HMWSoln.h

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/**
 *  @file HMWSoln.h
 *    Headers for the HMWSoln ThermoPhase object, which models concentrated
 *    electrolyte solutions
 *    (see \ref thermoprops and \link Cantera::HMWSoln HMWSoln \endlink) .
 *
 * Class HMWSoln represents a concentrated liquid electrolyte phase which
 * obeys the Pitzer formulation for nonideality using molality-based
 * standard states.
 */

// This file is part of Cantera. See License.txt in the top-level directory or
// at http://www.cantera.org/license.txt for license and copyright information.

#ifndef CT_HMWSOLN_H
#define CT_HMWSOLN_H

#include "MolalityVPSSTP.h"
#include "cantera/base/Array.h"

namespace Cantera
{

/**
 * Major Parameters:
 *   The form of the Pitzer expression refers to the form of the Gibbs free
 *   energy expression. The temperature dependence of the Pitzer coefficients
 *   are handled by another parameter.
 *
 * m_formPitzer = Form of the Pitzer expression
 *
 *  PITZERFORM_BASE = 0
 *
 * Only one form is supported atm. This parameter is included for
 * future expansion.
 */
#define PITZERFORM_BASE 0

/*!
 * @name Temperature Dependence of the Pitzer Coefficients
 *
 * Note, the temperature dependence of the Gibbs free energy also depends on the
 * temperature dependence of the standard state and the temperature dependence
 * of the Debye-Huckel constant, which includes the dielectric constant and the
 * density. Therefore, this expression defines only part of the temperature
 * dependence for the mixture thermodynamic functions.
 *
 *  PITZER_TEMP_CONSTANT
 *     All coefficients are considered constant wrt temperature
 *  PITZER_TEMP_LINEAR
 *     All coefficients are assumed to have a linear dependence
 *     wrt to temperature.
 *  PITZER_TEMP_COMPLEX1
 *     All coefficients are assumed to have a complex functional
 *     based dependence wrt temperature;  See:
 *    (Silvester, Pitzer, J. Phys. Chem. 81, 19 1822 (1977)).
 *
 *       beta0 = q0 + q3(1/T - 1/Tr) + q4(ln(T/Tr)) +
 *               q1(T - Tr) + q2(T**2 - Tr**2)
 */
//@{
#define PITZER_TEMP_CONSTANT 0
#define PITZER_TEMP_LINEAR 1
#define PITZER_TEMP_COMPLEX1 2
//@}

/*
 * @name ways to calculate the value of A_Debye
 *
 * These defines determine the way A_Debye is calculated
 */
//@{
#define A_DEBYE_CONST 0
#define A_DEBYE_WATER 1
//@}

class WaterProps;

