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 https://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
{
 
//! @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 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, that is,
 * 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).
 *
 * ### 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
 * sulfates 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 Silvester and Pitzer @cite silvester1977, 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 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.
 *
 * 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 @cite harvie1980. 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 value, specified for pair of cations
 * or a pair of anions.
 *
 * ### 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.
 *
 * ### 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 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.
 *
 * An example phase definition specifying a number of the above species interaction
 * parameters is given in the
 * [YAML API Reference](../yaml/phases.html#hmw-electrolyte).
 *
 * ### 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
 *
 * ### 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.
 *
 * @ingroup thermoprops
 */
class HMWSoln : public MolalityVPSSTP
{
public:
    ~HMWSoln();
 
    //! Construct and initialize an HMWSoln ThermoPhase object
    //! directly from an input file
    /*!
     *  This constructor is a shell that calls the routine initThermo(), with
     *  a reference to the parsed input file to get the info for the phase.
     *
     * @param inputFile Name of the input file containing the phase definition
     *                  to set up the object. If blank, an empty phase will be
     *                  created.
     * @param id        ID of the phase in the input file. Defaults to the
     *                  empty string.
     */
    explicit HMWSoln(const string& inputFile="", const string& id="");
 
    //! @name  Utilities
    //! @{
 
    string type() const override {
        return "HMW-electrolyte";
    }
 
    //! @}
    //! @name  Molar Thermodynamic Properties of the Solution
    //! @{
 
    /**
     * Excess molar enthalpy of the solution from
     * the mixing process. Units: J/ kmol.
     *
     * Note this is kmol of the total solution.
     */
    virtual double 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 double relative_molal_enthalpy() const;
 
    double cv_mole() const override;
 
    //! @}
    //! @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() will cause an exception to be thrown.
    //! @{
 
protected:
    void calcDensity() override;
 
public:
    //! @}
    //! @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 \ln 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
     */
    void getActivityConcentrations(span<double> c) const override;
 
    //! Return the standard concentration for the kth species
    /*!
     * The standard concentration @f$ C^0_k @f$ used to normalize the activity
     * (that is, 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.
     */
    double standardConcentration(size_t k=0) const override;
 
    //! 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.
     */
    void getActivities(span<double> ac) const override;
 
    //! @}
    //! @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
     */
    void getChemPotentials(span<double> mu) const override;
 
    //! 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.
     */
    void getPartialMolarEnthalpies(span<double> hbar) const override;
 
    //! 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.
     */
    void getPartialMolarEntropies(span<double> sbar) const override;
 
    //! 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.
     */
    void getPartialMolarVolumes(span<double> vbar) const override;
 
    //! 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.
     */
    void getPartialMolarCp(span<double> cpbar) const override;
 
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)
     */
    double satPressure(double T) override;
 
    /*
     *  -------------- Utilities -------------------------------
     */
 
    void setBinarySalt(const string& sp1, const string& sp2,
        span<const double> beta0, span<const double> beta1,
        span<const double> beta2, span<const double> Cphi, double alpha1,
        double alpha2);
    void setTheta(const string& sp1, const string& sp2,
        span<const double> theta);
    void setPsi(const string& sp1, const string& sp2,
        const string& sp3, span<const double> psi);
    void setLambda(const string& sp1, const string& sp2,
        span<const double> lambda);
    void setMunnn(const string& sp, span<const double> munnn);
    void setZeta(const string& sp1, const string& sp2,
        const string& sp3, span<const double> psi);
 
    void setPitzerTempModel(const string& model);
    void setPitzerRefTemperature(double Tref) {
        m_TempPitzerRef = Tref;
    }
 
    //! Set the A_Debye parameter. If a negative value is provided, enables
    //! calculation of A_Debye using the detailed water equation of state.
    void setA_Debye(double A);
 
    void setMaxIonicStrength(double Imax) {
        m_maxIionicStrength = Imax;
    }
 
    void setCroppingCoefficients(double ln_gamma_k_min, double ln_gamma_k_max,
                                 double ln_gamma_o_min, double ln_gamma_o_max);
 
    void initThermo() override;
    void getParameters(AnyMap& phaseNode) const override;
 
