Initial reactant ratio determines which integrated rate equation is valid. For the batch reaction (A+2B\rightarrow\text{products}), with (-r_A=kC_AC_B), define (M=C_{B0}/C_{A0}). When (M\neq2), the concentrations of (A) and (B) do not decline in a fixed ratio, so the integrated rate law is logarithmic; when (M=2), the stoichiometric relationship (C_B=2C_A) persists and the integration becomes a reciprocal expression.

The feed ratio is not merely an operating detail: it determines whether the two-reactant rate law must be integrated with changing concentration ratios or can be reduced to a single-species second-order form.

Why the Initial Ratio Controls the Integration

Stoichiometric concentration changes

For a constant-volume batch reactor, the stoichiometry gives

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

and

[ C_B=C_{B0}-2C_{A0}X_A =C_{A0}(M-2X_A). ]

The factor of 2 means that (B) is consumed twice as rapidly as (A) on a concentration basis.

The ratio (M) identifies the operating condition

The initial ratio

[ M=\frac{C_{B0}}{C_{A0}} ]

can be compared with the stoichiometric requirement (2).

If (M=2), the feed contains exactly the required amount of (B). If (M\neq2), one reactant is initially in excess, and the concentration ratio changes during the reaction.

Integration When (M\neq2)

Both concentrations must remain in the rate law

For first-order dependence on each reactant,

[ -r_A=kC_AC_B. ]

Using the batch balance,

[ -\frac{dC_A}{dt}=kC_AC_B, ]

while the stoichiometric relationship is

[ C_B-2C_A=C_{A0}(M-2). ]

When (M\neq2), (C_B) cannot be replaced by a constant multiple of (C_A). The changing concentration ratio must therefore be retained during integration.

Resulting logarithmic equation

The integrated expression is

[ \ln\left(\frac{C_BC_{A0}}{C_{B0}C_A}\right)

C_{A0}(M-2)kt. ]

Equivalently, using conversion,

[ \ln\left[ \frac{M-2X_A}{M(1-X_A)} \right]

C_{A0}(M-2)kt. ]

This form is used for kinetic analysis when the initial feed is non-stoichiometric.

Interpretation of the sign

If (M>2), (B) is initially in excess and the coefficient (M-2) is positive. If (M<2), (A) is initially in excess and the coefficient is negative.

The sign is not an error; it reflects how the ratio (C_B/C_A) evolves as the reaction proceeds.

Integration When (M=2)

The concentration ratio remains fixed

For a stoichiometric feed,

[ C_{B0}=2C_{A0}. ]

The batch stoichiometry then gives

[ C_B=2C_A. ]

This relationship remains valid throughout the run, allowing the two-reactant rate law to be reduced to

[ -r_A=kC_A(2C_A)=2kC_A^2. ]

In terms of conversion,

[ -r_A=2kC_{A0}^2(1-X_A)^2. ]

Resulting reciprocal equation

The batch balance becomes

[ -\frac{dC_A}{dt}=2kC_A^2. ]

Integration gives

[ \frac{1}{C_A}-\frac{1}{C_{A0}}=2kt. ]

In conversion form, this is

[ \frac{X_A}{C_{A0}(1-X_A)}=2kt. ]

Thus, the stoichiometric feed produces the familiar reciprocal integrated form for a second-order dependence on (C_A).

How Ratio Selection Affects Data Analysis

The ratio determines the plotting method

For (M=2), plotting

[ \frac{1}{C_A}-\frac{1}{C_{A0}} ]

against time should produce a straight line with slope (2k).

For (M\neq2), the appropriate linearized quantity is

[ \ln\left(\frac{C_BC_{A0}}{C_{B0}C_A}\right), ]

plotted against time. Its slope is

[ C_{A0}(M-2)k. ]

Using the reciprocal plot for a non-stoichiometric feed, or the logarithmic plot for a stoichiometric feed, imposes the wrong concentration relationship and can bias the estimated rate constant.

Feed metering affects the model

Accurate initial measurements of (C_{A0}) and (C_{B0}) are essential because (M) determines the integration structure. A small ratio error near (M=2) can also matter: the data may appear nearly stoichiometric while technically requiring the non-stoichiometric expression.

The limiting reactant affects the usable data range

When (M<2), (B) is limiting. When (M>2), (A) is limiting. The experiment should be interpreted only over the region where both reactant concentrations remain physically meaningful and measurable.

Understanding the Trade-offs

Stoichiometric feeds simplify the mathematics

With (M=2), only one reactant concentration needs to be tracked because (C_B=2C_A). This simplifies parameter fitting and makes the reciprocal diagnostic plot easy to interpret.

Stoichiometric feeds can reduce diagnostic flexibility

Because the concentration ratio does not change, a stoichiometric experiment provides less information about the separate concentration dependence of the two reactants. It confirms the combined model under one constrained trajectory rather than testing a broad range of (C_A)-to-(C_B) ratios.

Non-stoichiometric feeds provide broader information

Changing (M) changes how the reactant concentrations evolve relative to one another. Experiments with different ratios can therefore help test whether the assumed rate law (kC_AC_B) is consistent with the data.

Excess reactant does not make the exact model first order

A large excess of one reactant may permit an approximate pseudo-first-order treatment over a limited conversion range. However, that is an approximation; the exact batch integration for (M\neq2) remains the logarithmic expression above.

Making the Right Choice for Your Goal

The initial ratio should be selected based on the kinetic question the experiment must answer.

If your primary focus is mathematical simplicity: Use (M=2), maintain (C_B=2C_A), and analyze the data with (\frac{1}{C_A}-\frac{1}{C_{A0}}=2kt).
If your primary focus is testing the full two-reactant rate law: Use (M\neq2), measure both concentrations, and apply the logarithmic integrated equation.
If your primary focus is identifying limiting-reactant behavior: Choose a deliberate excess of either (A) or (B), while accounting for the changing concentration ratio.
If your primary focus is reliable parameter estimation: Record the actual initial concentrations rather than assuming the intended feed ratio, because (M) appears directly in the integrated model.

Choosing the initial reactant ratio deliberately ensures that the integrated rate equation matches the reactor’s actual concentration trajectory and that the fitted kinetic constant has a defensible physical meaning.

Summary Table:

Initial ratio	Concentration relationship	Integrated form	Best use
`M = 2`	`C_B = 2C_A` remains constant	`1/C_A - 1/C_A0 = 2kt`	Simple stoichiometric analysis
`M > 2`	`B` is initially in excess	Logarithmic equation	Analyze excess-`B` kinetics
`M < 2`	`A` is initially in excess	Logarithmic equation	Analyze excess-`A` kinetics
Any `M != 2`	Concentration ratio changes	`ln(C_BC_A0 / C_B0C_A) = C_A0(M - 2)kt`	Test the full two-reactant rate law

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