/**
 * Class HMWSoln represents a dilute or concentrated liquid electrolyte
 * phase which obeys the Pitzer formulation for nonideality.
 *
 * As a prerequisite to the specification of thermodynamic quantities,
 * The concentrations of the ionic species are assumed to obey the
 * electroneutrality condition.
 *
 * ## Specification of Species Standard State Properties
 *
 * The solvent is assumed to be liquid water. A real model for liquid water
 * (IAPWS 1995 formulation) is used as its standard state. All standard state
 * properties for the solvent are based on this real model for water, and
 * involve function calls to the object that handles the real water model,
 * #Cantera::WaterPropsIAPWS.
 *
 * The standard states for solutes are on the unit molality basis. Therefore, in
 * the documentation below, the normal \f$ o \f$ superscript is replaced with
 * the \f$ \triangle \f$ symbol. The reference state symbol is now
 * \f$ \triangle, ref \f$.
 *
 * It is assumed that the reference state thermodynamics may be obtained by a
 * pointer to a populated species thermodynamic property manager class (see
 * ThermoPhase::m_spthermo). How to relate pressure changes to the reference
 * state thermodynamics is resolved at this level.
 *
 * For solutes that rely on ThermoPhase::m_spthermo, are assumed to have an
 * incompressible standard state mechanical property. In other words, the molar
 * volumes are independent of temperature and pressure.
 *
 * For these incompressible, standard states, the molar internal energy is
 * independent of pressure. Since the thermodynamic properties are specified by
 * giving the standard-state enthalpy, the term \f$ P_0 \hat v\f$ is subtracted
 * from the specified molar enthalpy to compute the molar internal energy. The
 * entropy is assumed to be independent of the pressure.
 *
 * The enthalpy function is given by the following relation.
 *
 * \f[
 *    h^\triangle_k(T,P) = h^{\triangle,ref}_k(T)
 *        + \tilde{v}_k \left( P - P_{ref} \right)
 * \f]
 *
 * For an incompressible, stoichiometric substance, the molar internal energy is
 * independent of pressure. Since the thermodynamic properties are specified by
 * giving the standard-state enthalpy, the term \f$ P_{ref} \tilde v\f$ is
 * subtracted from the specified reference molar enthalpy to compute the molar
 * internal energy.
 *
 * \f[
 *      u^\triangle_k(T,P) = h^{\triangle,ref}_k(T) - P_{ref} \tilde{v}_k
 * \f]
 *
 * The solute standard state heat capacity and entropy are independent of
 * pressure. The solute standard state Gibbs free energy is obtained from the
 * enthalpy and entropy functions.
 *
 * The vector Phase::m_speciesSize[] is used to hold the base values of species
 * sizes. These are defined as the molar volumes of species at infinite dilution
 * at 300 K and 1 atm of water. m_speciesSize are calculated during the
 * initialization of the HMWSoln object and are then not touched.
 *
 * The current model assumes that an incompressible molar volume for all
 * solutes. The molar volume for the water solvent, however, is obtained from a
 * pure water equation of state, waterSS. Therefore, the water standard state
 * varies with both T and P. It is an error to request standard state water
 * properties at a T and P where the water phase is not a stable phase, i.e.,
 * beyond its spinodal curve.
 *
 * ## Specification of Solution Thermodynamic Properties
 *
 * Chemical potentials of the solutes, \f$ \mu_k \f$, and the solvent, \f$ \mu_o
 * \f$, which are based on the molality form, have the following general format:
 *
 * \f[
 *    \mu_k = \mu^{\triangle}_k(T,P) + R T ln(\gamma_k^{\triangle} \frac{m_k}{m^\triangle})
 * \f]
 * \f[
 *    \mu_o = \mu^o_o(T,P) + RT ln(a_o)
 * \f]
 *
 * where \f$ \gamma_k^{\triangle} \f$ is the molality based activity coefficient
 * for species \f$k\f$.
 *
 * Individual activity coefficients of ions can not be independently measured.
 * Instead, only binary pairs forming electroneutral solutions can be measured.
 * This problem leads to a redundancy in the evaluation of species standard
 * state properties. The redundancy issue is resolved by setting the standard
 * state chemical potential enthalpy, entropy, and volume for the hydrogen ion,
 * H+, to zero, for every temperature and pressure. After this convention is
 * applied, all other standard state properties of ionic species contain
 * meaningful information.
 *
 * ### Ionic Strength
 *
 * Most of the parameterizations within the model use the ionic strength as a
 * key variable. The ionic strength, \f$ I\f$ is defined as follows
 *
 * \f[
 *    I = \frac{1}{2} \sum_k{m_k  z_k^2}
 * \f]
 *
 * \f$ m_k \f$ is the molality of the kth species. \f$ z_k \f$ is the charge of
 * the kth species. Note, the ionic strength is a defined units quantity. The
 * molality has defined units of gmol kg-1, and therefore the ionic strength has
 * units of sqrt(gmol/kg).
 *
 * In some instances, from some authors, a different formulation is used for the
 * ionic strength in the equations below. The different formulation is due to
 * the possibility of the existence of weak acids and how association wrt to the
 * weak acid equilibrium relation affects the calculation of the activity
 * coefficients via the assumed value of the ionic strength.
 *
 * If we are to assume that the association reaction doesn't have an effect on
 * the ionic strength, then we will want to consider the associated weak acid as
 * in effect being fully dissociated, when we calculate an effective value for
 * the ionic strength. We will call this calculated value, the stoichiometric
 * ionic strength, \f$ I_s \f$, putting a subscript s to denote it from the more
 * straightforward calculation of \f$ I \f$.
 *
 * \f[
 *    I_s = \frac{1}{2} \sum_k{m_k^s  z_k^2}
 * \f]
 *
 * Here, \f$ m_k^s \f$ is the value of the molalities calculated assuming that
 * all weak acid-base pairs are in their fully dissociated states. This
 * calculation may be simplified by considering that the weakly associated acid
 * may be made up of two charged species, k1 and k2, each with their own
 * charges, obeying the following relationship:
 *
 * \f[
 *      z_k = z_{k1} +  z_{k2}
 * \f]
 * Then, we may only need to specify one charge value, say, \f$  z_{k1}\f$, the
 * cation charge number, in order to get both numbers, since we have already
 * specified \f$ z_k \f$ in the definition of original species. Then, the
 * stoichiometric ionic strength may be calculated via the following formula.
 *
 * \f[
 *    I_s = \frac{1}{2} \left(\sum_{k,ions}{m_k  z_k^2}+
 *               \sum_{k,weak_assoc}(m_k  z_{k1}^2 + m_k  z_{k2}^2) \right)
 * \f]
 *
 * The specification of which species are weakly associated acids is made in the
 * input file via the `stoichIsMods` XML block, where the charge for k1 is also
 * specified. An example is given below:
 *
 * @code
 * <stoichIsMods>
 *       NaCl(aq):-1.0
 * </stoichIsMods>
 * @endcode
 *
 * Because we need the concept of a weakly associated acid in order to calculated
 * \f$ I_s \f$ we need to catalog all species in the phase. This is done using
 * the following categories:
 *
 *  -  `cEST_solvent`                Solvent species (neutral)
 *  -  `cEST_chargedSpecies`         Charged species (charged)
 *  -  `cEST_weakAcidAssociated`     Species which can break apart into charged species.
 *                                   It may or may not be charged.  These may or
 *                                   may not be be included in the
 *                                   species solution vector.
 *  -  `cEST_strongAcidAssociated`   Species which always breaks apart into charged species.
 *                                   It may or may not be charged. Normally, these
 *                                   aren't included in the speciation vector.
 *  -  `cEST_polarNeutral`           Polar neutral species
 *  -  `cEST_nonpolarNeutral`        Non polar neutral species
 *
 * Polar and non-polar neutral species are differentiated, because some
 * additions to the activity coefficient expressions distinguish between these
 * two types of solutes. This is the so-called salt-out effect.
 *
 * The type of species is specified in the `electrolyteSpeciesType` XML block.
 * Note, this is not considered a part of the specification of the standard
 * state for the species, at this time. Therefore, this information is put under
 * the `activityCoefficient` XML block. An example is given below
 *
 * @code
 * <electrolyteSpeciesType>
 *        H2L(L):solvent
 *        H+:chargedSpecies
 *        NaOH(aq):weakAcidAssociated
 *        NaCl(aq):strongAcidAssociated
 *        NH3(aq):polarNeutral
 *        O2(aq):nonpolarNeutral
 * </electrolyteSpeciesType>
 * @endcode
 *
 * Much of the species electrolyte type information is inferred from other
 * information in the input file. For example, as species which is charged is
 * given the "chargedSpecies" default category. A neutral solute species is put
 * into the "nonpolarNeutral" category by default.
 *
 * ### Specification of the Excess Gibbs Free Energy
 *
 * Pitzer's formulation may best be represented as a specification of the excess
 * Gibbs free energy, \f$ G^{ex} \f$, defined as the deviation of the total
 * Gibbs free energy from that of an ideal molal solution.
 * \f[
 *     G = G^{id} + G^{ex}
 * \f]
 *
 * The ideal molal solution contribution, not equal to an ideal solution
 * contribution and in fact containing a singularity at the zero solvent mole
 * fraction limit, is given below.
 * \f[
 *     G^{id} = n_o \mu^o_o + \sum_{k\ne o} n_k \mu_k^{\triangle}
 *           + \tilde{M}_o n_o ( RT (\sum{m_i(\ln(m_i)-1)}))
 * \f]
 *
 * From the excess Gibbs free energy formulation, the activity coefficient
 * expression and the osmotic coefficient expression for the solvent may be
 * defined, by taking the appropriate derivatives. Using this approach
 * guarantees that the entire system will obey the Gibbs-Duhem relations.
 *
 * Pitzer employs the following general expression for the excess Gibbs free
 * energy
 *
 *  \f[
 *    \begin{array}{cclc}
 *     \frac{G^{ex}}{\tilde{M}_o  n_o RT} &= &
 *          \left( \frac{4A_{Debye}I}{3b} \right) \ln(1 + b \sqrt{I})
 *        +   2 \sum_c \sum_a m_c m_a B_{ca}
 *        +     \sum_c \sum_a m_c m_a Z C_{ca}
 *          \\&&
 *        +   \sum_{c < c'} \sum m_c m_{c'} \left[ 2 \Phi_{c{c'}} + \sum_a m_a \Psi_{c{c'}a} \right]
 *        +   \sum_{a < a'} \sum m_a m_{a'} \left[ 2 \Phi_{a{a'}} + \sum_c m_c \Psi_{a{a'}c} \right]
 *          \\&&
 *        + 2 \sum_n \sum_c m_n m_c \lambda_{nc} + 2 \sum_n \sum_a m_n m_a \lambda_{na}
 *        + 2 \sum_{n < n'} \sum m_n m_{n'} \lambda_{n{n'}}
 *        +  \sum_n m^2_n \lambda_{nn}
 *    \end{array}
 *  \f]
 *
 * *a* is a subscript over all anions, *c* is a subscript extending over all
 * cations, and  *i* is a subscript that extends over all anions and cations.
 * *n* is a subscript that extends only over neutral solute molecules. The
 * second line contains cross terms where cations affect cations and/or
 * cation/anion pairs, and anions affect anions or cation/anion pairs. Note part
 * of the coefficients, \f$ \Phi_{c{c'}} \f$ and  \f$ \Phi_{a{a'}} \f$ stem from
 * the theory of unsymmetrical mixing of electrolytes with different charges.
 * This theory depends on the total ionic strength of the solution, and
 * therefore, \f$ \Phi_{c{c'}} \f$ and  \f$ \Phi_{a{a'}} \f$  will depend on
 * *I*, the ionic strength.  \f$ B_{ca}\f$ is a strong function of the
 * total ionic strength, *I*, of the electrolyte. The rest of the coefficients
 * are assumed to be independent of the molalities or ionic strengths. However,
 * all coefficients are potentially functions of the temperature and pressure
 * of the solution.
 *
 * *A* is the Debye-Huckel constant. Its specification is described in its
 * own section below.
 *
 * \f$ I\f$ is the ionic strength of the solution, and is given by:
 *
 * \f[
 *     I = \frac{1}{2} \sum_k{m_k  z_k^2}
 * \f]
 *
 * In contrast to several other Debye-Huckel implementations (see \ref
 * DebyeHuckel), the parameter \f$ b\f$ in the above equation is a constant that
 * does not vary with respect to ion identity. This is an important
 * simplification as it avoids troubles with satisfaction of the Gibbs-Duhem
 * analysis.
 *
 * The function \f$ Z \f$ is given by
 *
 * \f[
 *     Z = \sum_i m_i \left| z_i \right|
 * \f]
 *
 * The value of \f$ B_{ca}\f$ is given by the following function
 *
 * \f[
 *     B_{ca} = \beta^{(0)}_{ca} + \beta^{(1)}_{ca} g(\alpha^{(1)}_{ca} \sqrt{I})
 *            + \beta^{(2)}_{ca} g(\alpha^{(2)}_{ca} \sqrt{I})
 * \f]
 *
 * where
 *
 * \f[
 *     g(x) = 2 \frac{(1 - (1 + x)\exp[-x])}{x^2}
 * \f]
 *
 * The formulation for \f$ B_{ca}\f$ combined with the formulation of the Debye-
 * Huckel term in the eqn. for the excess Gibbs free energy stems essentially
 * from an empirical fit to the ionic strength dependent data based over a wide
 * sampling of binary electrolyte systems. \f$ C_{ca} \f$, \f$ \lambda_{nc} \f$,
 * \f$ \lambda_{na} \f$, \f$ \lambda_{nn} \f$, \f$ \Psi_{c{c'}a} \f$, \f$
 * \Psi_{a{a'}c} \f$ are experimentally derived coefficients that may have
 * pressure and/or temperature dependencies.
 *
 * The \f$ \Phi_{c{c'}} \f$ and \f$ \Phi_{a{a'}} \f$ formulations are slightly
 * more complicated. \f$ b \f$ is a universal constant defined to be equal to
 * \f$ 1.2\ kg^{1/2}\ gmol^{-1/2} \f$. The exponential coefficient \f$
 * \alpha^{(1)}_{ca} \f$ is usually fixed at \f$ \alpha^{(1)}_{ca} = 2.0\
 * kg^{1/2} gmol^{-1/2}\f$ except for 2-2 electrolytes, while other parameters
 * were fit to experimental data. For 2-2 electrolytes, \f$ \alpha^{(1)}_{ca} =
 * 1.4\ kg^{1/2}\ gmol^{-1/2}\f$ is used in combination with either \f$
 * \alpha^{(2)}_{ca} = 12\ kg^{1/2}\ gmol^{-1/2}\f$ or \f$ \alpha^{(2)}_{ca} = k
 * A_\psi \f$, where *k* is a constant. For electrolytes other than 2-2
 * electrolytes the \f$ \beta^{(2)}_{ca} g(\alpha^{(2)}_{ca} \sqrt{I}) \f$  term
 * is not used in the fitting procedure; it is only used for divalent metal
 * solfates and other high-valence electrolytes which exhibit significant
 * association at low ionic strengths.
 *
 * The \f$ \beta^{(0)}_{ca} \f$, \f$ \beta^{(1)}_{ca}\f$, \f$ \beta^{(2)}_{ca}
 * \f$, and \f$ C_{ca} \f$ binary coefficients are referred to as ion-
 * interaction or Pitzer parameters. These Pitzer parameters may vary with
 * temperature and pressure but they do not depend on the ionic strength. Their
 * values and temperature derivatives of their values have been tabulated for a
 * range of electrolytes
 *
 * The \f$ \Phi_{c{c'}} \f$ and \f$ \Phi_{a{a'}} \f$ contributions, which
 * capture cation-cation and anion-anion interactions, also have an ionic
 * strength dependence.
 *
 * Ternary contributions \f$ \Psi_{c{c'}a} \f$ and \f$ \Psi_{a{a'}c} \f$ have
 * been measured also for some systems. The success of the Pitzer method lies in
 * its ability to model nonlinear activity coefficients of complex
 * multicomponent systems with just binary and minor ternary contributions,
 * which can be independently measured in binary or ternary subsystems.
 *
 * ### Multicomponent Activity Coefficients for Solutes
 *
 * The formulas for activity coefficients of solutes may be obtained by taking
 * the following derivative of the excess Gibbs Free Energy formulation
 * described above:
 *
 * \f[
 *    \ln(\gamma_k^\triangle) = \frac{d\left( \frac{G^{ex}}{M_o n_o RT} \right)}{d(m_k)}\Bigg|_{n_i}
 * \f]
 *
 * In the formulas below the following conventions are used. The subscript *M*
 * refers to a particular cation. The subscript X refers to a particular anion,
 * whose activity is being currently evaluated. the subscript *a* refers to a
 * summation over all anions in the solution, while the subscript *c* refers to
 * a summation over all cations in the solutions.
 *
 * The activity coefficient for a particular cation *M* is given by
 *
 * \f[
 *     \ln(\gamma_M^\triangle) = -z_M^2(F) + \sum_a m_a \left( 2 B_{Ma} + Z C_{Ma} \right)
 *     + z_M   \left( \sum_a  \sum_c m_a m_c C_{ca} \right)
 *            + \sum_c m_c \left[ 2 \Phi_{Mc} + \sum_a m_a \Psi_{Mca} \right]
 *            + \sum_{a < a'} \sum m_a m_{a'} \Psi_{Ma{a'}}
 *            +  2 \sum_n m_n \lambda_{nM}
 * \f]
 *
 * The activity coefficient for a particular anion *X* is given by
 *
 * \f[
 *     \ln(\gamma_X^\triangle) = -z_X^2(F) + \sum_a m_c \left( 2 B_{cX} + Z C_{cX} \right)
 *     + \left|z_X \right|  \left( \sum_a  \sum_c m_a m_c C_{ca} \right)
 *            + \sum_a m_a \left[ 2 \Phi_{Xa} + \sum_c m_c \Psi_{cXa} \right]
 *            + \sum_{c < c'} \sum m_c m_{c'} \Psi_{c{c'}X}
 *            +  2 \sum_n m_n \lambda_{nM}
 * \f]
 * where the function \f$ F \f$ is given by
 *
 * \f[
 *      F = - A_{\phi} \left[ \frac{\sqrt{I}}{1 + b \sqrt{I}}
 *                + \frac{2}{b} \ln{\left(1 + b\sqrt{I}\right)} \right]
 *                + \sum_a \sum_c m_a m_c B'_{ca}
 *                + \sum_{c < c'} \sum m_c m_{c'} \Phi'_{c{c'}}
 *                + \sum_{a < a'} \sum m_a m_{a'} \Phi'_{a{a'}}
 * \f]
 *
 * We have employed the definition of \f$ A_{\phi} \f$, also used by Pitzer
 * which is equal to
 *
 * \f[
 *   A_{\phi} = \frac{A_{Debye}}{3}
 * \f]
 *
 * In the above formulas, \f$ \Phi'_{c{c'}} \f$  and \f$ \Phi'_{a{a'}} \f$ are the
 * ionic strength derivatives of \f$ \Phi_{c{c'}} \f$  and \f$  \Phi_{a{a'}} \f$,
 * respectively.
 *
 * The function \f$ B'_{MX} \f$ is defined as:
 *
 * \f[
 *      B'_{MX} = \left( \frac{\beta^{(1)}_{MX} h(\alpha^{(1)}_{MX} \sqrt{I})}{I}  \right)
 *                \left( \frac{\beta^{(2)}_{MX} h(\alpha^{(2)}_{MX} \sqrt{I})}{I}  \right)
 * \f]
 *
 * where \f$ h(x) \f$ is defined as
 *
 * \f[
 *     h(x) = g'(x) \frac{x}{2} =
 *      \frac{2\left(1 - \left(1 + x + \frac{x^2}{2} \right)\exp(-x) \right)}{x^2}
 * \f]
 *
 * The activity coefficient for neutral species *N* is given by
 *
 * \f[
 *     \ln(\gamma_N^\triangle) = 2 \left( \sum_i m_i \lambda_{iN}\right)
 * \f]
 *
 * ### Activity of the Water Solvent
 *
 * The activity for the solvent water,\f$ a_o \f$, is not independent and must
 * be determined either from the Gibbs-Duhem relation or from taking the
 * appropriate derivative of the same excess Gibbs free energy function as was
 * used to formulate the solvent activity coefficients. Pitzer's description
 * follows the later approach to derive a formula for the osmotic coefficient,
 * \f$ \phi \f$.
 *
 * \f[
 *      \phi - 1 = - \left( \frac{d\left(\frac{G^{ex}}{RT} \right)}{d(\tilde{M}_o n_o)}  \right)
 *               \frac{1}{\sum_{i \ne 0} m_i}
 * \f]
 *
 * The osmotic coefficient may be related to the water activity by the following relation:
 *
 * \f[
 *      \phi = - \frac{1}{\tilde{M}_o \sum_{i \neq o} m_i} \ln(a_o)
 *          = - \frac{n_o}{\sum_{i \neq o}n_i} \ln(a_o)
 * \f]
 *
 * The result is the following
 *
 * \f[
 *   \begin{array}{ccclc}
 *     \phi - 1 &= &
 *         \frac{2}{\sum_{i \ne 0} m_i}
 *          \bigg[ &
 *       -  A_{\phi} \frac{I^{3/2}}{1 + b \sqrt{I}}
 *       +   \sum_c  \sum_a m_c m_a \left( B^{\phi}_{ca} + Z C_{ca}\right)
 *         \\&&&
 *       +   \sum_{c < c'} \sum m_c m_{c'} \left[ \Phi^{\phi}_{c{c'}} + \sum_a m_a \Psi_{c{c'}a} \right]
 *       +   \sum_{a < a'} \sum m_a m_{a'} \left[ \Phi^{\phi}_{a{a'}} + \sum_c m_c \Psi_{a{a'}c} \right]
 *         \\&&&
 *       + \sum_n \sum_c m_n m_c \lambda_{nc} +  \sum_n \sum_a m_n m_a \lambda_{na}
 *       + \sum_{n < n'} \sum m_n m_{n'} \lambda_{n{n'}}
 *       + \frac{1}{2} \left( \sum_n m^2_n \lambda_{nn}\right)
 *         \bigg]
 *   \end{array}
 * \f]
 *
 * It can be shown that the expression
 *
 * \f[
 *    B^{\phi}_{ca} = \beta^{(0)}_{ca} + \beta^{(1)}_{ca} \exp{(- \alpha^{(1)}_{ca} \sqrt{I})}
 *            + \beta^{(2)}_{ca} \exp{(- \alpha^{(2)}_{ca} \sqrt{I} )}
 * \f]
 *
 * is consistent with the expression \f$ B_{ca} \f$ in the \f$ G^{ex} \f$
 * expression after carrying out the derivative wrt \f$ m_M \f$.
 *
 * Also taking into account that  \f$ {\Phi}_{c{c'}} \f$ and
 * \f$ {\Phi}_{a{a'}} \f$ has an ionic strength dependence.
 *
 * \f[
 *   \Phi^{\phi}_{c{c'}} = {\Phi}_{c{c'}} + I \frac{d{\Phi}_{c{c'}}}{dI}
 * \f]
 *
 * \f[
 *   \Phi^{\phi}_{a{a'}} = \Phi_{a{a'}} + I \frac{d\Phi_{a{a'}}}{dI}
 * \f]
 *
 * ### Temperature and Pressure Dependence of the Pitzer Parameters
 *
 * In general most of the coefficients introduced in the previous section may
 * have a temperature and pressure dependence. The temperature and pressure
 * dependence of these coefficients strongly influence the value of the excess
 * Enthalpy and excess Volumes of Pitzer solutions. Therefore, these are readily
 * measurable quantities. HMWSoln provides several different methods for putting
 * these dependencies into the coefficients. HMWSoln has an implementation
 * described by Silverter and Pitzer (1977), which was used to fit experimental
 * data for NaCl over an extensive range, below the critical temperature of
 * water. They found a temperature functional form for fitting the 3 following
 * coefficients that describe the Pitzer parameterization for a single salt to
 * be adequate to describe how the excess Gibbs free energy values for the
 * binary salt changes with respect to temperature. The following functional
 * form was used to fit the temperature dependence of the Pitzer Coefficients
 * for each cation - anion pair, M X.
 *
 * \f[
 *     \beta^{(0)}_{MX} = q^{b0}_0
 *                      + q^{b0}_1 \left( T - T_r \right)
 *                      + q^{b0}_2 \left( T^2 - T_r^2 \right)
 *                      + q^{b0}_3 \left( \frac{1}{T} - \frac{1}{T_r}\right)
 *                      + q^{b0}_4 \ln \left( \frac{T}{T_r} \right)
 * \f]
 * \f[
 *     \beta^{(1)}_{MX} = q^{b1}_0  + q^{b1}_1 \left( T - T_r \right)
 *                      + q^{b1}_{2} \left( T^2 - T_r^2 \right)
 * \f]
 * \f[
 *    C^{\phi}_{MX} = q^{Cphi}_0
 *                  + q^{Cphi}_1 \left( T - T_r \right)
 *                  + q^{Cphi}_2 \left( T^2 - T_r^2 \right)
 *                  + q^{Cphi}_3 \left( \frac{1}{T} - \frac{1}{T_r}\right)
 *                  + q^{Cphi}_4 \ln \left( \frac{T}{T_r} \right)
 * \f]
 *
 * where
 *
 * \f[
 *     C^{\phi}_{MX} =  2 {\left| z_M z_X \right|}^{1/2} C_{MX}
 * \f]
 *
