The extent of reaction framework calculates continuous-reactor outlet composition by assigning one reaction extent to each independent reaction, then applying all stoichiometric changes to the inlet molar flow rates. For reaction (j), the extent (\dot{\xi}_j) is the molar rate at which that reaction proceeds. The outlet flow of species (i) is therefore

[ \dot{n}_{i,\text{out}}

\dot{n}{i,\text{in}} + \sum{j=1}^{R}\nu_{ij}\dot{\xi}_j ]

where (\nu_{ij}) is the stoichiometric coefficient of species (i) in reaction (j).

Core takeaway: Side reactions require separate reaction extents. Once those extents are determined from conversion, yield, selectivity, or measured outlet data, the complete reactor effluent and all material balances follow directly from the stoichiometric coefficients.

Why Separate Extents Are Needed

One extent represents one independent reaction

For a single reaction, every species is linked to the same extent:

[ \dot{n}_{i,\text{out}}

\dot{n}_{i,\text{in}} + \nu_i\dot{\xi} ]

Reactants have negative coefficients, products have positive coefficients, and inert species have zero coefficients.

For multiple independent reactions, each reaction has its own extent:

[ \dot{n}_{i,\text{out}}

\dot{n}{i,\text{in}} + \nu{i1}\dot{\xi}1 + \nu{i2}\dot{\xi}2 +\cdots+ \nu{iR}\dot{\xi}_R ]

The extents must be expressed as molar rates in a continuous steady-state reactor, typically in (\mathrm{mol/s}), (\mathrm{kmol/h}), or a comparable unit.

Side reactions create additional material pathways

Consider a primary reaction and an unwanted side reaction:

[ \text{Primary:}\qquad A \rightarrow B ]

[ \text{Side:}\qquad A \rightarrow C ]

If (\dot{\xi}_1) is the primary-reaction extent and (\dot{\xi}_2) is the side-reaction extent, then

[ \dot{n}_{A,\text{out}}

\dot{n}_{A,\text{in}}

\dot{\xi}_1

\dot{\xi}_2 ]

[ \dot{n}_{B,\text{out}}

\dot{n}_{B,\text{in}} + \dot{\xi}_1 ]

[ \dot{n}_{C,\text{out}}

\dot{n}_{C,\text{in}} + \dot{\xi}_2 ]

An inert component (I) is unchanged:

[ \dot{n}_{I,\text{out}}

\dot{n}_{I,\text{in}} ]

This separates the total consumption of (A) from the way that consumption is distributed between the desired and undesired pathways.

How Conversion Determines the Extents

Total reactant consumption gives a combined extent

If the fractional conversion of (A) is (X_A), the total molar rate of (A) consumed is

[ \dot{n}{A,\text{in}}-\dot{n}{A,\text{out}}

X_A\dot{n}_{A,\text{in}} ]

For the two reactions above,

[ \dot{\xi}_1+\dot{\xi}_2

X_A\dot{n}_{A,\text{in}} ]

Conversion alone determines the sum of the reaction extents. It does not reveal how much (A) formed (B) versus (C).

Product yield or selectivity separates the pathways

Suppose the yield of (B), based on the inlet flow of (A), is defined as

[ Y_B

\frac{\dot{n}{B,\text{out}}-\dot{n}{B,\text{in}}} {\dot{n}_{A,\text{in}}} ]

For the unit-stoichiometry example,

[ \dot{\xi}_1

Y_B\dot{n}_{A,\text{in}} ]

The side-reaction extent then follows from the total conversion:

[ \dot{\xi}_2

X_A\dot{n}_{A,\text{in}}

\dot{\xi}_1 ]

or

[ \dot{\xi}_2

(X_A-Y_B)\dot{n}_{A,\text{in}} ]

This calculation is valid when (B) is formed only by the primary reaction and is not subsequently consumed.

Stoichiometric coefficients must be included

If the reactions are instead

[ \text{Primary:}\qquad 2A \rightarrow B ]

[ \text{Side:}\qquad A \rightarrow C ]

then

[ \dot{n}_{A,\text{out}}

\dot{n}_{A,\text{in}}

2\dot{\xi}_1

\dot{\xi}_2 ]

and

[ \dot{n}_{B,\text{out}}

\dot{n}_{B,\text{in}} + \dot{\xi}_1 ]

A measured production rate of (B) gives

[ \dot{\xi}_1

\frac{\dot{n}{B,\text{out}}-\dot{n}{B,\text{in}}}{1} ]

The corresponding consumption of (A) through the primary pathway is (2\dot{\xi}_1), not (\dot{\xi}_1). This is why the stoichiometric coefficient must remain explicit in every balance.

How to Build the Complete Material Balance

Start with the inlet stream

List the inlet molar flow rate of every component:

[ \dot{\mathbf n}_{\text{in}}

\begin{bmatrix} \dot n_{A,\text{in}}\ \dot n_{B,\text{in}}\ \dot n_{C,\text{in}}\ \dot n_{I,\text{in}} \end{bmatrix} ]

Include reactants, products already present in the feed, diluents, and inert species.

