For a batch reaction (A+B\rightarrow\text{Products}) with (C_{A0}\neq C_{B0}), evaluate the kinetic constant by fitting the integrated second-order rate equation to concentration-time data. Assuming constant volume, isothermal operation, and an elementary rate law (-r_A=kC_AC_B), measure (C_A) and (C_B) at several times and calculate (\ln(C_B/C_A)). A plot of this quantity against time should be linear, with slope ((C_{B0}-C_{A0})k), allowing (k) to be determined from the slope.

The unequal initial concentrations must be retained in the integrated balance. The correct parameter estimate is (k=\text{slope}/(C_{B0}-C_{A0})), obtained from a regression of the integrated model rather than from a single half-life measurement.

Set Up the Batch Reactor Model

Start with the Rate Law

For the irreversible reaction

[ A+B\rightarrow\text{Products}, ]

assume the second-order rate law

[ -r_A=kC_AC_B. ]

For a constant-volume batch reactor, the mole balance is

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

Because (A) and (B) react in a 1:1 stoichiometric ratio,

[ C_B-C_{B0}=C_A-C_{A0}. ]

Therefore,

[ C_B=C_A+(C_{B0}-C_{A0}). ]

This concentration relationship is essential when the two reactants do not begin at equal concentrations.

Measure Concentration Over Time

Conduct the batch reaction at controlled temperature and constant volume. Withdraw samples, or use an appropriate in situ measurement method, to determine (C_A(t)) and (C_B(t)).

The initial concentrations (C_{A0}) and (C_{B0}) should be measured or established accurately because they appear directly in the integrated rate equation.

Use the Integrated Rate Expression

Integrated Form in Concentrations

For (C_{A0}\neq C_{B0}), integration gives

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

(C_{B0}-C_{A0})kt. ]

Equivalently,

[ \ln\left(\frac{C_B}{C_A}\right)

\ln\left(\frac{C_{B0}}{C_{A0}}\right) + (C_{B0}-C_{A0})kt. ]

This has the linear form

[ y=b+mt, ]

where

[ y=\ln\left(\frac{C_B}{C_A}\right), ]

[ b=\ln\left(\frac{C_{B0}}{C_{A0}}\right), ]

and

[ m=(C_{B0}-C_{A0})k. ]

Determine the Rate Constant from the Slope

Fit a straight line to the calculated values of (\ln(C_B/C_A)) versus (t). The rate constant is then

[ \boxed{ k=\frac{m}{C_{B0}-C_{A0}} } ]

where (m) is the fitted slope.

For a second-order rate law involving two concentrations, the units of (k) are typically

[ \mathrm{volume,mol^{-1},time^{-1}}, ]

such as (\mathrm{L,mol^{-1},s^{-1}}).

Interpret the Intercept

The fitted intercept should approximately equal

[ \ln\left(\frac{C_{B0}}{C_{A0}}\right). ]

Agreement between the measured intercept and this theoretical value provides a useful check on the initial-concentration measurements and the assumed reaction model.

Apply the Conversion Form

Define Conversion and the Initial Concentration Ratio

Define conversion of (A) as

[ X_A=\frac{C_{A0}-C_A}{C_{A0}}, ]

and define the initial concentration ratio as

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

The reactant concentrations become

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

and

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

Fit the Conversion-Time Data

Substitution into the integrated rate expression gives

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

C_{A0}(M-1)kt. ]

Thus, plotting

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

against (t) produces a straight line with slope

[ C_{A0}(M-1)k. ]

The rate constant can therefore be calculated from

[ \boxed{ k= \frac{\text{slope}}{C_{A0}(M-1)} } ]

provided (M\neq 1).

Fit the Full Concentration Model

Avoid Numerical Differentiation

Although the instantaneous rate could theoretically be estimated from (-dC_A/dt), numerical differentiation amplifies experimental noise. This often produces less reliable kinetic parameters.

A better method is to fit the integrated reactor model directly to the concentration-time measurements.