 * In later papers, Pitzer has added additional temperature dependencies to all
 * of the other remaining second and third order virial coefficients. Some of
 * these dependencies are justified and motivated by theory. Therefore, a
 * formalism wherein all of the coefficients in the base theory have temperature
 * dependencies associated with them has been implemented within the HMWSoln
 * object. Much of the formalism, however, has been unexercised.
 *
 * In the HMWSoln object, the temperature dependence of the Pitzer parameters
 * are specified in the following way.
 *
 *    - PIZTER_TEMP_CONSTANT      - string name "CONSTANT"
 *      - Assumes that all coefficients are independent of temperature
 *        and pressure
 *    - PIZTER_TEMP_COMPLEX1     - string name "COMPLEX" or "COMPLEX1"
 *      - Uses the full temperature dependence for the
 *        \f$\beta^{(0)}_{MX} \f$ (5 coeffs),
 *        the  \f$\beta^{(1)}_{MX} \f$ (3 coeffs),
 *        and \f$ C^{\phi}_{MX} \f$ (5 coeffs) parameters described above.
 *    - PITZER_TEMP_LINEAR        - string name "LINEAR"
 *      - Uses just the temperature dependence for the
 *        \f$\beta^{(0)}_{MX} \f$, the \f$\beta^{(1)}_{MX} \f$,
 *        and \f$ C^{\phi}_{MX} \f$ coefficients described above.
 *        There are 2 coefficients for each term.
 *
 * The temperature dependence is specified in an attributes field in the
 * `activityCoefficients` XML block, called `TempModel`. Permissible values for
 * that attribute are `CONSTANT`, `COMPLEX1`, and `LINEAR`.
 *
 * The specification of the binary interaction between a cation and an anion is
 * given by the coefficients, \f$ B_{MX}\f$ and \f$ C_{MX}\f$ The specification
 * of \f$ B_{MX}\f$ is a function of \f$\beta^{(0)}_{MX} \f$,
 * \f$\beta^{(1)}_{MX} \f$, \f$\beta^{(2)}_{MX} \f$, \f$\alpha^{(1)}_{MX} \f$,
 * and \f$\alpha^{(2)}_{MX} \f$. \f$ C_{MX}\f$ is calculated from
 * \f$C^{\phi}_{MX} \f$ from the formula above. All of the underlying
 * coefficients are specified in the XML element block `binarySaltParameters`,
 * which has the attribute `cation` and `anion` to identify the interaction. XML
 * elements named `beta0, beta1, beta2, Cphi, Alpha1, Alpha2` within each
 * `binarySaltParameters` block specify the parameters. Within each of these
 * blocks multiple parameters describing temperature or pressure dependence are
 * serially listed in the order that they appear in the equation in this
 * document. An example of the `beta0` block that fits the `COMPLEX1`
 * temperature dependence given above is
 *
 * @code
 *  <binarySaltParameters cation="Na+" anion="OH-">
 *    <beta0> q0, q1, q2, q3, q4  </beta0>
 *  </binarySaltParameters>
 * @endcode
 *
 * The parameters for \f$ \beta^{(0)}\f$ fit the following equation:
 *
 * \f[
 *     \beta^{(0)} = q_0^{{\beta}0} + q_1^{{\beta}0} \left( T - T_r \right)
 *            + q_2^{{\beta}0} \left( T^2 - T_r^2 \right)
 *            + q_3^{{\beta}0} \left( \frac{1}{T} - \frac{1}{T_r} \right)
 *            + q_4^{{\beta}0} \ln \left( \frac{T}{T_r} \right)
 * \f]
 *
 * This same `COMPLEX1` temperature dependence given above is used for the
 * following parameters:
 * \f$ \beta^{(0)}_{MX} \f$, \f$ \beta^{(1)}_{MX} \f$,
 * \f$ \beta^{(2)}_{MX} \f$, \f$ \Theta_{cc'} \f$, \f$\Theta_{aa'} \f$,
 * \f$ \Psi_{c{c'}a} \f$ and \f$ \Psi_{ca{a'}} \f$.
 *
 * ### Like-Charged Binary Ion Parameters and the Mixing Parameters
 *
 * The previous section contained the functions, \f$ \Phi_{c{c'}} \f$,
 * \f$ \Phi_{a{a'}} \f$ and their derivatives wrt the ionic strength, \f$
 * \Phi'_{c{c'}} \f$ and \f$ \Phi'_{a{a'}} \f$. Part of these terms come from
 * theory.
 *
 * Since like charged ions repel each other and are generally not near each
 * other, the virial coefficients for same-charged ions are small. However,
 * Pitzer doesn't ignore these in his formulation. Relatively larger and longer
 * range terms between like-charged ions exist however, which appear only for
 * unsymmetrical mixing of same-sign charged ions with different charges. \f$
 * \Phi_{ij} \f$, where \f$ ij \f$ is either \f$ a{a'} \f$ or \f$ c{c'} \f$ is
 * given by
 *
 * \f[
 *     {\Phi}_{ij} = \Theta_{ij} + \,^E \Theta_{ij}(I)
 * \f]
 *
 * \f$ \Theta_{ij} \f$ is the small virial coefficient expansion term. Dependent
 * in general on temperature and pressure, its ionic strength dependence is
 * ignored in Pitzer's approach. \f$ \,^E\Theta_{ij}(I) \f$ accounts for the
 * electrostatic unsymmetrical mixing effects and is dependent only on the
 * charges of the ions i, j, the total ionic strength and on the dielectric
 * constant and density of the solvent. This seems to be a relatively well-
 * documented part of the theory. They theory below comes from Pitzer summation
 * (Pitzer) in the appendix. It's also mentioned in Bethke's book (Bethke), and
 * the equations are summarized in Harvie & Weare (1980). Within the code, \f$
 * \,^E\Theta_{ij}(I) \f$ is evaluated according to the algorithm described in
 * Appendix B [Pitzer] as
 *
 * \f[
 *    \,^E\Theta_{ij}(I) = \left( \frac{z_i z_j}{4I} \right)
 *       \left( J(x_{ij})  - \frac{1}{2} J(x_{ii})
 *                         - \frac{1}{2} J(x_{jj})  \right)
 * \f]
 *
 * where \f$ x_{ij} = 6 z_i z_j A_{\phi} \sqrt{I} \f$ and
 *
 *  \f[
 *     J(x) = \frac{1}{x} \int_0^{\infty}{\left( 1 + q +
 *            \frac{1}{2} q^2 - e^q \right) y^2 dy}
 *  \f]
 *
 * and \f$ q = - (\frac{x}{y}) e^{-y} \f$. \f$ J(x) \f$ is evaluated by
 * numerical integration.
 *
 * The \f$  \Theta_{ij} \f$ term is a constant that is specified by the XML
 * element `thetaCation` and `thetaAnion`, which has the attribute `cation1`,
 * `cation2` and `anion1`, `anion2` respectively to identify the interaction. No
 * temperature or pressure dependence of this parameter is currently allowed. An
 * example of the block is presented below.
 *
 * @code
 *   <thetaCation cation1="Na+" cation2="H+">
 *              <Theta> 0.036 </Theta>
 *   </thetaCation>
 * @endcode
 *
 * ### Ternary Pitzer Parameters
 *
 * The \f$  \Psi_{c{c'}a} \f$ and \f$  \Psi_{ca{a'}} \f$ terms represent ternary
 * interactions between two cations and an anion and two anions and a cation,
 * respectively. In Pitzer's implementation these terms are usually small in
 * absolute size. Currently these parameters do not have any dependence on
 * temperature, pressure, or ionic strength.
 *
 * Their values are input using the XML element `psiCommonCation` and
 * `psiCommonAnion`. The species id's are specified in attribute fields in the
 * XML element. The fields `cation`, `anion1`, and `anion2` are used for
 * `psiCommonCation`. The fields `anion`, `cation1` and `cation2` are used for
 * `psiCommonAnion`. An example block is given below. The `Theta` field below is
 * a duplicate of the `thetaAnion` field mentioned above. The two fields are
 * input into the same block for convenience, and because their data are highly
 * correlated, in practice. It is an error for the two blocks to specify
 * different information about thetaAnion (or thetaCation) in different blocks.
 * It's ok to specify duplicate but consistent information in multiple blocks.
 *
 * @code
 * <psiCommonCation cation="Na+" anion1="Cl-" anion2="OH-">
 *     <Theta> -0.05 </Theta>
 *     <Psi> -0.006 </Psi>
 * </psiCommonCation>
 * @endcode
 *
 * ### Treatment of Neutral Species
 *
 * Binary virial-coefficient-like interactions between two neutral species may
 * be specified in the \f$ \lambda_{mn} \f$ terms that appear in the formulas
 * above. Currently these interactions are independent of temperature, pressure,
 * and ionic strength. Also, currently, the neutrality of the species are not
 * checked. Therefore, this interaction may involve charged species in the
 * solution as well. The identity of the species is specified by the `species1`
 * and `species2` attributes to the XML `lambdaNeutral` node. These terms are
 * symmetrical; `species1` and `species2` may be reversed and the term will be
 * the same. An example is given below.
 *
 * @code
 * <lambdaNeutral species1="CO2" species2="CH4">
 *     <lambda> 0.05 </lambda>
 * </lambdaNeutral>
 * @endcode
 *
 * ## Example of the Specification of Parameters for the Activity Coefficients
 *
 * An example is given below.
 *
 * An example `activityCoefficients` XML block for this formulation is supplied
 * below
 *
 * @code
 * <activityCoefficients model="Pitzer" TempModel="complex1">
 *     <!-- Pitzer Coefficients
 *          These coefficients are from Pitzer's main
 *          paper, in his book.
 *       -->
 *     <A_Debye model="water" />
 *     <ionicRadius default="3.042843"  units="Angstroms">
 *     </ionicRadius>
 *     <binarySaltParameters cation="Na+" anion="Cl-">
 *       <beta0> 0.0765, 0.008946, -3.3158E-6,
 *               -777.03, -4.4706
 *       </beta0>
 *       <beta1> 0.2664, 6.1608E-5, 1.0715E-6, 0.0, 0.0 </beta1>
 *       <beta2> 0.0,   0.0, 0.0, 0.0, 0.0  </beta2>
 *       <Cphi> 0.00127, -4.655E-5, 0.0,
 *              33.317, 0.09421
 *       </Cphi>
 *       <Alpha1> 2.0 </Alpha1>
 *     </binarySaltParameters>
 *
 *     <binarySaltParameters cation="H+" anion="Cl-">
 *       <beta0> 0.1775, 0.0, 0.0, 0.0, 0.0 </beta0>
 *       <beta1> 0.2945, 0.0, 0.0, 0.0, 0.0 </beta1>
 *       <beta2> 0.0,    0.0, 0.0, 0.0, 0.0 </beta2>
 *       <Cphi> 0.0008, 0.0, 0.0, 0.0, 0.0 </Cphi>
 *       <Alpha1> 2.0 </Alpha1>
 *     </binarySaltParameters>
 *
 *     <binarySaltParameters cation="Na+" anion="OH-">
 *       <beta0> 0.0864, 0.0, 0.0, 0.0, 0.0 </beta0>
 *       <beta1> 0.253,  0.0, 0.0  0.0, 0.0 </beta1>
 *       <beta2> 0.0     0.0, 0.0, 0.0, 0.0 </beta2>
 *       <Cphi> 0.0044,  0.0, 0.0, 0.0, 0.0 </Cphi>
 *       <Alpha1> 2.0 </Alpha1>
 *     </binarySaltParameters>
 *
 *     <thetaAnion anion1="Cl-" anion2="OH-">
 *       <Theta> -0.05,  0.0, 0.0, 0.0, 0.0 </Theta>
 *     </thetaAnion>
 *
 *     <psiCommonCation cation="Na+" anion1="Cl-" anion2="OH-">
 *       <Theta> -0.05,  0.0, 0.0, 0.0, 0.0 </Theta>
 *       <Psi> -0.006 </Psi>
 *     </psiCommonCation>
 *
 *     <thetaCation cation1="Na+" cation2="H+">
 *       <Theta> 0.036,  0.0, 0.0, 0.0, 0.0 </Theta>
 *     </thetaCation>
 *
 *     <psiCommonAnion anion="Cl-" cation1="Na+" cation2="H+">
 *       <Theta> 0.036,  0.0, 0.0, 0.0, 0.0 </Theta>
 *       <Psi> -0.004 </Psi>
 *     </psiCommonAnion>
 *   </activityCoefficients>
 * @endcode
 *
 * ### Specification of the Debye-Huckel Constant
 *
 * In the equations above, the formula for  \f$  A_{Debye} \f$ is needed. The
 * HMWSoln object uses two methods for specifying these quantities. The default
 * method is to assume that \f$  A_{Debye} \f$  is a constant, given in the
 * initialization process, and stored in the member double, m_A_Debye.
 * Optionally, a full water treatment may be employed that makes
 * \f$ A_{Debye} \f$ a full function of *T* and *P* and creates nontrivial
 * entries for the excess heat capacity, enthalpy, and excess volumes of
 * solution.
 *
 * \f[
 *     A_{Debye} = \frac{F e B_{Debye}}{8 \pi \epsilon R T} {\left( C_o \tilde{M}_o \right)}^{1/2}
 * \f]
 * where
 *
 * \f[
 *     B_{Debye} = \frac{F} {{(\frac{\epsilon R T}{2})}^{1/2}}
 * \f]
 * Therefore:
 * \f[
 *     A_{Debye} = \frac{1}{8 \pi}
 *                 {\left(\frac{2 N_a \rho_o}{1000}\right)}^{1/2}
 *                 {\left(\frac{N_a e^2}{\epsilon R T }\right)}^{3/2}
 * \f]
 *
 * Units = sqrt(kg/gmol)
 *
 * where
 *  - \f$ N_a \f$ is Avogadro's number
 *  - \f$ \rho_w \f$ is the density of water
 *  - \f$ e \f$ is the electronic charge
 *  - \f$ \epsilon = K \epsilon_o \f$ is the permittivity of water
 *  - \f$ K \f$ is the dielectric constant of water,
 *  - \f$ \epsilon_o \f$ is the permittivity of free space.
 *  - \f$ \rho_o \f$ is the density of the solvent in its standard state.
 *
 * Nominal value at 298 K and 1 atm = 1.172576 (kg/gmol)^(1/2)
 * based on:
 *  - \f$ \epsilon / \epsilon_0 \f$ = 78.54 (water at 25C)
 *  - T = 298.15 K
 *  - B_Debye = 3.28640E9 (kg/gmol)^(1/2) / m
 *
 * An example of a fixed value implementation is given below.
 * @code
 *   <activityCoefficients model="Pitzer">
 *         <!-- A_Debye units = sqrt(kg/gmol)  -->
 *         <A_Debye> 1.172576 </A_Debye>
 *         <!-- object description continues -->
 *   </activityCoefficients>
 * @endcode
 *
 * An example of a variable value implementation within the HMWSoln object is
 * given below. The model attribute, "water", triggers the full implementation.
 *
 * @code
 *   <activityCoefficients model="Pitzer">
 *         <!-- A_Debye units = sqrt(kg/gmol)  -->
 *         <A_Debye model="water" />
 *         <!-- object description continues -->
 *   </activityCoefficients>
 * @endcode
 *
 * ### Temperature and Pressure Dependence of the Activity Coefficients
 *
 * Temperature dependence of the activity coefficients leads to nonzero terms
 * for the excess enthalpy and entropy of solution. This means that the partial
 * molar enthalpies, entropies, and heat capacities are all non-trivial to
 * compute. The following formulas are used.
 *
 * The partial molar enthalpy, \f$ \bar s_k(T,P) \f$:
 *
 * \f[
 * \bar h_k(T,P) = h^{\triangle}_k(T,P)
 *               - R T^2 \frac{d \ln(\gamma_k^\triangle)}{dT}
 * \f]
 * The solvent partial molar enthalpy is equal to
 * \f[
 * \bar h_o(T,P) = h^{o}_o(T,P) - R T^2 \frac{d \ln(a_o)}{dT}
 *       = h^{o}_o(T,P)
 *       + R T^2 (\sum_{k \neq o} m_k)  \tilde{M_o} (\frac{d \phi}{dT})
 * \f]
 *
 * The partial molar entropy, \f$ \bar s_k(T,P) \f$:
 *
 * \f[
 *     \bar s_k(T,P) =  s^{\triangle}_k(T,P)
 *             - R \ln( \gamma^{\triangle}_k \frac{m_k}{m^{\triangle}}))
 *                    - R T \frac{d \ln(\gamma^{\triangle}_k) }{dT}
 * \f]
 * \f[
 *      \bar s_o(T,P) = s^o_o(T,P) - R \ln(a_o)
 *                    - R T \frac{d \ln(a_o)}{dT}
 * \f]
 *
 * The partial molar heat capacity, \f$ C_{p,k}(T,P)\f$:
 *
 * \f[
 *     \bar C_{p,k}(T,P) =  C^{\triangle}_{p,k}(T,P)
 *             - 2 R T \frac{d \ln( \gamma^{\triangle}_k)}{dT}
 *                    - R T^2 \frac{d^2 \ln(\gamma^{\triangle}_k) }{{dT}^2}
 * \f]
 * \f[
 *      \bar C_{p,o}(T,P) = C^o_{p,o}(T,P)
 *                   - 2 R T \frac{d \ln(a_o)}{dT}
 *                    - R T^2 \frac{d^2 \ln(a_o)}{{dT}^2}
 * \f]
 *
 * The pressure dependence of the activity coefficients leads to non-zero terms
 * for the excess Volume of the solution. Therefore, the partial molar volumes
 * are functions of the pressure derivatives of the activity coefficients.
 * \f[
 *     \bar V_k(T,P) =  V^{\triangle}_k(T,P)
 *                    + R T \frac{d \ln(\gamma^{\triangle}_k) }{dP}
 * \f]
 * \f[
 *      \bar V_o(T,P) = V^o_o(T,P)
 *                    + R T \frac{d \ln(a_o)}{dP}
 * \f]
 *
 * The majority of work for these functions take place in the internal routines
 * that calculate the first and second derivatives of the log of the activity
 * coefficients wrt temperature, s_update_dlnMolalityActCoeff_dT(),
 * s_update_d2lnMolalityActCoeff_dT2(), and the first derivative of the log
 * activity coefficients wrt pressure, s_update_dlnMolalityActCoeff_dP().
 *
 * ## %Application within Kinetics Managers
 *
 * For the time being, we have set the standard concentration for all solute
 * species in this phase equal to the default concentration of the solvent at
 * the system temperature and pressure multiplied by Mnaught (kg solvent / gmol
 * solvent). The solvent standard concentration is just equal to its standard
 * state concentration.
 *
 * This means that the kinetics operator essentially works on an generalized
 * concentration basis (kmol / m3), with units for the kinetic rate constant
 * specified as if all reactants (solvent or solute) are on a concentration
 * basis (kmol /m3). The concentration will be modified by the activity
 * coefficients.
 *
 * For example, a bulk-phase binary reaction between liquid solute species *j*
 * and *k*, producing a new liquid solute species *l* would have the following
 * equation for its rate of progress variable, \f$ R^1 \f$, which has units of
 * kmol m-3 s-1.
 *
 * \f[
 *    R^1 = k^1 C_j^a C_k^a =  k^1 (C^o_o \tilde{M}_o a_j) (C^o_o \tilde{M}_o a_k)
 * \f]
 *
 * where
 *
 * \f[
 *    C_j^a = C^o_o \tilde{M}_o a_j \quad and \quad C_k^a = C^o_o \tilde{M}_o a_k
 * \f]
 *
 * \f$ C_j^a \f$ is the activity concentration of species *j*, and
 * \f$ C_k^a \f$ is the activity concentration of species *k*. \f$ C^o_o \f$ is
 * the concentration of water at 298 K and 1 atm. \f$ \tilde{M}_o \f$ has units
 * of kg solvent per gmol solvent and is equal to
 *
 * \f[
 *     \tilde{M}_o = \frac{M_o}{1000}
 * \f]
 *
 * \f$ a_j \f$ is the activity of species *j* at the current temperature and
 * pressure and concentration of the liquid phase is given by the molality based
 * activity coefficient multiplied by the molality of the jth species.
 *
 * \f[
 *      a_j  =  \gamma_j^\triangle m_j = \gamma_j^\triangle \frac{n_j}{\tilde{M}_o n_o}
 * \f]
 *
 * \f$k^1 \f$ has units of m^3/kmol/s.
 *
 * Therefore the generalized activity concentration of a solute species has the following form
 *
 * \f[
 *      C_j^a = C^o_o \frac{\gamma_j^\triangle n_j}{n_o}
 * \f]
 *
 * The generalized activity concentration of the solvent has the same units, but it's a simpler form
 *
 * \f[
 *      C_o^a = C^o_o a_o
 * \f]
 *
 * The reverse rate constant can then be obtained from the law of microscopic reversibility
 * and the equilibrium expression for the system.
 *
 * \f[
 *      \frac{a_j a_k}{ a_l} = K^{o,1} = \exp(\frac{\mu^o_l - \mu^o_j - \mu^o_k}{R T} )
 * \f]
 *
 * \f$ K^{o,1} \f$ is the dimensionless form of the equilibrium constant.
 *
 * \f[
 *       R^{-1} = k^{-1} C_l^a =  k^{-1} (C_o  \tilde{M}_o a_l)
 * \f]
 *
 * where
 *
 * \f[
 *       k^{-1} =  k^1 K^{o,1} C_o \tilde{M}_o
 * \f]
 *
 * \f$ k^{-1} \f$ has units of 1/s.
 *
 * Note, this treatment may be modified in the future, as events dictate.
 *
 * ## Instantiation of the Class
 *
 * The constructor for this phase is now located in the default ThermoFactory
 * for %Cantera. The following code snippet may be used to initialize the phase
 * using the default construction technique within %Cantera.
 *
 * @code
 *      ThermoPhase *HMW = newPhase("HMW_NaCl.xml", "NaCl_electrolyte");
 * @endcode
 *
 * A new HMWSoln object may be created by the following code snippets:
 *
 * @code
 *      HMWSoln *HMW = new HMWSoln("HMW_NaCl.xml", "NaCl_electrolyte");
 * @endcode
 *
 * or
 *
 * @code
 *    XML_Node *xm = get_XML_NameID("phase", "HMW_NaCl.xml#NaCl_electrolyte", 0);
 *    HMWSoln *dh = new HMWSoln(*xm);
 * @endcode
 *
 * or by the following call to importPhase():
 *
 * @code
 *    XML_Node *xm = get_XML_NameID("phase", "HMW_NaCl.xml#NaCl_electrolyte", 0);
 *    HMWSoln dhphase;
 *    importPhase(*xm, &dhphase);
 * @endcode
 *
 * ## XML Example
 *
 * The phase model name for this is called StoichSubstance. It must be supplied
 * as the model attribute of the thermo XML element entry. Within the phase XML
 * block, the density of the phase must be specified. An example of an XML file
 * this phase is given below.
 *
 * @code
 * <phase id="NaCl_electrolyte" dim="3">
 *   <speciesArray datasrc="#species_waterSolution">
 *              H2O(L) Na+ Cl- H+ OH-
 *   </speciesArray>
 *   <state>
 *     <temperature units="K"> 300  </temperature>
 *     <pressure units="Pa">101325.0</pressure>
 *     <soluteMolalities>
 *            Na+:3.0
 *            Cl-:3.0
 *            H+:1.0499E-8
 *            OH-:1.3765E-6
 *     </soluteMolalities>
 *   </state>
 *   <!-- thermo model identifies the inherited class
 *        from ThermoPhase that will handle the thermodynamics.
 *     -->
 *   <thermo model="HMW">
 *      <standardConc model="solvent_volume" />
 *    <activityCoefficients model="Pitzer" TempModel="complex1">
 *               <!-- Pitzer Coefficients
 *                    These coefficients are from Pitzer's main
 *                    paper, in his book.
 *                 -->
 *               <A_Debye model="water" />
 *               <ionicRadius default="3.042843"  units="Angstroms">
 *               </ionicRadius>
 *               <binarySaltParameters cation="Na+" anion="Cl-">
 *                 <beta0> 0.0765, 0.008946, -3.3158E-6,
 *                         -777.03, -4.4706
 *                 </beta0>
 *                 <beta1> 0.2664, 6.1608E-5, 1.0715E-6 </beta1>
 *                 <beta2> 0.0    </beta2>
 *                 <Cphi> 0.00127, -4.655E-5, 0.0,
 *                        33.317, 0.09421
 *                 </Cphi>
 *                 <Alpha1> 2.0 </Alpha1>
 *               </binarySaltParameters>
 *
 *               <binarySaltParameters cation="H+" anion="Cl-">
 *                 <beta0> 0.1775, 0.0, 0.0, 0.0, 0.0</beta0>
 *                 <beta1> 0.2945, 0.0, 0.0 </beta1>
 *                 <beta2> 0.0    </beta2>
 *                 <Cphi> 0.0008, 0.0, 0.0, 0.0, 0.0 </Cphi>
 *                 <Alpha1> 2.0 </Alpha1>
 *               </binarySaltParameters>
 *
 *               <binarySaltParameters cation="Na+" anion="OH-">
 *                 <beta0> 0.0864, 0.0, 0.0, 0.0, 0.0 </beta0>
 *                 <beta1> 0.253, 0.0, 0.0 </beta1>
 *                 <beta2> 0.0    </beta2>
 *                 <Cphi> 0.0044, 0.0, 0.0, 0.0, 0.0 </Cphi>
 *                 <Alpha1> 2.0 </Alpha1>
 *               </binarySaltParameters>
 *
 *               <thetaAnion anion1="Cl-" anion2="OH-">
 *                 <Theta> -0.05 </Theta>
 *               </thetaAnion>
 *
 *               <psiCommonCation cation="Na+" anion1="Cl-" anion2="OH-">
 *                 <Theta> -0.05 </Theta>
 *                 <Psi> -0.006 </Psi>
 *               </psiCommonCation>
 *
 *               <thetaCation cation1="Na+" cation2="H+">
 *                 <Theta> 0.036 </Theta>
 *               </thetaCation>
 *
 *               <psiCommonAnion anion="Cl-" cation1="Na+" cation2="H+">
 *                 <Theta> 0.036 </Theta>
 *                 <Psi> -0.004 </Psi>
 *               </psiCommonAnion>
 *
 *      </activityCoefficients>
 *
 *      <solvent> H2O(L) </solvent>
 *   </thermo>
 *   <elementArray datasrc="elements.xml"> O H Na Cl </elementArray>
 *   <kinetics model="none" >
 *   </kinetics>
 * </phase>
 * @endcode
 * @ingroup thermoprops
 */
class HMWSoln : public MolalityVPSSTP
{
public:
    //! Default Constructor
    HMWSoln();