Construct the stoichiometric matrix

For the reactions

[ A\rightarrow B \qquad\text{and}\qquad A\rightarrow C ]

the stoichiometric matrix is

[ \boldsymbol{\nu}

\begin{bmatrix} -1 & -1\ +1 & 0\ 0 & +1\ 0 & 0 \end{bmatrix} ]

Rows correspond to species, and columns correspond to reactions.

Apply the extent equation

The stream calculation becomes

[ \dot{\mathbf n}_{\text{out}}

\dot{\mathbf n}_{\text{in}} + \boldsymbol{\nu} \begin{bmatrix} \dot{\xi}_1\ \dot{\xi}_2 \end{bmatrix} ]

For each component, this produces an individual outlet flow rate without writing a separate reaction-generation expression from scratch.

Calculate total outlet flow and composition

After determining every component flow,

[ \dot n_{\text{out,total}}

\sum_i \dot n_{i,\text{out}} ]

The outlet mole fraction of species (i) is

[ y_{i,\text{out}}

\frac{\dot n_{i,\text{out}}} {\dot n_{\text{out,total}}} ]

Mass fractions can be calculated from

[ w_{i,\text{out}}

\frac{\dot n_{i,\text{out}}M_i} {\sum_k \dot n_{k,\text{out}}M_k} ]

where (M_i) is the molecular weight of component (i).

How Measurements Identify Reaction Extents

Use a directly measured product

If a product is generated by only one reaction, its outlet flow can identify that reaction's extent. For a product (P) formed with coefficient (\nu_{Pj}),

[ \dot{\xi}_j

\frac{\dot n_{P,\text{out}}-\dot n_{P,\text{in}}} {\nu_{Pj}} ]

This is often the simplest approach when product concentrations are available from gas chromatography, liquid chromatography, or another analyzer.

Use conversion and selectivity together

Conversion measures how much reactant disappeared. Selectivity or product yield measures where that consumed reactant went.

For example, with (A) converted through two pathways,

[ \text{total consumption of }A

\sum_j \left|\nu_{Aj}\right|\dot{\xi}_j ]

A second independent specification, such as the production rate of the desired product, is required to determine the individual extents.

Solve a linear system when several reactions occur

Measured outlet flows can provide equations of the form

[ \dot n_{i,\text{out}}-\dot n_{i,\text{in}}

\sum_j \nu_{ij}\dot{\xi}_j ]

If there are (R) independent reaction extents, at least (R) independent reliable equations are needed. The equations can be solved directly as a linear algebra problem, provided the reaction stoichiometry is known and the selected measurements are independent.

Understanding the Trade-offs

Conversion alone is insufficient for pathway analysis

A high reactant conversion does not necessarily mean a high desired-product yield. The same reactant disappearance may result from the primary reaction, side reactions, or sequential reactions that consume an intermediate or desired product.

The extents expose this distinction, but only when enough independent data or specifications are available.

Reaction independence matters

The extent method requires a set of independent reactions. Including redundant reactions can make the system underdetermined or produce nonunique extents, even though the overall species balances remain valid.

Reaction independence is a stoichiometric issue, not a question of whether the reactions occur in separate physical locations.

Measurement errors affect inferred extents

When extents are calculated from differences between large inlet and outlet flows, small measurement errors can produce substantial uncertainty, particularly for low-conversion reactions or minor side products.

A practical balance should therefore compare calculated flows with measured outlet data and report physically impossible results, such as negative flow rates or negative extents, as evidence of inconsistent data or an incomplete reaction network.

The limiting-reactant concept needs care with side reactions

For a single reaction, the theoretical maximum extent can be estimated for each reactant, and the smallest value identifies the limiting reactant. With multiple competing reactions, however, the maximum extent depends on how the feed is allocated among pathways.

It is more reliable to apply the full reaction network and check reactant availability across all extents rather than assign one limiting-reactant extent without considering the side reactions.

Overall mole flow may change

The total molar flow is not necessarily constant. It changes whenever the reactions change the total number of moles, such as reactions involving association, dissociation, or oxidation with gaseous oxygen.

Mole fractions must therefore be calculated from the actual summed outlet flow, not from the inlet total or an assumed constant flow rate.

Making the Right Choice for Your Goal

Use the following approach when setting up a continuous-reactor material balance:

• If your primary focus is desired-product production: Determine the primary reaction extent from the measured desired-product formation rate, then use conversion or side-product data to determine the remaining extents.
• If your primary focus is total reactant conversion: Use the conversion to calculate total stoichiometric reactant consumption, but obtain selectivity, yield, or additional outlet measurements before assigning that consumption to individual reactions.
• If your primary focus is complete effluent composition: Define every independent reaction, determine all extents from independent specifications, and calculate each outlet flow with the stoichiometric matrix.
• If your primary focus is material-balance verification: Compare calculated outlet flows, total mass flow, and measured compositions, while checking that all extents and component flows are physically plausible.
• If your primary focus is reactor heat duty: Use the determined extents with each reaction's heat of reaction, then add sensible enthalpy changes between inlet and outlet streams.

Once the independent reaction extents are known, side reactions become explicit stoichiometric pathways rather than hidden sources of material-balance error.

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