Fractional conversion provides the link between reaction progress and component concentrations in a constant-volume batch reactor pilot plant. For reactant A, the conversion (X_A) is the fraction of its initial moles that has reacted, so its concentration is (C_A=C_{A0}(1-X_A)). Because the reactor volume remains fixed, the concentration changes follow directly from the stoichiometric changes in moles.

In a constant-volume batch reactor, fractional conversion acts as a common coordinate for calculating every component concentration. Once (X_A) is known, stoichiometry determines the concentrations of unreacted reactants, products, and inerts.

How Conversion Defines Reactant Concentration

Reactant A as the Conversion Basis

Fractional conversion of A is defined as

[ X_A=\frac{N_{A0}-N_A}{N_{A0}} ]

where (N_{A0}) is the initial amount of A and (N_A) is the amount remaining at time (t).

Rearranging gives

[ N_A=N_{A0}(1-X_A) ]

For a constant-volume reactor, concentration is proportional to moles because (C_A=N_A/V). Therefore,

[ \boxed{C_A=C_{A0}(1-X_A)} ]

where (C_{A0}) is the initial concentration of A.

Interpreting the Relationship

If (X_A=0), no A has reacted and (C_A=C_{A0}). If (X_A=1), A has been completely consumed and (C_A=0), assuming A is the limiting reactant and the reaction proceeds to completion.

For example, (X_A=0.60) means that 60% of the initial A has reacted, leaving (40%) of its initial concentration:

[ C_A=0.40C_{A0} ]

How Stoichiometry Determines Other Concentrations

Consider the reaction

[ aA+bB\rightarrow cC+dD ]

The conversion of A determines how much of every other species is consumed or produced.

Co-Reactant B

The amount of B consumed per mole of A reacted is (b/a). Thus,

[ N_B=N_{B0}-\left(\frac{b}{a}\right)N_{A0}X_A ]

Dividing by the constant reactor volume gives

[ \boxed{C_B=C_{B0}-\left(\frac{b}{a}\right)C_{A0}X_A} ]

This equation also accounts for the possibility that B has a different initial concentration from A.

Products C and D

Products increase as A reacts. Their concentrations are

[ \boxed{C_C=C_{C0}+\left(\frac{c}{a}\right)C_{A0}X_A} ]

and

[ \boxed{C_D=C_{D0}+\left(\frac{d}{a}\right)C_{A0}X_A} ]

If the products are absent initially, then (C_{C0}=0) and (C_{D0}=0).

Inert Components

An inert species does not participate in the reaction, so its moles remain constant:

[ N_I=N_{I0} ]

Because the volume is also constant, its concentration remains constant:

[ \boxed{C_I=C_{I0}} ]

Why Constant Volume Matters

Concentration Changes Come Only From Reaction

In a constant-volume batch reactor, the volume (V) does not change as the reaction proceeds. Consequently,

[ C_j=\frac{N_j}{V} ]

and every concentration change is caused solely by the stoichiometric change in the number of moles of species (j).

This makes the conversion-concentration relationships linear in (X_A).

Total Moles May Still Change

Constant volume does not mean that the total number of moles remains constant. For the general reaction above, the total moles can change according to

[ N_T=N_{T0}+\delta N_{A0}X_A ]

where

[ \delta=\frac{c}{a}+\frac{d}{a}-\frac{b}{a}-1 ]

represents the net stoichiometric change in total moles per mole of A reacted.

The total concentration may therefore change even though the reactor volume does not:

[ C_T=\frac{N_T}{V} ]

This distinction is especially important for gas-phase reactions and rate laws based on total pressure or mole fraction.

Using Conversion in Kinetic Analysis

First-Order Reaction

For an irreversible first-order reaction in A,

[ -r_A=kC_A ]

Using (C_A=C_{A0}(1-X_A)), the integrated batch-reactor relationship becomes

[ \boxed{\ln\left(\frac{1}{1-X_A}\right)=kt} ]

A plot of (\ln[1/(1-X_A)]) against time should be linear, with slope (k), when the first-order model is appropriate.

Second-Order Reaction

For a second-order reaction based on A, the corresponding relationship is

[ \boxed{\frac{X_A}{1-X_A}=kC_{A0}t} ]

Experimental conversion-time data can therefore be transformed into a form that tests the assumed reaction order and estimates the rate constant.

Rate Laws With Multiple Components

For a rate law involving multiple species, conversion is used to calculate each concentration before evaluating the rate. For example,

[ -r_A=kC_A^\alpha C_B^\beta ]

becomes

[ -r_A

k\left[C_{A0}(1-X_A)\right]^\alpha \left[ C_{B0}-\left(\frac{b}{a}\right)C_{A0}X_A \right]^\beta ]

This allows the reactor material balance and kinetic model to be written using a single reaction-progress variable, (X_A).

Understanding the Trade-Offs

Constant-Volume Assumptions Simplify the Model

The main advantage is that concentration expressions are direct and algebraically simple. This is useful for pilot-plant experiments, kinetic fitting, and comparing runs with different initial feed compositions.

Constant Volume Does Not Apply to Every Batch System

A constant-volume model is appropriate when the reactor volume is controlled or effectively unchanged. It is not appropriate for a constant-pressure gas-phase batch system where the reaction changes the total number of moles and causes expansion or contraction.

Under constant pressure, the reactant concentration generally follows

[ C_A= \frac{C_{A0}(1-X_A)}{1+\varepsilon_A X_A} ]

where (\varepsilon_A) represents the fractional volume change. Omitting this term can produce incorrect concentrations and misleading kinetic parameters.

Conversion Must Respect Stoichiometric Limits

The calculated concentration of every reactant must remain nonnegative. If B is depleted before A reaches complete conversion, then the maximum physically possible (X_A) is limited by the available amount of B:

[ X_{A,\max}

\frac{aN_{B0}}{bN_{A0}} ]

when B is the limiting co-reactant.

Making the Right Choice for Your Goal

Use fractional conversion as the central variable, then calculate each component concentration from the reaction stoichiometry and the constant reactor volume.

If your primary focus is calculating species concentrations: Use (C_A=C_{A0}(1-X_A)), then apply the stoichiometric coefficients to determine reactant and product concentrations.
If your primary focus is estimating reaction kinetics: Measure (X_A) over time, convert it to (C_A) and other relevant concentrations, and test the resulting data against integrated rate expressions.
If your primary focus is modeling a gas-phase batch reaction: Confirm whether the reactor is truly constant volume; if the volume changes under constant pressure, include the fractional volume-change correction.
If your primary focus is comparing pilot-plant runs: Express the material balances in terms of (X_A) so different initial feed compositions can be analyzed using the same stoichiometric framework.

Fractional conversion is the practical bridge between measured reaction progress, stoichiometric material balances, and the component concentrations required for kinetic modeling.

Summary Table:

Topic	Key relationship or insight
Reactant A	`C_A = C_A0(1 - X_A)`
Co-reactant B	`C_B = C_B0 - (b/a)C_A0X_A`
Products C and D	`C_C = C_C0 + (c/a)C_A0X_A`; `C_D = C_D0 + (d/a)C_A0X_A`
Inert species	`C_I = C_I0` because its moles and the reactor volume remain constant
Total moles	May change despite constant volume when reaction stoichiometry changes the total number of moles
Kinetic analysis	Conversion-time data can be transformed to evaluate first-order, second-order, and multi-component rate laws
Conversion limit	Reactant availability sets the maximum physically achievable conversion
Model selection	Use constant-volume equations only when reactor volume remains effectively unchanged; account for volume change in constant-pressure gas reactions

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