    //! Construct and initialize an HMWSoln ThermoPhase object
    //! directly from an ASCII input file
    /*!
     *  This constructor is a shell that calls the routine initThermo(), with
     *  a reference to the XML database to get the info for the phase.
     *
     * @param inputFile Name of the input file containing the phase XML data
     *                  to set up the object
     * @param id        ID of the phase in the input file. Defaults to the
     *                  empty string.
     */
    HMWSoln(const std::string& inputFile, const std::string& id = "");

    //! Construct and initialize an HMWSoln ThermoPhase object
    //! directly from an XML database
    /*!
     *  @param phaseRef XML phase node containing the description of the phase
     *  @param id     id attribute containing the name of the phase.
     *                (default is the empty string)
     */
    HMWSoln(XML_Node& phaseRef, const std::string& id = "");

    HMWSoln(const HMWSoln& right);
    HMWSoln& operator=(const HMWSoln& right);
    virtual ~HMWSoln();
    virtual ThermoPhase* duplMyselfAsThermoPhase() const;

    //! Import, construct, and initialize a HMWSoln phase
    //! specification from an XML tree into the current object.
    /*
     * This routine is a precursor to constructPhaseXML(XML_Node*)
     * routine, which does most of the work.
     *
     * @param inputfile XML file containing the description of the phase
     * @param id  Optional parameter identifying the name of the
     *            phase. If none is given, the first XML
     *            phase element will be used.
     * @deprecated Use initThermoFile() instead. To be removed after Cantera 2.3.
     */
    void constructPhaseFile(std::string inputFile, std::string id);

    //! Import and initialize a HMWSoln phase specification in an XML tree
    //! into the current object.
    /*!
     * Here we read an XML description of the phase. We import descriptions of
     * the elements that make up the species in a phase. We import information
     * about the species, including their reference state thermodynamic
     * polynomials. We then freeze the state of the species.
     *
     * Then, we read the species molar volumes from the XML tree to finish the
     * initialization.
     *
     * @param phaseNode This object must be the phase node of a complete XML
     *             tree description of the phase, including all of the species
     *             data. In other words while "phase" must point to an XML phase
     *             object, it must have sibling nodes "speciesData" that
     *             describe the species in the phase.
     * @param id   ID of the phase. If nonnull, a check is done to see if
     *             phaseNode is pointing to the phase with the correct id.
     * @deprecated Use importPhase() instead. To be removed after Cantera 2.3.
     */
    void constructPhaseXML(XML_Node& phaseNode, std::string id);