    //! 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;
 
    //! 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 @cite denbigh1981 for a thorough discussion. This method 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(span<double> acMolality) const override;
 
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.
     */
    double 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.
     */
    double 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.
     */
    double 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.
     */
    double s_NBS_CLM_dlnMolalityActCoeff_dP() const;
 
private:
    /**
     * 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 = PITZER_TEMP_CONSTANT;
 
    //! 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 = 0.0;
 
    //! 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 = 298.15;
 
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 = A_DEBYE_CONST;
 
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 = 1.172576;
 
    //! Water standard state calculator
    /*!
     *  derived from the equation of state for water.
     */
    PDSS* m_waterSS = nullptr;
 
    //! Pointer to the water property calculator
    unique_ptr<WaterProps> m_waterProps;
 
    /**
     *  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<double> m_Beta0MX_ij;
 
    //! Derivative of Beta0_ij[i][j] wrt T. Vector index is counterIJ
    mutable vector<double> m_Beta0MX_ij_L;
 
    //! Derivative of Beta0_ij[i][j] wrt TT. Vector index is counterIJ
    mutable vector<double> m_Beta0MX_ij_LL;
 
    //! Derivative of Beta0_ij[i][j] wrt P. Vector index is counterIJ
    mutable vector<double> m_Beta0MX_ij_P;
 
    //! Array of coefficients for Beta0, a variable in Pitzer's papers
    /*!
     * Column index is counterIJ. m_Beta0MX_ij_coeff.col(counterIJ) is a
     * `span` containing the 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<double> m_Beta1MX_ij;
 
    //! Derivative of Beta1_ij[i][j] wrt T. Vector index is counterIJ
    mutable vector<double> m_Beta1MX_ij_L;
 
    //! Derivative of Beta1_ij[i][j] wrt TT. Vector index is counterIJ
    mutable vector<double> m_Beta1MX_ij_LL;
 
    //! Derivative of Beta1_ij[i][j] wrt P. Vector index is counterIJ
    mutable vector<double> m_Beta1MX_ij_P;
 
    //! Array of coefficients for Beta1, a variable in Pitzer's papers
    /*!
     * Column index is counterIJ. m_Beta1MX_ij_coeff.col(counterIJ) is a
     * `span` 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<double> m_Beta2MX_ij;
 
    //! Derivative of Beta2_ij[i][j] wrt T. Vector index is counterIJ
    mutable vector<double> m_Beta2MX_ij_L;
 
    //! Derivative of Beta2_ij[i][j] wrt TT. Vector index is counterIJ
    mutable vector<double> m_Beta2MX_ij_LL;
 
    //! Derivative of Beta2_ij[i][j] wrt P. Vector index is counterIJ
    mutable vector<double> m_Beta2MX_ij_P;
 
    //! Array of coefficients for Beta2, a variable in Pitzer's papers
    /*!
     * column index is counterIJ. m_Beta2MX_ij_coeff.col(counterIJ) is a
     *  `span` 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<double> 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<double> 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<double> m_CphiMX_ij;
 
    //! Derivative of Cphi_ij[i][j] wrt T. Vector index is counterIJ
    mutable vector<double> m_CphiMX_ij_L;
 
    //! Derivative of Cphi_ij[i][j] wrt TT. Vector index is counterIJ
    mutable vector<double> m_CphiMX_ij_LL;
 
    //! Derivative of Cphi_ij[i][j] wrt P. Vector index is counterIJ
    mutable vector<double> m_CphiMX_ij_P;
 
    //! Array of coefficients for CphiMX, a parameter in the activity
    //! coefficient formulation
    /*!
     *  Column index is counterIJ. m_CphiMX_ij_coeff.col(counterIJ) is a
     *  `span` 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<double> m_Theta_ij;
 
    //! Derivative of Theta_ij[i][j] wrt T. Vector index is counterIJ
    mutable vector<double> m_Theta_ij_L;
 
    //! Derivative of Theta_ij[i][j] wrt TT. Vector index is counterIJ
    mutable vector<double> m_Theta_ij_LL;
 