Use Nonlinear Regression When Appropriate

For each measured time (t_j), calculate the model-predicted concentrations using the integrated balance and compare them with the measured values. Estimate (k) by minimizing the sum of squared residuals, for example,

[ SS=\sum_j\left(C_{A,\text{data},j}-C_{A,\text{model},j}\right)^2. ]

This approach is especially useful when concentration measurements have unequal uncertainties, when data are noisy, or when both reactant concentrations are measured.

The linearized plot is convenient for an initial estimate and model check, while direct nonlinear fitting generally preserves the original measurement scale more faithfully.

Check Whether the Second-Order Model Is Valid

Examine Linearity

A suitable second-order model should produce an approximately straight plot of

[ \ln(C_B/C_A) ]

against (t). The coefficient of determination, residual pattern, and consistency of the fitted slope should all be examined.

A high correlation alone is insufficient if the residuals show curvature or systematic deviation.

Compare Both Reactants

For a 1:1 reaction at constant volume,

[ C_{B0}-C_B=C_{A0}-C_A. ]

The measured concentration changes should satisfy this stoichiometric relationship within experimental uncertainty. Significant disagreement may indicate sampling errors, side reactions, volume changes, or incorrect concentration analysis.

Confirm Temperature Control

The rate constant depends on temperature. Kinetic runs should therefore be conducted at a controlled, documented temperature, and results from different temperatures should not be combined into one estimate of (k) without an appropriate temperature model.

Understanding the Trade-offs

Why the Simple Half-Life Method Fails

The usual second-order half-life relation cannot be applied directly when (C_{A0}\neq C_{B0}). Reactants (A) and (B) do not reach half of their respective initial concentrations at the same time.

The limiting reactant is also consumed first, so the reaction cannot be treated as though both concentrations began at the same value.

Special Case When Initial Concentrations Are Equal

If (C_{A0}=C_{B0}), the unequal-concentration expression becomes indeterminate in its difference form and must be replaced by the equal-initial-concentration result:

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

The unequal-concentration equation should therefore be used only when the initial concentrations are genuinely different.

Limitations of Linearized Regression

Taking logarithms changes the distribution of measurement errors and can give disproportionate influence to low-concentration data. It may also exclude or distort data near the detection limit.

For final reporting, direct regression of the untransformed concentration data is often preferable, with the integrated linear plot retained as a diagnostic check.

How to Apply This to Your Experiment

Use the following procedure to evaluate the kinetic parameter:

If your primary focus is a quick graphical estimate: Calculate (\ln(C_B/C_A)), plot it against (t), determine the slope, and calculate (k=m/(C_{B0}-C_{A0})).
If your primary focus is conversion-based analysis: Calculate (M=C_{B0}/C_{A0}) and fit (\ln[(M-X_A)/(M(1-X_A))]) against (t), using (k=\text{slope}/[C_{A0}(M-1)]).
If your primary focus is the most reliable parameter estimate: Fit the integrated concentration-time model directly to the measured (C_A) and (C_B) data by minimizing the residual sum of squares.
If your primary focus is validating the reaction model: Check the plot linearity, intercept, stoichiometric concentration relationship, residuals, and temperature consistency before accepting the calculated (k).

A reliable value of (k) comes from fitting the complete unequal-concentration batch-reactor balance while verifying that the data satisfy the assumed second-order stoichiometric model.

Summary Table:

Key aspect	Recommended approach	Result or check
Rate law	Use -rA = kCACB for A + B -> products	Second-order kinetics
Unequal concentrations	Plot ln(CB/CA) against time	Linear slope m = (CB0 - CA0)k
Rate constant	Calculate k = m/(CB0 - CA0)	Units: volume mol^-1 time^-1
Conversion method	Plot ln[(M - XA)/(M(1 - XA))] against time	M = CB0/CA0
Best final estimate	Fit the integrated concentration model by nonlinear regression	Reduces distortion from log-transformed errors
Model validation	Check residuals, intercept, stoichiometry, and temperature	Confirms whether the model is suitable
Equal-concentration case	Use 1/CA - 1/CA0 = kt when CA0 = CB0	Do not use the unequal-concentration form

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