    //! @name  Utilities
    //! @{

    virtual int eosType() const;
    virtual std::string type() const {
        return "HMWSoln";
    }

    //! @}
    //! @name  Molar Thermodynamic Properties of the Solution
    //! @{

    /// Molar enthalpy. Units: J/kmol.
    /**
     * Molar enthalpy of the solution. Units: J/kmol.
     *      (HKM -> Bump up to Parent object)
     */
    virtual doublereal enthalpy_mole() const;

    /**
     * Excess molar enthalpy of the solution from
     * the mixing process. Units: J/ kmol.
     *
     * Note this is kmol of the total solution.
     */
    virtual doublereal relative_enthalpy() const;

    /**
     * Excess molar enthalpy of the solution from
     * the mixing process on a molality basis.
     *  Units: J/ (kmol add salt).
     *
     * Note this is kmol of the guessed at salt composition
     */
    virtual doublereal relative_molal_enthalpy() const;

    /// Molar entropy. Units: J/kmol/K.
    /**
     * Molar entropy of the solution. Units: J/kmol/K. For an ideal, constant
     * partial molar volume solution mixture with pure species phases which
     * exhibit zero volume expansivity:
     * \f[
     * \hat s(T, P, X_k) = \sum_k X_k \hat s^0_k(T)
     *      - \hat R  \sum_k X_k log(X_k)
     * \f]
     * The reference-state pure-species entropies \f$ \hat s^0_k(T,p_{ref}) \f$
     * are computed by the species thermodynamic property manager. The pure
     * species entropies are independent of temperature since the volume
     * expansivities are equal to zero.
     * @see MultiSpeciesThermo
     *
     *      (HKM -> Bump up to Parent object)
     */
    virtual doublereal entropy_mole() const;

    /// Molar Gibbs function. Units: J/kmol.
    /*!
     *      (HKM -> Bump up to Parent object)
     */
    virtual doublereal gibbs_mole() const;

    virtual doublereal cp_mole() const;

    /// Molar heat capacity at constant volume. Units: J/kmol/K.
    /*!
     *      (HKM -> Bump up to Parent object)
     */
    virtual doublereal cv_mole() const;

    //!@}
    //! @name Mechanical Equation of State Properties
    /*!
     * In this equation of state implementation, the density is a function
     * only of the mole fractions. Therefore, it can't be an independent
     * variable. Instead, the pressure is used as the independent variable.
     * Functions which try to set the thermodynamic state by calling
     * setDensity() may cause an exception to be thrown.
     */
    //!@{

protected:
    /**
     * Calculate the density of the mixture using the partial
     * molar volumes and mole fractions as input
     *
     * The formula for this is
     *
     * \f[
     * \rho = \frac{\sum_k{X_k W_k}}{\sum_k{X_k V_k}}
     * \f]
     *
     * where \f$X_k\f$ are the mole fractions, \f$W_k\f$ are the molecular
     * weights, and \f$V_k\f$ are the pure species molar volumes.
     *
     * Note, the basis behind this formula is that in an ideal solution the
     * partial molar volumes are equal to the pure species molar volumes. We
     * have additionally specified in this class that the pure species molar
     * volumes are independent of temperature and pressure.
     *
     * NOTE: This is a non-virtual function, which is not a member of the
     *       ThermoPhase base class.
     */
    void calcDensity();

public:
    //! Set the internally stored density (kg/m^3) of the phase.
    /*!
     * Overridden setDensity() function is necessary because the density is not
     * an independent variable.
     *
     * This function will now throw an error condition.
     *
     * Note, in general, setting the phase density is now a nonlinear
     * calculation. P and T are the fundamental variables. This routine should
     * be revamped to do the nonlinear problem.
     *
     * @todo May have to adjust the strategy here to make the eos for these
     *     materials slightly compressible, in order to create a condition where
     *     the density is a function of the pressure.
     * @todo Now have a compressible ss equation for liquid water. Therefore,
     *       this phase is compressible. May still want to change the
     *       independent variable however.
     *
     * @param rho Input density (kg/m^3).
     */
    virtual void setDensity(const doublereal rho);

    //! Set the internally stored molar density (kmol/m^3) for the phase.
    /**
     * Overridden setMolarDensity() function is necessary because of the
     * underlying water model.
     *
     * This function will now throw an error condition if the input isn't
     * exactly equal to the current molar density.
     *
     * @param conc   Input molar density (kmol/m^3).
     */
    virtual void setMolarDensity(const doublereal conc);

    /**
     * @}
     * @name Activities, Standard States, and Activity Concentrations
     *
     * The activity \f$a_k\f$ of a species in solution is related to the
     * chemical potential by \f[ \mu_k = \mu_k^0(T) + \hat R T \log a_k. \f] The
     * quantity \f$\mu_k^0(T,P)\f$ is the chemical potential at unit activity,
     * which depends only on temperature and the pressure. Activity is assumed
     * to be molality-based here.
     * @{
     */

    //! This method returns an array of generalized activity concentrations
    /*!
     * The generalized activity concentrations, \f$ C_k^a\f$, are defined such
     * that \f$ a_k = C^a_k / C^0_k, \f$ where \f$ C^0_k \f$ is a standard
     * concentration defined below.  These generalized concentrations are used
     * by kinetics manager classes to compute the forward and reverse rates of
     * elementary reactions.
     *
     * The generalized activity concentration of a solute species has the
     * following form
     *
     *  \f[
     *      C_j^a = C^o_o \frac{\gamma_j^\triangle n_j}{n_o}
     *  \f]
     *
     * The generalized activity concentration of the solvent has the same units,
     * but it's a simpler form
     *
     *  \f[
     *      C_o^a = C^o_o a_o
     *  \f]
     *
     * @param c Array of generalized concentrations. The
     *          units are kmol m-3 for both the solvent and the solute species
     */
    virtual void getActivityConcentrations(doublereal* c) const;

    //! Return the standard concentration for the kth species
    /*!
     * The standard concentration \f$ C^0_k \f$ used to normalize the activity
     * (i.e., generalized) concentration for use
     *
     * We have set the standard concentration for all solute species in this
     * phase equal to the default concentration of the solvent at the system
     * temperature and pressure multiplied by Mnaught (kg solvent / gmol
     * solvent). The solvent standard concentration is just equal to its
     * standard state concentration.
     *
     *   \f[
     *      C_j^0 = C^o_o \tilde{M}_o \quad and  C_o^0 = C^o_o
     *   \f]
     *
     * The consequence of this is that the standard concentrations have unequal
     * units between the solvent and the solute. However, both the solvent and
     * the solute activity concentrations will have the same units of kmol/kg^3.
     *
     * This means that the kinetics operator essentially works on an generalized
     * concentration basis (kmol / m3), with units for the kinetic rate constant
     * specified as if all reactants (solvent or solute) are on a concentration
     * basis (kmol /m3). The concentration will be modified by the activity
     * coefficients.
     *
     * For example, a bulk-phase binary reaction between liquid solute species
     * *j* and *k*, producing a new liquid solute species *l* would have the
     * following equation for its rate of progress variable, \f$ R^1 \f$, which
     * has units of kmol m-3 s-1.
     *
     * \f[
     *    R^1 = k^1 C_j^a C_k^a =  k^1 (C^o_o \tilde{M}_o a_j) (C^o_o \tilde{M}_o a_k)
     * \f]
     *
     * where
     *
     * \f[
     *      C_j^a = C^o_o \tilde{M}_o a_j \quad and \quad C_k^a = C^o_o \tilde{M}_o a_k
     * \f]
     *
     * \f$ C_j^a \f$ is the activity concentration of species *j*, and
     * \f$ C_k^a \f$ is the activity concentration of species *k*. \f$ C^o_o \f$
     * is the concentration of water at 298 K and 1 atm. \f$ \tilde{M}_o \f$ has
     * units of kg solvent per gmol solvent and is equal to
     *
     * \f[
     *     \tilde{M}_o = \frac{M_o}{1000}
     * \f]
     *
     * \f$ a_j \f$ is
     *  the activity of species *j* at the current temperature and pressure
     *  and concentration of the liquid phase is given by the molality based
     *  activity coefficient multiplied by the molality of the jth species.
     *
     * \f[
     *      a_j  =  \gamma_j^\triangle m_j = \gamma_j^\triangle \frac{n_j}{\tilde{M}_o n_o}
     * \f]
     *
     * \f$k^1 \f$ has units of m^3/kmol/s.
     *
     * Therefore the generalized activity concentration of a solute species has
     * the following form
     *
     * \f[
     *     C_j^a = C^o_o \frac{\gamma_j^\triangle n_j}{n_o}
     * \f]
     *
     * The generalized activity concentration of the solvent has the same units,
     * but it's a simpler form
     *
     * \f[
     *     C_o^a = C^o_o a_o
     * \f]
     *
     * @param k Optional parameter indicating the species. The default is to
     *         assume this refers to species 0.
     * @returns the standard Concentration in units of m^3/kmol.
     *
     * @param k Species index
     */
    virtual doublereal standardConcentration(size_t k=0) const;

    //! Get the array of non-dimensional activities at the current solution
    //! temperature, pressure, and solution concentration.
    /*!
     *
     * We resolve this function at this level by calling on the
     * activityConcentration function. However, derived classes may want to
     * override this default implementation.
     *
     * (note solvent is on molar scale).
     *
     * @param ac  Output vector of activities. Length: m_kk.
     */
    virtual void getActivities(doublereal* ac) const;

    //! @}
    //! @name  Partial Molar Properties of the Solution
    //! @{

    //! Get the species chemical potentials. Units: J/kmol.
    /*!
     *
     * This function returns a vector of chemical potentials of the
     * species in solution.
     *
     * \f[
     *    \mu_k = \mu^{\triangle}_k(T,P) + R T ln(\gamma_k^{\triangle} m_k)
     * \f]
     *
     * @param mu  Output vector of species chemical
     *            potentials. Length: m_kk. Units: J/kmol
     */
    virtual void getChemPotentials(doublereal* mu) const;

    //! Returns an array of partial molar enthalpies for the species
    //! in the mixture. Units (J/kmol)
    /*!
     * For this phase, the partial molar enthalpies are equal to the standard
     * state enthalpies modified by the derivative of the molality-based
     * activity coefficient wrt temperature
     *
     *  \f[
     * \bar h_k(T,P) = h^{\triangle}_k(T,P)
     *               - R T^2 \frac{d \ln(\gamma_k^\triangle)}{dT}
     * \f]
     * The solvent partial molar enthalpy is equal to
     *  \f[
     * \bar h_o(T,P) = h^{o}_o(T,P) - R T^2 \frac{d \ln(a_o)}{dT}
     *       = h^{o}_o(T,P)
     *       + R T^2 (\sum_{k \neq o} m_k)  \tilde{M_o} (\frac{d \phi}{dT})
     * \f]
     *
     * @param hbar    Output vector of species partial molar enthalpies.
     *                Length: m_kk. units are J/kmol.
     */
    virtual void getPartialMolarEnthalpies(doublereal* hbar) const;

    //! Returns an array of partial molar entropies of the species in the
    //! solution. Units: J/kmol/K.
    /*!
     * Maxwell's equations provide an answer for how calculate this
     *   (p.215 Smith and Van Ness)
     *
     *      d(chemPot_i)/dT = -sbar_i
     *
     * For this phase, the partial molar entropies are equal to the SS species
     * entropies plus the ideal solution contribution plus complicated functions
     * of the temperature derivative of the activity coefficients.
     *
     *  \f[
     *     \bar s_k(T,P) =  s^{\triangle}_k(T,P)
     *             - R \ln( \gamma^{\triangle}_k \frac{m_k}{m^{\triangle}}))
     *                    - R T \frac{d \ln(\gamma^{\triangle}_k) }{dT}
     * \f]
     * \f[
     *      \bar s_o(T,P) = s^o_o(T,P) - R \ln(a_o)
     *                    - R T \frac{d \ln(a_o)}{dT}
     * \f]
     *
     *  @param sbar    Output vector of species partial molar entropies.
     *                 Length = m_kk. units are J/kmol/K.
     */
    virtual void getPartialMolarEntropies(doublereal* sbar) const;

    //! Return an array of partial molar volumes for the species in the mixture.
    //! Units: m^3/kmol.
    /*!
     * For this solution, the partial molar volumes are functions of the
     * pressure derivatives of the activity coefficients.
     *
     * \f[
     *     \bar V_k(T,P)  = V^{\triangle}_k(T,P)
     *                    + R T \frac{d \ln(\gamma^{\triangle}_k) }{dP}
     * \f]
     * \f[
     *      \bar V_o(T,P) = V^o_o(T,P)
     *                    + R T \frac{d \ln(a_o)}{dP}
     * \f]
     *
     * @param vbar   Output vector of species partial molar volumes.
     *               Length = m_kk. units are m^3/kmol.
     */
    virtual void getPartialMolarVolumes(doublereal* vbar) const;

    //! Return an array of partial molar heat capacities for the species in the
    //! mixture.  Units: J/kmol/K
    /*!
     * The following formulas are implemented within the code.
     *
     * \f[
     *     \bar C_{p,k}(T,P) =  C^{\triangle}_{p,k}(T,P)
     *             - 2 R T \frac{d \ln( \gamma^{\triangle}_k)}{dT}
     *                    - R T^2 \frac{d^2 \ln(\gamma^{\triangle}_k) }{{dT}^2}
     * \f]
     * \f[
     *      \bar C_{p,o}(T,P) = C^o_{p,o}(T,P)
     *                   - 2 R T \frac{d \ln(a_o)}{dT}
     *                    - R T^2 \frac{d^2 \ln(a_o)}{{dT}^2}
     * \f]
     *
     * @param cpbar   Output vector of species partial molar heat capacities at
     *                constant pressure. Length = m_kk. units are J/kmol/K.
     */
    virtual void getPartialMolarCp(doublereal* cpbar) const;

public:
    //@}

    //! Get the saturation pressure for a given temperature.
    /*!
     * Note the limitations of this function. Stability considerations
     * concerning multiphase equilibrium are ignored in this calculation.
     * Therefore, the call is made directly to the SS of water underneath. The
     * object is put back into its original state at the end of the call.
     *
     * @todo This is probably not implemented correctly. The stability of the
     *       salt should be added into this calculation. The underlying water
     *       model may be called to get the stability of the pure water
     *       solution, if needed.
     *
     * @param T  Temperature (kelvin)
     */
    virtual doublereal satPressure(doublereal T);