    //! Derivative of Theta_ij[i][j] wrt P. Vector index is counterIJ
    mutable vector<double> 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.col(counterIJ) is a `span` 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<double> m_Psi_ijk;
 
    //! Derivative of Psi_ijk[n] wrt T. See m_Psi_ijk for reference on the
    //! indexing into this variable.
    mutable vector<double> 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<double> 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<double> 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.col(n) is a `span` 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, that is, 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, that is, i. The charged species occupy the j
     *  coordinate. Neutral, neutral interactions are also included here.
     *
     *      n = j + m_kk * i
     *
     * m_Lambda_nj_coeff.col(n) is a `span` 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<double> 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<double> 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<double> 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<double> 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<double> 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<double> m_lnActCoeffMolal_Unscaled;
 
    //! Derivative of the Logarithm of the activity coefficients on the molality
    //! scale wrt T. Index is the species index
    mutable vector<double> m_dlnActCoeffMolaldT_Scaled;
 
    //! Derivative of the Logarithm of the activity coefficients on the molality
    //! scale wrt T. Index is the species index
    mutable vector<double> m_dlnActCoeffMolaldT_Unscaled;
 
    //! Derivative of the Logarithm of the activity coefficients on the molality
    //! scale wrt TT. Index is the species index.
    mutable vector<double> m_d2lnActCoeffMolaldT2_Scaled;
 
    //! Derivative of the Logarithm of the activity coefficients on the molality
    //! scale wrt TT. Index is the species index
    mutable vector<double> m_d2lnActCoeffMolaldT2_Unscaled;
 
    //! Derivative of the Logarithm of the activity coefficients on the
    //! molality scale wrt P. Index is the species index
    mutable vector<double> m_dlnActCoeffMolaldP_Scaled;
 
    //! Derivative of the Logarithm of the activity coefficients on the
    //! molality scale wrt P. Index is the species index
    mutable vector<double> 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<double> m_molalitiesCropped;
 
    //! Boolean indicating whether the molalities are cropped or are modified
    mutable bool m_molalitiesAreCropped = false;
 
    //! 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<double> m_gfunc_IJ;
 
    //! This is the value of g2(x2) in Pitzer's papers. Vector index is counterIJ
    mutable vector<double> 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<double> 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<double> m_h2func_IJ;
 
    //! Intermediate variable called BMX in Pitzer's paper. This is the basic
    //! cation - anion interaction. Vector index is counterIJ
    mutable vector<double> m_BMX_IJ;
 
    //! Derivative of BMX_IJ wrt T. Vector index is counterIJ
    mutable vector<double> m_BMX_IJ_L;
 
    //! Derivative of BMX_IJ wrt TT. Vector index is counterIJ
    mutable vector<double> m_BMX_IJ_LL;
 
    //! Derivative of BMX_IJ wrt P. Vector index is counterIJ
    mutable vector<double> m_BMX_IJ_P;
 
    //! Intermediate variable called BprimeMX in Pitzer's paper. Vector index is
    //! counterIJ
    mutable vector<double> m_BprimeMX_IJ;
 
    //! Derivative of BprimeMX wrt T. Vector index is counterIJ
    mutable vector<double> m_BprimeMX_IJ_L;
 
    //! Derivative of BprimeMX wrt TT. Vector index is counterIJ
    mutable vector<double> m_BprimeMX_IJ_LL;
 
    //! Derivative of BprimeMX wrt P. Vector index is counterIJ
    mutable vector<double> m_BprimeMX_IJ_P;
 
    //! Intermediate variable called BphiMX in Pitzer's paper. Vector index is
    //! counterIJ
    mutable vector<double> m_BphiMX_IJ;
 
    //! Derivative of BphiMX_IJ wrt T. Vector index is counterIJ
    mutable vector<double> m_BphiMX_IJ_L;
 
    //! Derivative of BphiMX_IJ wrt TT. Vector index is counterIJ
    mutable vector<double> m_BphiMX_IJ_LL;
 
    //! Derivative of BphiMX_IJ wrt P. Vector index is counterIJ
    mutable vector<double> m_BphiMX_IJ_P;
 
