Theory Manual Version 3.4
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Subsection 2.10.3: Mixture with Negligible Solute Volume Fraction Up Section 2.10: Chemical Reactions Section 2.11: Fluid Mechanics

### 2.10.4 Chemical Kinetics

Productions rates are described by constitutive relations which are functions of the state variables. In a biological mixture under isothermal conditions, the minimum set of state variables needed to describe reactive mixtures that include a solid matrix are: the (uniform) temperature , the solid matrix deformation gradient (or related strain measures), and the molar content of the various constituents. This set differs from the classical treatment of chemical kinetics in fluid mixtures by the inclusion of and the subset of constituents bound to the solid matrix. To maintain a consistent notation in this section, solid-bound molecular species are described by their molar concentrations and molar supplies which may be related to their referential mass density and referential mass supply according to Consider a general chemical reaction, where is the chemical species representing constituent ; and represent stoichiometric coefficients of the reactants and products, respectively. Since the molar supply of reactants and products is constrained by stoichiometry, it follows that all molar supplies in a specific chemical reaction may be related to a production rate according to where represents the net stoichiometric coefficient for , Thus, formulating constitutive relations for is equivalent to providing a single relation for . When the chemical reaction is reversible, the relations of (2.10.4-3)-(2.10.4-4) still apply but the form of would be different.
Using the relations of (2.10.2-1), (2.10.4-1) and (2.10.4-3), it follows in general that , so that the constraint of (2.10-2) is equivalent to enforcing stoichiometry, namely, Thus, properly balancing a chemical reaction satisfies this constraint.
The mixture mass balance in (2.10.3-3) may now be rewritten as where and is the molar volume of . (Currently in FEBio, is specified independently of , because users may choose to neglect the contribution from in (2.10.4-7); therefore, if one desires to model chemical reactions, (2.10.4-2) or (2.10.4-5), that involve the solvent, it is necessary to explicitly provide a solvent supply function compatible with the above relations, namely .) Similarly, the solute mass balance in (2.10.2-3) becomes These mass balance equations reduce to those of non-reactive mixtures when .
Subsection 2.10.3: Mixture with Negligible Solute Volume Fraction Up Section 2.10: Chemical Reactions Section 2.11: Fluid Mechanics