Competing Reaction Pathways From a Shared Reactant

Why a Shared Reactant Creates Branching

A reactant can be consumed by more than one reaction at the same time. For example, a compound may be converted into either product \(P_1\) or product \(P_2\). The amount of reactant available to each pathway is shared: as either reaction proceeds, it reduces the amount that remains for both. A useful first model asks how the competing rates determine the split of reactant between the products.

The key distinction is between the instantaneous rate split and the cumulative product split. Under constant first-order rate constants, these splits are equal and remain fixed over time. That convenient result depends on specific assumptions; it does not automatically apply when co-reactants change, products are removed, or pathways are reversible.

Here \([A]\), \([P_1]\), and \([P_2]\) denote concentrations, and \(k_1\) and \(k_2\) have units of inverse time. Assume a constant volume, constant rate constants, no input of \(A\), no reverse reactions, no other routes of consumption, and no loss or further reaction of either product. These conditions allow product formation to be counted directly from reactant consumption.

Deriving the First-Order Branching Equations

Both reactions remove \(A\), so their rates add in the balance for the shared reactant. Each reaction forms its own product at the rate at which it consumes \(A\). The resulting system is

Let \(K=k_1+k_2\), and let the initial concentrations be \([A](0)=[A]_0\), \([P_1](0)=[P_1]_0\), and \([P_2](0)=[P_2]_0\). Separating variables in the reactant equation gives

To derive product formation, substitute this reactant solution into each product equation and integrate from 0 to \(t\). For pathway 1,

The same calculation for pathway 2 replaces \(k_1\) with \(k_2\). Thus, the amount of \(A\) consumed by time \(t\) is \([A]_0(1-e^{-Kt})\), and the two product increments are fixed shares of that amount. Dividing the instantaneous rates gives the same shares:

The cancellation of \([A]\) is the central reason that the branching fractions do not change as \(A\) falls. The fractions are nonnegative and sum to one when \(k_1,k_2\geq 0\) and \(k_1+k_2>0\). If both rate constants are zero, no reaction occurs and branching fractions are undefined.

Pseudo-First-Order Competing Pathways

A pathway may be bimolecular rather than first-order in \(A\). Suppose the reactions are \(A+B\to P_1\) and \(A+C\to P_2\), with mass-action rates \(v_1=k_1[A][B]\) and \(v_2=k_2[A][C]\). If \([B]\) and \([C]\) are held constant, or are in such excess that their changes are negligible over the interval of interest, define \(k'_1=k_1[B]\) and \(k'_2=k_2[C]\). Then \(v_1=k'_1[A]\) and \(v_2=k'_2[A]\), which has the same mathematical form as the first-order model.

For this case, \(k_1\) and \(k_2\) have units of concentration\(^{-1}\) time\(^{-1}\), while the pseudo-first-order constants \(k'_1\) and \(k'_2\) have units of inverse time. The total disappearance constant is \(K'=k'_1+k'_2\). The reactant decays exponentially with \(K'\), and the product fractions are \(k'_1/K'\) and \(k'_2/K'\), provided the co-reactant concentrations stay effectively constant.

When the Branching Fractions Can Change

If the co-reactants are not effectively constant, the pseudo-first-order constants vary with time. The instantaneous fraction into pathway 1 is then

assuming \([A](t)>0\). This expression can change as \(B\) or \(C\) is consumed or supplied. The cumulative fraction is not generally the instantaneous fraction at the endpoint; it depends on the integrated amounts through each pathway. In that situation, use the full coupled mass-action equations rather than assigning a fixed product fraction.

A useful check under the simple one-to-one model is the balance \([A]+[P_1]+[P_2]=[A]_0+[P_1]_0+[P_2]_0\). It follows by adding the three differential equations, whose sum is zero. If data substantially violate this balance, possible explanations include unmeasured pathways, product degradation, measurement error, or a mistaken stoichiometric model. Likewise, curvature in a plot of \(\ln([A](t)/[A]_0)\) against time is inconsistent with a single constant total first-order rate over that interval, though noisy measurements can also produce apparent curvature.

The mass-action model is mechanistic: it retains both pathways and predicts each product. A simpler lumped description may fit only the total disappearance of \(A\), using \(K=k_1+k_2\); that alone cannot identify the individual pathway constants or predict the product split. Conversely, if either pathway is reversible, saturable, or has changing co-reactants, the constant first-order branch formula is generally not adequate. Those cases require rate laws that represent the relevant mechanism rather than treating the fractions as fixed.

Common Pitfall and Practical Checks

A common error is to treat \(k_1\) and \(k_2\) themselves as product fractions. Rate constants have units of inverse time in the first-order model, whereas fractions are dimensionless. The correct comparison is \(k_1/(k_1+k_2)\), not \(k_1\) alone. For pseudo-first-order pathways, compare \(k_1[B]\) and \(k_2[C]\), not the underlying bimolecular constants without their co-reactant concentrations.

Before using a fixed branching fraction, check that the pathways consume the same shared reactant, follow the stated rate laws, and have constant rate coefficients over the analyzed interval. Confirm that the product stoichiometry supports the assumed one-to-one concentration balance. If measured product increments do not sum to reactant consumed, do not force them into fractions that sum to one without accounting for missing routes, losses, or stoichiometric differences.