    /*
     *  -------------- Utilities -------------------------------
     */

    virtual void initThermo();

    //! Initialize the phase parameters from an XML file.
    /*!
     * This gets called from importPhase(). It processes the XML file after the
     * species are set up. This is the main routine for reading in activity
     * coefficient parameters.
     *
     * @param phaseNode This object must be the phase node of a complete XML
     *             tree description of the phase, including all of the species
     *             data. In other words while "phase" must point to an XML phase
     *             object, it must have sibling nodes "speciesData" that
     *             describe the species in the phase.
     * @param id   ID of the phase. If nonnull, a check is done to see if
     *             phaseNode is pointing to the phase with the correct id.
     */
    virtual void initThermoXML(XML_Node& phaseNode, const std::string& id);

    //! Value of the Debye Huckel constant as a function of temperature
    //! and pressure.
    /*!
     *            A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
     *
     *            Units = sqrt(kg/gmol)
     *
     * @param temperature  Temperature of the derivative calculation
     *                     or -1 to indicate the current temperature
     * @param pressure    Pressure of the derivative calculation
     *                    or -1 to indicate the current pressure
     */
    virtual double A_Debye_TP(double temperature = -1.0,
                              double pressure = -1.0) const;

    //! Value of the derivative of the Debye Huckel constant with respect to
    //! temperature as a function of temperature and pressure.
    /*!
     *            A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
     *
     *            Units = sqrt(kg/gmol)
     *
     * @param temperature  Temperature of the derivative calculation
     *                     or -1 to indicate the current temperature
     * @param pressure    Pressure of the derivative calculation
     *                    or -1 to indicate the current pressure
     */
    virtual double dA_DebyedT_TP(double temperature = -1.0,
                                 double pressure = -1.0) const;

    /**
     * Value of the derivative of the Debye Huckel constant with respect to
     * pressure, as a function of temperature and pressure.
     *
     *      A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
     *
     *  Units = sqrt(kg/gmol)
     *
     * @param temperature  Temperature of the derivative calculation
     *                     or -1 to indicate the current temperature
     * @param pressure    Pressure of the derivative calculation
     *                    or -1 to indicate the current pressure
     */
    virtual double dA_DebyedP_TP(double temperature = -1.0,
                                 double pressure = -1.0) const;

    /**
     * Return Pitzer's definition of A_L. This is basically the
     * derivative of the A_phi multiplied by 4 R T**2
     *
     *            A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
     *            dA_phidT = d(A_Debye)/dT / 3.0
     *            A_L = dA_phidT * (4 * R * T * T)
     *
     *            Units = sqrt(kg/gmol) (RT)
     *
     * @param temperature  Temperature of the derivative calculation
     *                     or -1 to indicate the current temperature
     * @param pressure    Pressure of the derivative calculation
     *                    or -1 to indicate the current pressure
     */
    double ADebye_L(double temperature = -1.0,
                    double pressure = -1.0) const;

    /**
     * Return Pitzer's definition of A_J. This is basically the temperature
     * derivative of A_L, and the second derivative of A_phi
     *
     *            A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
     *            dA_phidT = d(A_Debye)/dT / 3.0
     *            A_J = 2 A_L/T + 4 * R * T * T * d2(A_phi)/dT2
     *
     *            Units = sqrt(kg/gmol) (R)
     *
     * @param temperature  Temperature of the derivative calculation
     *                     or -1 to indicate the current temperature
     * @param pressure    Pressure of the derivative calculation
     *                    or -1 to indicate the current pressure
     */
    double ADebye_J(double temperature = -1.0,
                    double pressure = -1.0) const;

    /**
     * Return Pitzer's definition of A_V. This is the derivative wrt pressure of
     * A_phi multiplied by - 4 R T
     *
     *            A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
     *            dA_phidT = d(A_Debye)/dP / 3.0
     *            A_V = - dA_phidP * (4 * R * T)
     *
     *            Units = sqrt(kg/gmol) (RT) / Pascal
     *
     * @param temperature  Temperature of the derivative calculation
     *                     or -1 to indicate the current temperature
     * @param pressure    Pressure of the derivative calculation
     *                    or -1 to indicate the current pressure
     */
    double ADebye_V(double temperature = -1.0,
                    double pressure = -1.0) const;

    //! Value of the 2nd derivative of the Debye Huckel constant with respect to
    //! temperature as a function of temperature and pressure.
    /*!
     *            A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
     *
     *            Units = sqrt(kg/gmol)
     *
     * @param temperature  Temperature of the derivative calculation
     *                     or -1 to indicate the current temperature
     * @param pressure    Pressure of the derivative calculation
     *                    or -1 to indicate the current pressure
     */
    virtual double d2A_DebyedT2_TP(double temperature = -1.0,
                                   double pressure = -1.0) const;

    //! Reports the ionic radius of the kth species
    /*!
     *  @param k Species index
     */
    double AionicRadius(int k = 0) const;

    /**
     * formPitzer():
     *
     *  Returns the form of the Pitzer parameterization used
     */
    int formPitzer() const {
        return m_formPitzer;
    }

    //! Print out all of the input Pitzer coefficients.
    void printCoeffs() const;

    //!  Get the array of unscaled non-dimensional molality based
    //!  activity coefficients at the current solution temperature,
    //!  pressure, and solution concentration.
    /*!
     *  See Denbigh p. 278 for a thorough discussion. This class must be
     *  overridden in classes which derive from MolalityVPSSTP. This function
     *  takes over from the molar-based activity coefficient calculation,
     *  getActivityCoefficients(), in derived classes.
     *
     * @param acMolality Output vector containing the molality based activity coefficients.
     *                   length: m_kk.
     */
    void getUnscaledMolalityActivityCoefficients(doublereal* acMolality) const;

private:
    //! Apply the current phScale to a set of activity Coefficients
    /*!
     *  See the Eq3/6 Manual for a thorough discussion.
     */
    void s_updateScaling_pHScaling() const;

    //! Apply the current phScale to a set of derivatives of the activity
    //! Coefficients wrt temperature
    /*!
     *  See the Eq3/6 Manual for a thorough discussion of the need
     */
    void s_updateScaling_pHScaling_dT() const;

    //! Apply the current phScale to a set of 2nd derivatives of the activity
    //! Coefficients wrt temperature
    /*!
     *  See the Eq3/6 Manual for a thorough discussion of the need
     */
    void s_updateScaling_pHScaling_dT2() const;

    //! Apply the current phScale to a set of derivatives of the activity
    //! Coefficients wrt pressure
    /*!
     *  See the Eq3/6 Manual for a thorough discussion of the need
     */
    void s_updateScaling_pHScaling_dP() const;

    //! Calculate the Chlorine activity coefficient on the NBS scale
    /*!
     *  We assume here that the m_IionicMolality variable is up to date.
     */
    doublereal s_NBS_CLM_lnMolalityActCoeff() const;

    //! Calculate the temperature derivative of the Chlorine activity
    //! coefficient on the NBS scale
    /*!
     *  We assume here that the m_IionicMolality variable is up to date.
     */
    doublereal s_NBS_CLM_dlnMolalityActCoeff_dT() const;

    //! Calculate the second temperature derivative of the Chlorine activity
    //! coefficient on the NBS scale
    /*!
     *  We assume here that the m_IionicMolality variable is up to date.
     */
    doublereal s_NBS_CLM_d2lnMolalityActCoeff_dT2() const;

    //! Calculate the pressure derivative of the Chlorine activity coefficient
    /*!
     *  We assume here that the m_IionicMolality variable is up to date.
     */
    doublereal s_NBS_CLM_dlnMolalityActCoeff_dP() const;

    //@}

private:
    /**
     * This is the form of the Pitzer parameterization used in this model. The
     * options are described at the top of this document, and in the general
     * documentation. The list is repeated here:
     *
     * PITZERFORM_BASE  = 0    (only one supported atm)
     */
    int m_formPitzer;

    /**
     * This is the form of the temperature dependence of Pitzer parameterization
     * used in the model.
     *
     *       PITZER_TEMP_CONSTANT   0
     *       PITZER_TEMP_LINEAR     1
     *       PITZER_TEMP_COMPLEX1   2
     */
    int m_formPitzerTemp;

    /**
     * Format for the generalized concentration:
     *
     *  0 = unity
     *  1 = molar_volume
     *  2 = solvent_volume    (default)
     *
     * The generalized concentrations can have three different forms
     * depending on the value of the member attribute m_formGC, which
     * is supplied in the constructor.
     *
     * | m_formGC | GeneralizedConc | StandardConc |
     * | -------- | --------------- | ------------ |
     * | 0        | X_k             | 1.0          |
     * | 1        | X_k / V_k       | 1.0 / V_k    |
     * | 2        | X_k / V_N       | 1.0 / V_N    |
     *
     * The value and form of the generalized concentration will affect reaction
     * rate constants involving species in this phase.
     *
     * (HKM Note: Using option #1 may lead to spurious results and has been
     *  included only with warnings. The reason is that it molar volumes of
     *  electrolytes may often be negative. The molar volume of H+ is defined to
     *  be zero too. Either options 0 or 2 are the appropriate choice. Option 0
     *  leads to bulk reaction rate constants which have units of s-1. Option 2
     *  leads to bulk reaction rate constants for bimolecular rxns which have
     *  units of m-3 kmol-1 s-1.)
     */
    int m_formGC;

    //! Vector containing the electrolyte species type
    /*!
     * The possible types are:
     *  - solvent
     *  - Charged Species
     *  - weakAcidAssociated
     *  - strongAcidAssociated
     *  - polarNeutral
     *  - nonpolarNeutral
     */
    vector_int m_electrolyteSpeciesType;

    //! a_k = Size of the ionic species in the DH formulation. units = meters
    vector_fp m_Aionic;

    //! Current value of the ionic strength on the molality scale Associated
    //! Salts, if present in the mechanism, don't contribute to the value of the
    //! ionic strength in this version of the Ionic strength.
    mutable double m_IionicMolality;

    //! Maximum value of the ionic strength allowed in the calculation of the
    //! activity coefficients.
    double m_maxIionicStrength;

    //! Reference Temperature for the Pitzer formulations.
    double m_TempPitzerRef;

    //! Stoichiometric ionic strength on the molality scale. This differs from
    //! m_IionicMolality in the sense that associated salts are treated as
    //! unassociated salts, when calculating the Ionic strength by this method.
    mutable double m_IionicMolalityStoich;

public:
    /**
     * Form of the constant outside the Debye-Huckel term called A. It's
     * normally a function of temperature and pressure. However, it can be set
     * from the input file in order to aid in numerical comparisons. Acceptable
     * forms:
     *
     *       A_DEBYE_CONST  0
     *       A_DEBYE_WATER  1
     *
     * The A_DEBYE_WATER form may be used for water solvents with needs to cover
     * varying temperatures and pressures. Note, the dielectric constant of
     * water is a relatively strong function of T, and its variability must be
     * accounted for,
     */
    int m_form_A_Debye;

private:
    /**
     * A_Debye: this expression appears on the top of the ln actCoeff term in
     * the general Debye-Huckel expression It depends on temperature.
     * And, therefore, most be recalculated whenever T or P changes.
     * This variable is a local copy of the calculation.
     *
     *  A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
     *
     *       where B_Debye = F / sqrt(epsilon R T/2)
     *                       (dw/1000)^(1/2)
     *
     *  A_Debye = (1/ (8 Pi)) (2 Na * dw/1000)^(1/2)
     *             (e * e / (epsilon * kb * T))^(3/2)
     *
     *  Units = sqrt(kg/gmol)
     *
     *  Nominal value = 1.172576 sqrt(kg/gmol)
     *        based on:
     *          epsilon/epsilon_0 = 78.54
     *                 (water at 25C)
     *          epsilon_0 = 8.854187817E-12 C2 N-1 m-2
     *          e = 1.60217653 E-19 C
     *          F = 9.6485309E7 C kmol-1
     *          R = 8.314472E3 kg m2 s-2 kmol-1 K-1
     *          T = 298.15 K
     *          B_Debye = 3.28640E9 sqrt(kg/gmol)/m
     *          dw = C_0 * M_0 (density of water) (kg/m3)
     *             = 1.0E3 at 25C
     */
    mutable double m_A_Debye;

    //! Water standard state calculator
    /*!
     *  derived from the equation of state for water.
     */
    PDSS* m_waterSS;

    //! density of standard-state water
    /*!
     * internal temporary variable
     */
    double m_densWaterSS;

    //! Pointer to the water property calculator
    std::unique_ptr<WaterProps> m_waterProps;

    //! vector of size m_kk, used as a temporary holding area.
    mutable vector_fp m_tmpV;

    /**
     * Stoichiometric species charge -> This is for calculations of the ionic
     * strength which ignore ion-ion pairing into neutral molecules. The
     * Stoichiometric species charge is the charge of one of the ion that would
     * occur if the species broke into two charged ion pairs.
     *
     *  NaCl ->   m_speciesCharge_Stoich = -1;
     *  HSO4- -> H+ + SO42-              = -2
     *      -> The other charge is calculated.
     *
     * For species that aren't ion pairs, its equal to the m_speciesCharge[]
     * value.
     */
    vector_fp m_speciesCharge_Stoich;

    /**
     *  Array of 2D data used in the Pitzer/HMW formulation. Beta0_ij[i][j] is
     *  the value of the Beta0 coefficient for the ij salt. It will be nonzero
     *  iff i and j are both charged and have opposite sign. The array is also
     *  symmetric. counterIJ where counterIJ = m_counterIJ[i][j] is used to
     *  access this array.
     */
    mutable vector_fp m_Beta0MX_ij;

    //! Derivative of Beta0_ij[i][j] wrt T. Vector index is counterIJ
    mutable vector_fp m_Beta0MX_ij_L;

    //! Derivative of Beta0_ij[i][j] wrt TT. Vector index is counterIJ
    mutable vector_fp m_Beta0MX_ij_LL;

    //! Derivative of Beta0_ij[i][j] wrt P. Vector index is counterIJ
    mutable vector_fp m_Beta0MX_ij_P;

    //! Array of coefficients for Beta0, a variable in Pitzer's papers
    /*!
     * Column index is counterIJ. m_Beta0MX_ij_coeff.ptrColumn(counterIJ) is a
     * double* containing the vector of coefficients for the counterIJ
     * interaction.
     */
    mutable Array2D m_Beta0MX_ij_coeff;

