Dissolved Reversible Reactions

Equilibrium reactions within a condensed phase, following the form:

\[\text{Reactants} \rightleftharpoons \text{Products}\]

Equilibrium Constant

Temperature-dependent:

\[K_\text{eq}(T) = A \cdot \exp\!\left(C \left(\frac{1}{T_0} - \frac{1}{T}\right)\right)\]

At each time step, the forward rate constant is derived from:

\[k_f = K_\text{eq} \cdot k_r\]

where \(k_r\) is specified directly (e.g., via an Arrhenius rate constant). Alternatively, \(k_f\) can be provided explicitly.

Rate Expression

For a generic reaction with reactants \(R_1, R_2, \ldots\) and products \(P_1, P_2, \ldots\):

\[r_f = k_f \prod_i [R_i], \qquad r_r = k_r \prod_j [P_j]\]

Special treatment of solvents

When a species is designated as the solvent, its concentration is not included as a separate multiplier in the rate expression. Instead, the solvent’s activity is implicitly set to unity (dilute solution approximation). This means that reactions like:

\[\text{H₂O} \rightleftharpoons \text{OH⁻} + \text{H⁺}\]

use \(r_f = k_f\) (not \(r_f = k_f [\text{H₂O}]\)), because water is the overwhelmingly dominant species and its activity ≈ 1.

Net Rate and Forcing

\[R_\text{net} = r_f - r_r = k_f \prod_i [R_i] - k_r \prod_j [P_j]\]

For each reactant species:

\[\frac{d[R_i]}{dt} = -R_\text{net}\]

For each product species:

\[\frac{d[P_j]}{dt} = +R_\text{net}\]

Jacobian

The MICM solver stores −J. For each reactant–variable pair:

\[\text{stored } {-J[R_i, x]} = -\frac{\partial (-R_\text{net})}{\partial x}\]

For a first-order forward reaction (\(R_\text{net} = k_f - k_r [P]\)):

Entry	Mathematical \(J\)	Stored \(-J\)
\(J[P, P]\)	\(-k_r\)	\(+k_r\)

For a second-order reverse reaction (\(r_r = k_r [P_1][P_2]\)):

Entry	Mathematical \(J\)	Stored \(-J\)
\(J[R, P_1]\)	\(+k_r [P_2]\)	\(-k_r [P_2]\)
\(J[R, P_2]\)	\(+k_r [P_1]\)	\(-k_r [P_1]\)
\(J[P_1, P_1]\)	\(-k_r [P_2]\)	\(+k_r [P_2]\)
\(J[P_1, P_2]\)	\(-k_r [P_1]\)	\(+k_r [P_1]\)

The general pattern follows standard chemical kinetics Jacobian entries.

No Aerosol Properties Required

Unlike Henry’s Law phase transfer, dissolved reversible reactions are purely chemical and do not depend on particle size or number. They produce no indirect Jacobian entries through aerosol properties.

Multiple Instances

When the same reaction occurs in multiple condensed-phase instances (e.g., the same aqueous chemistry in several aerosol modes), each instance operates independently on its own set of fully-qualified state variables (e.g., AITKEN.AQUEOUS.H2O vs ACCUMULATION.AQUEOUS.H2O).