moiety conservation constrain valid reaction networks before kinetic fitting. Matrix null relations reveal conserved pools in closed systems, while feed, outflow and electrical transfer explain changes across explicitly open boundaries.

Practice questions

1. If A, B and C each retain one labelled carbon atom through A → B → C, what total is conserved in a closed vessel? Answer: [A] + [B] + [C], assuming no other labelled-carbon species. 2. Why include electrons in an electrochemical half-reaction balance? Answer: They account for charge transferred through the electrical boundary. 3. What does 85% carbon recovery suggest? Answer: An unmeasured carbon pool, loss or analytical problem should be investigated. 4. Can reaction stoichiometry conserve atoms while reactor concentrations change due to feed? Answer: Yes. Reaction columns conserve atoms, while boundary flows change amounts inside the reactor.