    //! Array of 2D data used in the Pitzer/HMW formulation. Beta1_ij[i][j] is
    //! the value of the Beta1 coefficient for the ij salt. It will be nonzero
    //! iff i and j are both charged and have opposite sign. The array is also
    //! symmetric. counterIJ where counterIJ = m_counterIJ[i][j] is used to
    //! access this array.
    mutable vector_fp m_Beta1MX_ij;

    //! Derivative of Beta1_ij[i][j] wrt T. Vector index is counterIJ
    mutable vector_fp m_Beta1MX_ij_L;

    //! Derivative of Beta1_ij[i][j] wrt TT. Vector index is counterIJ
    mutable vector_fp m_Beta1MX_ij_LL;

    //! Derivative of Beta1_ij[i][j] wrt P. Vector index is counterIJ
    mutable vector_fp m_Beta1MX_ij_P;

    //! Array of coefficients for Beta1, a variable in Pitzer's papers
    /*!
     * Column index is counterIJ. m_Beta1MX_ij_coeff.ptrColumn(counterIJ) is a
     * double* containing the vector of coefficients for the counterIJ
     * interaction.
     */
    mutable Array2D m_Beta1MX_ij_coeff;

    //! Array of 2D data used in the Pitzer/HMW formulation. Beta2_ij[i][j] is
    //! the value of the Beta2 coefficient for the ij salt. It will be nonzero
    //! iff i and j are both charged and have opposite sign, and i and j both
    //! have charges of 2 or more. The array is also symmetric. counterIJ where
    //! counterIJ = m_counterIJ[i][j] is used to access this array.
    mutable vector_fp m_Beta2MX_ij;

    //! Derivative of Beta2_ij[i][j] wrt T. Vector index is counterIJ
    mutable vector_fp m_Beta2MX_ij_L;

    //! Derivative of Beta2_ij[i][j] wrt TT. Vector index is counterIJ
    mutable vector_fp m_Beta2MX_ij_LL;

    //! Derivative of Beta2_ij[i][j] wrt P. Vector index is counterIJ
    mutable vector_fp m_Beta2MX_ij_P;

    //! Array of coefficients for Beta2, a variable in Pitzer's papers
    /*!
     * column index is counterIJ. m_Beta2MX_ij_coeff.ptrColumn(counterIJ) is a
     *  double* containing the vector of coefficients for the counterIJ
     *  interaction. This was added for the YMP database version of the code
     *  since it contains temperature-dependent parameters for some 2-2
     *  electrolytes.
     */
    mutable Array2D m_Beta2MX_ij_coeff;

    // Array of 2D data used in the Pitzer/HMW formulation. Alpha1MX_ij[i][j] is
    // the value of the alpha1 coefficient for the ij interaction. It will be
    // nonzero iff i and j are both charged and have opposite sign. It is
    // symmetric wrt i, j. counterIJ where counterIJ = m_counterIJ[i][j] is used
    // to access this array.
    vector_fp m_Alpha1MX_ij;

    //! Array of 2D data used in the Pitzer/HMW formulation. Alpha2MX_ij[i][j]
    //! is the value of the alpha2 coefficient for the ij interaction. It will
    //! be nonzero iff i and j are both charged and have opposite sign, and i
    //! and j both have charges of 2 or more, usually. It is symmetric wrt i, j.
    //! counterIJ, where counterIJ = m_counterIJ[i][j], is used to access this
    //! array.
    vector_fp m_Alpha2MX_ij;

    //! Array of 2D data used in the Pitzer/HMW formulation. CphiMX_ij[i][j] is
    //! the value of the Cphi coefficient for the ij interaction. It will be
    //! nonzero iff i and j are both charged and have opposite sign, and i and j
    //! both have charges of 2 or more. The array is also symmetric. counterIJ
    //! where counterIJ = m_counterIJ[i][j] is used to access this array.
    mutable vector_fp m_CphiMX_ij;

    //! Derivative of Cphi_ij[i][j] wrt T. Vector index is counterIJ
    mutable vector_fp m_CphiMX_ij_L;

    //! Derivative of Cphi_ij[i][j] wrt TT. Vector index is counterIJ
    mutable vector_fp m_CphiMX_ij_LL;

    //! Derivative of Cphi_ij[i][j] wrt P. Vector index is counterIJ
    mutable vector_fp m_CphiMX_ij_P;

    //! Array of coefficients for CphiMX, a parameter in the activity
    //! coefficient formulation
    /*!
     *  Column index is counterIJ. m_CphiMX_ij_coeff.ptrColumn(counterIJ) is a
     *  double* containing the vector of coefficients for the counterIJ
     *  interaction.
     */
    mutable Array2D m_CphiMX_ij_coeff;

    //! Array of 2D data for Theta_ij[i][j] in the Pitzer/HMW formulation.
    /*!
     *  Array of 2D data used in the Pitzer/HMW formulation. Theta_ij[i][j] is
     *  the value of the theta coefficient for the ij interaction. It will be
     *  nonzero for charged ions with the same sign. It is symmetric. counterIJ
     *  where counterIJ = m_counterIJ[i][j] is used to access this array.
     *
     *  HKM Recent Pitzer papers have used a functional form for Theta_ij, which
     *      depends on the ionic strength.
     */
    mutable vector_fp m_Theta_ij;

    //! Derivative of Theta_ij[i][j] wrt T. Vector index is counterIJ
    mutable vector_fp m_Theta_ij_L;

    //! Derivative of Theta_ij[i][j] wrt TT. Vector index is counterIJ
    mutable vector_fp m_Theta_ij_LL;

    //! Derivative of Theta_ij[i][j] wrt P. Vector index is counterIJ
    mutable vector_fp m_Theta_ij_P;

    //! Array of coefficients for Theta_ij[i][j] in the Pitzer/HMW formulation.
    /*!
     *  Theta_ij[i][j] is the value of the theta coefficient for the ij
     *  interaction. It will be nonzero for charged ions with the same sign. It
     *  is symmetric. Column index is counterIJ. counterIJ where counterIJ =
     *  m_counterIJ[i][j] is used to access this array.
     *
     *  m_Theta_ij_coeff.ptrColumn(counterIJ) is a double* containing
     *  the vector of coefficients for the counterIJ interaction.
     */
    Array2D m_Theta_ij_coeff;

    //! Array of 3D data used in the Pitzer/HMW formulation.
    /*!
     * Psi_ijk[n] is the value of the psi coefficient for the
     * ijk interaction where
     *
     *   n = k + j * m_kk + i * m_kk * m_kk;
     *
     * It is potentially nonzero everywhere. The first two coordinates are
     * symmetric wrt cations, and the last two coordinates are symmetric wrt
     * anions.
     */
    mutable vector_fp m_Psi_ijk;

    //! Derivative of Psi_ijk[n] wrt T. See m_Psi_ijk for reference on the
    //! indexing into this variable.
    mutable vector_fp m_Psi_ijk_L;

    //! Derivative of Psi_ijk[n] wrt TT. See m_Psi_ijk for reference on the
    //! indexing into this variable.
    mutable vector_fp m_Psi_ijk_LL;

    //! Derivative of Psi_ijk[n] wrt P. See m_Psi_ijk for reference on the
    //! indexing into this variable.
    mutable vector_fp m_Psi_ijk_P;

    //! Array of coefficients for Psi_ijk[n] in the Pitzer/HMW formulation.
    /*!
     * Psi_ijk[n] is the value of the psi coefficient for the
     * ijk interaction where
     *
     *   n = k + j * m_kk + i * m_kk * m_kk;
     *
     * It is potentially nonzero everywhere. The first two coordinates are
     * symmetric wrt cations, and the last two coordinates are symmetric wrt
     * anions.
     *
     *  m_Psi_ijk_coeff.ptrColumn(n) is a double* containing the vector of
     *  coefficients for the n interaction.
     */
    Array2D m_Psi_ijk_coeff;

    //! Lambda coefficient for the ij interaction
    /*!
     * Array of 2D data used in the Pitzer/HMW formulation. Lambda_nj[n][j]
     * represents the lambda coefficient for the ij interaction. This is a
     * general interaction representing neutral species. The neutral species
     * occupy the first index, i.e., n. The charged species occupy the j
     * coordinate. neutral, neutral interactions are also included here.
     */
    mutable Array2D m_Lambda_nj;

    //! Derivative of Lambda_nj[i][j] wrt T. see m_Lambda_ij
    mutable Array2D m_Lambda_nj_L;

    //! Derivative of Lambda_nj[i][j] wrt TT
    mutable Array2D m_Lambda_nj_LL;

    //! Derivative of Lambda_nj[i][j] wrt P
    mutable Array2D m_Lambda_nj_P;

    //! Array of coefficients for Lambda_nj[i][j] in the Pitzer/HMW formulation.
    /*!
     *  Array of 2D data used in the Pitzer/HMW formulation. Lambda_ij[i][j]
     *  represents the lambda coefficient for the ij interaction. This is a
     *  general interaction representing neutral species. The neutral species
     *  occupy the first index, i.e., i. The charged species occupy the j
     *  coordinate. Neutral, neutral interactions are also included here.
     *
     *      n = j + m_kk * i
     *
     * m_Lambda_ij_coeff.ptrColumn(n) is a double* containing the vector of
     * coefficients for the (i,j) interaction.
     */
    Array2D m_Lambda_nj_coeff;

    //! Mu coefficient for the self-ternary neutral coefficient
    /*!
     * Array of 2D data used in the Pitzer/HMW formulation. Mu_nnn[i] represents
     * the Mu coefficient for the nnn interaction. This is a general interaction
     * representing neutral species interacting with itself.
     */
    mutable vector_fp m_Mu_nnn;

    //! Mu coefficient temperature derivative for the self-ternary neutral
    //! coefficient
    /*!
     * Array of 2D data used in the Pitzer/HMW formulation. Mu_nnn_L[i]
     * represents the Mu coefficient temperature derivative for the nnn
     * interaction. This is a general interaction representing neutral species
     * interacting with itself.
     */
    mutable vector_fp m_Mu_nnn_L;

    //! Mu coefficient 2nd temperature derivative for the self-ternary neutral
    //! coefficient
    /*!
     * Array of 2D data used in the Pitzer/HMW formulation. Mu_nnn_L[i]
     * represents the Mu coefficient 2nd temperature derivative for the nnn
     * interaction. This is a general interaction representing neutral species
     * interacting with itself.
     */
    mutable vector_fp m_Mu_nnn_LL;

    //! Mu coefficient pressure derivative for the self-ternary neutral
    //! coefficient
    /*!
     * Array of 2D data used in the Pitzer/HMW formulation. Mu_nnn_L[i]
     * represents the Mu coefficient pressure derivative for the nnn
     * interaction. This is a general interaction representing neutral species
     * interacting with itself.
     */
    mutable vector_fp m_Mu_nnn_P;

    //! Array of coefficients form_Mu_nnn term
    Array2D m_Mu_nnn_coeff;

    //! Logarithm of the activity coefficients on the molality scale.
    /*!
     * mutable because we change this if the composition or temperature or
     * pressure changes. Index is the species index
     */
    mutable vector_fp m_lnActCoeffMolal_Scaled;

    //! Logarithm of the activity coefficients on the molality scale.
    /*!
     * mutable because we change this if the composition or temperature or
     * pressure changes. Index is the species index
     */
    mutable vector_fp m_lnActCoeffMolal_Unscaled;

    //! Derivative of the Logarithm of the activity coefficients on the molality
    //! scale wrt T. Index is the species index
    mutable vector_fp m_dlnActCoeffMolaldT_Scaled;

    //! Derivative of the Logarithm of the activity coefficients on the molality
    //! scale wrt T. Index is the species index
    mutable vector_fp m_dlnActCoeffMolaldT_Unscaled;

    //! Derivative of the Logarithm of the activity coefficients on the molality
    //! scale wrt TT. Index is the species index.
    mutable vector_fp m_d2lnActCoeffMolaldT2_Scaled;

    //! Derivative of the Logarithm of the activity coefficients on the molality
    //! scale wrt TT. Index is the species index
    mutable vector_fp m_d2lnActCoeffMolaldT2_Unscaled;

    //! Derivative of the Logarithm of the activity coefficients on the
    //! molality scale wrt P. Index is the species index
    mutable vector_fp m_dlnActCoeffMolaldP_Scaled;

    //! Derivative of the Logarithm of the activity coefficients on the
    //! molality scale wrt P. Index is the species index
    mutable vector_fp m_dlnActCoeffMolaldP_Unscaled;

    // -------- Temporary Variables Used in the Activity Coeff Calc

    //! Cropped and modified values of the molalities used in activity
    //! coefficient calculations
    mutable vector_fp m_molalitiesCropped;

    //! Boolean indicating whether the molalities are cropped or are modified
    mutable bool m_molalitiesAreCropped;

    //! a counter variable for keeping track of symmetric binary
    //! interactions amongst the solute species.
    /*!
     * n = m_kk*i + j
     * m_CounterIJ[n] = counterIJ
     */
    mutable vector_int m_CounterIJ;

    //! This is elambda, MEC
    mutable double elambda[17];

    //! This is elambda1, MEC
    mutable double elambda1[17];

    /**
     *  Various temporary arrays used in the calculation of the Pitzer activity
     *  coefficients. The subscript, L, denotes the same quantity's derivative
     *  wrt temperature
     */

    //! This is the value of g(x) in Pitzer's papers. Vector index is counterIJ
    mutable vector_fp m_gfunc_IJ;

    //! This is the value of g2(x2) in Pitzer's papers. Vector index is counterIJ
    mutable vector_fp m_g2func_IJ;

    //! hfunc, was called gprime in Pitzer's paper. However, it's not the
    //! derivative of gfunc(x), so I renamed it. Vector index is counterIJ
    mutable vector_fp m_hfunc_IJ;

    //! hfunc2, was called gprime in Pitzer's paper. However, it's not the
    //! derivative of gfunc(x), so I renamed it. Vector index is counterIJ
    mutable vector_fp m_h2func_IJ;

    //! Intermediate variable called BMX in Pitzer's paper. This is the basic
    //! cation - anion interaction. Vector index is counterIJ
    mutable vector_fp m_BMX_IJ;

    //! Derivative of BMX_IJ wrt T. Vector index is counterIJ
    mutable vector_fp m_BMX_IJ_L;

    //! Derivative of BMX_IJ wrt TT. Vector index is counterIJ
    mutable vector_fp m_BMX_IJ_LL;

    //! Derivative of BMX_IJ wrt P. Vector index is counterIJ
    mutable vector_fp m_BMX_IJ_P;

    //! Intermediate variable called BprimeMX in Pitzer's paper. Vector index is
    //! counterIJ
    mutable vector_fp m_BprimeMX_IJ;

    //! Derivative of BprimeMX wrt T. Vector index is counterIJ
    mutable vector_fp m_BprimeMX_IJ_L;

    //! Derivative of BprimeMX wrt TT. Vector index is counterIJ
    mutable vector_fp m_BprimeMX_IJ_LL;

    //! Derivative of BprimeMX wrt P. Vector index is counterIJ
    mutable vector_fp m_BprimeMX_IJ_P;

    //! Intermediate variable called BphiMX in Pitzer's paper. Vector index is
    //! counterIJ
    mutable vector_fp m_BphiMX_IJ;

    //! Derivative of BphiMX_IJ wrt T. Vector index is counterIJ
    mutable vector_fp m_BphiMX_IJ_L;

    //! Derivative of BphiMX_IJ wrt TT. Vector index is counterIJ
    mutable vector_fp m_BphiMX_IJ_LL;

    //! Derivative of BphiMX_IJ wrt P. Vector index is counterIJ
    mutable vector_fp m_BphiMX_IJ_P;

    //! Intermediate variable called Phi in Pitzer's paper. Vector index is
    //! counterIJ
    mutable vector_fp m_Phi_IJ;

    //! Derivative of m_Phi_IJ wrt T. Vector index is counterIJ
    mutable vector_fp m_Phi_IJ_L;

    //! Derivative of m_Phi_IJ wrt TT. Vector index is counterIJ
    mutable vector_fp m_Phi_IJ_LL;

    //! Derivative of m_Phi_IJ wrt P. Vector index is counterIJ
    mutable vector_fp m_Phi_IJ_P;

    //! Intermediate variable called Phiprime in Pitzer's paper. Vector index is
    //! counterIJ
    mutable vector_fp m_Phiprime_IJ;

    //! Intermediate variable called PhiPhi in Pitzer's paper. Vector index is
    //! counterIJ
    mutable vector_fp m_PhiPhi_IJ;

    //! Derivative of m_PhiPhi_IJ wrt T. Vector index is counterIJ
    mutable vector_fp m_PhiPhi_IJ_L;

    //! Derivative of m_PhiPhi_IJ wrt TT. Vector index is counterIJ
    mutable vector_fp m_PhiPhi_IJ_LL;

    //! Derivative of m_PhiPhi_IJ wrt P. Vector index is counterIJ
    mutable vector_fp m_PhiPhi_IJ_P;

    //! Intermediate variable called CMX in Pitzer's paper. Vector index is
    //! counterIJ
    mutable vector_fp m_CMX_IJ;

    //! Derivative of m_CMX_IJ wrt T. Vector index is counterIJ
    mutable vector_fp m_CMX_IJ_L;

    //! Derivative of m_CMX_IJ wrt TT. Vector index is counterIJ
    mutable vector_fp m_CMX_IJ_LL;

    //! Derivative of m_CMX_IJ wrt P. Vector index is counterIJ
    mutable vector_fp m_CMX_IJ_P;

    //! Intermediate storage of the activity coefficient itself. Vector index is
    //! the species index
    mutable vector_fp m_gamma_tmp;

    //! Logarithm of the molal activity coefficients. Normally these are all
    //! one. However, stability schemes will change that
    mutable vector_fp IMS_lnActCoeffMolal_;

    //! IMS Cutoff type
    int IMS_typeCutoff_;

    //! value of the solute mole fraction that centers the cutoff polynomials
    //! for the cutoff =1 process;
    doublereal IMS_X_o_cutoff_;

    //! gamma_o value for the cutoff process at the zero solvent point
    doublereal IMS_gamma_o_min_;

    //! gamma_k minimum for the cutoff process at the zero solvent point
    doublereal IMS_gamma_k_min_;

    //! Parameter in the polyExp cutoff treatment having to do with rate of exp decay
    doublereal IMS_cCut_;

    //! Parameter in the polyExp cutoff treatment
    /*!
     *  This is the slope of the f function at the zero solvent point
     *  Default value is 0.6
     */
    doublereal IMS_slopefCut_;

    //! Parameter in the polyExp cutoff treatment
    /*!
     *  This is the slope of the g function at the zero solvent point
     *  Default value is 0.0
     */
    doublereal IMS_slopegCut_;

    //! @name Parameters in the polyExp cutoff treatment having to do with rate of exp decay
    //! @{
    doublereal IMS_dfCut_;
    doublereal IMS_efCut_;
    doublereal IMS_afCut_;
    doublereal IMS_bfCut_;
    doublereal IMS_dgCut_;
    doublereal IMS_egCut_;
    doublereal IMS_agCut_;
    doublereal IMS_bgCut_;
    //! @}

    //! value of the solvent mole fraction that centers the cutoff polynomials
    //! for the cutoff =1 process;
    doublereal MC_X_o_cutoff_;

    //! gamma_o value for the cutoff process at the zero solvent point
    doublereal MC_X_o_min_;

    //! Parameter in the Molality Exp cutoff treatment
    /*!
     *  This is the slope of the p function at the zero solvent point
     *  Default value is 0.0
     */
    doublereal MC_slopepCut_;

    //! @name Parameters in the Molality Exp cutoff treatment
    //! @{
    doublereal MC_dpCut_;
    doublereal MC_epCut_;
    doublereal MC_apCut_;
    doublereal MC_bpCut_;
    doublereal MC_cpCut_;
    doublereal CROP_ln_gamma_o_min;
    doublereal CROP_ln_gamma_o_max;
    doublereal CROP_ln_gamma_k_min;
    doublereal CROP_ln_gamma_k_max;

    //! This is a boolean-type vector indicating whether
    //! a species's activity coefficient is in the cropped regime
    /*!
     *  * 0 = Not in cropped regime
     *  * 1 = In a transition regime where it is altered but there
     *        still may be a temperature or pressure dependence
     *  * 2 = In a cropped regime where there is no temperature
     *        or pressure dependence
     */
    mutable vector_int CROP_speciesCropped_;
    //! @}

    //!  Initialize all of the species-dependent lengths in the object
    void initLengths();

    //! Apply the current phScale to a set of activity Coefficients or
    //! activities
    /*!
     *  See the Eq3/6 Manual for a thorough discussion.
     *
     * @param acMolality input/Output vector containing the molality based
     *                   activity coefficients. length: m_kk.
     */
    virtual void applyphScale(doublereal* acMolality) const;

private:
    /*
     * This function will be called to update the internally stored
     * natural logarithm of the molality activity coefficients
     */
    void s_update_lnMolalityActCoeff() const;

    //! This function calculates the temperature derivative of the
    //! natural logarithm of the molality activity coefficients.
    /*!
     * This function does all of the direct work. The solvent activity
     * coefficient is on the molality scale. It's derivative is too.
     */
    void s_update_dlnMolalityActCoeff_dT() const;

    /**
     * This function calculates the temperature second derivative of the natural
     * logarithm of the molality activity coefficients.
     */
    void s_update_d2lnMolalityActCoeff_dT2() const;

    /**
     * This function calculates the pressure derivative of the
     * natural logarithm of the molality activity coefficients.
     *
     * Assumes that the activity coefficients are current.
     */
    void s_update_dlnMolalityActCoeff_dP() const;

    //! This function will be called to update the internally stored
    //! natural logarithm of the molality activity coefficients
    /*
     * Normally they are all one. However, sometimes they are not,
     * due to stability schemes
     *
     *    gamma_k_molar =  gamma_k_molal / Xmol_solvent
     *
     *    gamma_o_molar = gamma_o_molal
     */
    void s_updateIMS_lnMolalityActCoeff() const;

private:
    //! Calculate the Pitzer portion of the activity coefficients.
    /**
     *  This is the main routine in the whole module. It calculates the molality
     *  based activity coefficients for the solutes, and the activity of water.
     */
    void s_updatePitzer_lnMolalityActCoeff() const;

    //! Calculates the temperature derivative of the natural logarithm of the
    //! molality activity coefficients.
    /*!
     * Public function makes sure that all dependent data is
     * up to date, before calling a private function
     */
    void s_updatePitzer_dlnMolalityActCoeff_dT() const;

    /**
     * This function calculates the temperature second derivative of the
     * natural logarithm of the molality activity coefficients.
     *
     * It is assumed that the Pitzer activity coefficient and first derivative
     * routine are called immediately preceding the call to this routine.
     */
    void s_updatePitzer_d2lnMolalityActCoeff_dT2() const;

    //! Calculates the Pressure derivative of the natural logarithm of the
    //! molality activity coefficients.
    /*!
     * It is assumed that the Pitzer activity coefficient and first derivative
     * routine are called immediately preceding the calling of this routine.
     */
    void s_updatePitzer_dlnMolalityActCoeff_dP() const;

    //! Calculates the Pitzer coefficients' dependence on the temperature.
    /*!
     * It will also calculate the temperature derivatives of the coefficients,
     * as they are important in the calculation of the latent heats and the heat
     * capacities of the mixtures.
     *
     * @param doDerivs If >= 1, then the routine will calculate the first
     *                 derivative. If >= 2, the routine will calculate the first
     *                 and second temperature derivative. default = 2
     */
    void s_updatePitzer_CoeffWRTemp(int doDerivs = 2) const;

    //! Calculate the lambda interactions.
    /*!
     * Calculate E-lambda terms for charge combinations of like sign, using
     * method of Pitzer (1975). This implementation is based on Bethke,
     * Appendix 2.
     *
     * @param is Ionic strength
     */
    void calc_lambdas(double is) const;
    mutable doublereal m_last_is;

    /**
     * Calculate etheta and etheta_prime
     *
     * This interaction accounts for the mixing effects of like-signed ions with
     * different charges. This interaction will be nonzero for species with the
     * same charge. this routine is not to be called for neutral species; it
     * core dumps or error exits.
     *
     * MEC implementation routine.
     *
     * @param z1 charge of the first molecule
     * @param z2 charge of the second molecule
     * @param etheta return pointer containing etheta
     * @param etheta_prime Return pointer containing etheta_prime.
     *
     * This routine uses the internal variables, elambda[] and elambda1[].
     */
    void calc_thetas(int z1, int z2,
                     double* etheta, double* etheta_prime) const;

    //! Set up a counter variable for keeping track of symmetric binary
    //! interactions amongst the solute species.
    /*!
     * The purpose of this is to squeeze the ij parameters into a
     * compressed single counter.
     *
     * n = m_kk*i + j
     * m_Counter[n] = counter
     */
    void counterIJ_setup() const;

    //! Calculate the cropped molalities
    /*!
     * This is an internal routine that calculates values of m_molalitiesCropped
     * from m_molalities
     */
    void calcMolalitiesCropped() const;

    //! Process an XML node called "binarySaltParameters"
    /*!
     * This node contains all of the parameters necessary to describe the Pitzer
     * model for that particular binary salt. This function reads the XML file
     * and writes the coefficients it finds to an internal data structures.
     *
     * @param BinSalt  reference to the XML_Node named binarySaltParameters
     *                 containing the anion - cation interaction
     */
    void readXMLBinarySalt(XML_Node& BinSalt);

    //! Process an XML node called "thetaAnion"
    /*!
     * This node contains all of the parameters necessary to describe the binary
     * interactions between two anions.
     *
     * @param BinSalt  reference to the XML_Node named thetaAnion containing the
     *                 anion - anion interaction
     */
    void readXMLThetaAnion(XML_Node& BinSalt);

    //! Process an XML node called "thetaCation"
    /*!
     * This node contains all of the parameters necessary to describe the binary
     * interactions between two cations.
     *
     * @param BinSalt  reference to the XML_Node named thetaCation containing
     *                 the cation - cation interaction
     */
    void readXMLThetaCation(XML_Node& BinSalt);

    //! Process an XML node called "psiCommonAnion"
    /*!
     * This node contains all of the parameters necessary to describe
     * the ternary interactions between one anion and two cations.
     *
     * @param BinSalt  reference to the XML_Node named psiCommonAnion containing
     *                 the anion - cation1 - cation2 interaction
     */
    void readXMLPsiCommonAnion(XML_Node& BinSalt);

    //! Process an XML node called "psiCommonCation"
    /*!
     * This node contains all of the parameters necessary to describe
     * the ternary interactions between one cation and two anions.
     *
     * @param BinSalt  reference to the XML_Node named psiCommonCation
     *                 containing the cation - anion1 - anion2 interaction
     */
    void readXMLPsiCommonCation(XML_Node& BinSalt);

    //! Process an XML node called "lambdaNeutral"
    /*!
     * This node contains all of the parameters necessary to describe the binary
     * interactions between one neutral species and any other species (neutral
     * or otherwise) in the mechanism.
     *
     * @param BinSalt  reference to the XML_Node named lambdaNeutral containing
     *                 multiple Neutral - species interactions
     */
    void readXMLLambdaNeutral(XML_Node& BinSalt);

    //! Process an XML node called "MunnnNeutral"
    /*!
     * This node contains all of the parameters necessary to describe
     * the self-ternary interactions for one neutral species.
     *
     * @param BinSalt  reference to the XML_Node named Munnn containing the
     *                 self-ternary interaction
     */
    void readXMLMunnnNeutral(XML_Node& BinSalt);

    //! Process an XML node called "zetaCation"
    /*!
     * This node contains all of the parameters necessary to describe
     * the ternary interactions between one neutral, one cation, and one anion.
     *
     * @param BinSalt  reference to the XML_Node named psiCommonCation
     *                 containing the neutral - cation - anion interaction
     */
    void readXMLZetaCation(const XML_Node& BinSalt);

    //! Process an XML node called "croppingCoefficients" for the cropping
    //! coefficients values
    /*!
     * @param acNode Activity Coefficient XML Node
     */
    void readXMLCroppingCoefficients(const XML_Node& acNode);

    //! Precalculate the IMS Cutoff parameters for typeCutoff = 2
    void calcIMSCutoffParams_();

    //! Calculate molality cut-off parameters
    void calcMCCutoffParams_();

    //! Utility function to assign an integer value from a string for the
    //! ElectrolyteSpeciesType field.
    /*!
     *  @param estString string name of the electrolyte species type
     */
    static int interp_est(const std::string& estString);

public:
    //! Turn on copious debug printing when this is true
    mutable int m_debugCalc;

    //! Return int specifying the amount of debug printing
    int debugPrinting();
};

}

#endif