    //! Intermediate variable called Phi in Pitzer's paper. Vector index is
    //! counterIJ
    mutable vector<double> m_Phi_IJ;
 
    //! Derivative of m_Phi_IJ wrt T. Vector index is counterIJ
    mutable vector<double> m_Phi_IJ_L;
 
    //! Derivative of m_Phi_IJ wrt TT. Vector index is counterIJ
    mutable vector<double> m_Phi_IJ_LL;
 
    //! Derivative of m_Phi_IJ wrt P. Vector index is counterIJ
    mutable vector<double> m_Phi_IJ_P;
 
    //! Intermediate variable called Phiprime in Pitzer's paper. Vector index is
    //! counterIJ
    mutable vector<double> m_Phiprime_IJ;
 
    //! Intermediate variable called PhiPhi in Pitzer's paper. Vector index is
    //! counterIJ
    mutable vector<double> m_PhiPhi_IJ;
 
    //! Derivative of m_PhiPhi_IJ wrt T. Vector index is counterIJ
    mutable vector<double> m_PhiPhi_IJ_L;
 
    //! Derivative of m_PhiPhi_IJ wrt TT. Vector index is counterIJ
    mutable vector<double> m_PhiPhi_IJ_LL;
 
    //! Derivative of m_PhiPhi_IJ wrt P. Vector index is counterIJ
    mutable vector<double> m_PhiPhi_IJ_P;
 
    //! Intermediate variable called CMX in Pitzer's paper. Vector index is
    //! counterIJ
    mutable vector<double> m_CMX_IJ;
 
    //! Derivative of m_CMX_IJ wrt T. Vector index is counterIJ
    mutable vector<double> m_CMX_IJ_L;
 
    //! Derivative of m_CMX_IJ wrt TT. Vector index is counterIJ
    mutable vector<double> m_CMX_IJ_LL;
 
    //! Derivative of m_CMX_IJ wrt P. Vector index is counterIJ
    mutable vector<double> m_CMX_IJ_P;
 
    //! Intermediate storage of the activity coefficient itself. Vector index is
    //! the species index
    mutable vector<double> m_gamma_tmp;
 
    //! Logarithm of the molal activity coefficients. Normally these are all
    //! one. However, stability schemes will change that
    mutable vector<double> IMS_lnActCoeffMolal_;
 
    //! value of the solute mole fraction that centers the cutoff polynomials
    //! for the cutoff =1 process;
    double IMS_X_o_cutoff_ = 0.2;
 
    //! Parameter in the polyExp cutoff treatment having to do with rate of exp decay
    double IMS_cCut_ = 0.05;
 
    //! Parameter in the polyExp cutoff treatment
    /*!
     *  This is the slope of the g function at the zero solvent point
     *  Default value is 0.0
     */
    double IMS_slopegCut_ = 0.0;
 
    //! @name Parameters in the polyExp cutoff treatment having to do with rate of exp decay
    //! @{
    double IMS_dfCut_ = 0.0;
    double IMS_efCut_ = 0.0;
    double IMS_afCut_ = 0.0;
    double IMS_bfCut_ = 0.0;
    double IMS_dgCut_ = 0.0;
    double IMS_egCut_ = 0.0;
    double IMS_agCut_ = 0.0;
    double IMS_bgCut_ = 0.0;
    //! @}
 
    //! value of the solvent mole fraction that centers the cutoff polynomials
    //! for the cutoff =1 process;
    double MC_X_o_cutoff_ = 0.0;
 
    //! @name Parameters in the Molality Exp cutoff treatment
    //! @{
    double MC_dpCut_ = 0.0;
    double MC_epCut_ = 0.0;
    double MC_apCut_ = 0.0;
    double MC_bpCut_ = 0.0;
    double MC_cpCut_ = 0.0;
    double CROP_ln_gamma_o_min;
    double CROP_ln_gamma_o_max;
    double CROP_ln_gamma_k_min;
    double 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.
     */
    void applyphScale(span<double> acMolality) const override;
 
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 @cite pitzer1975. This implementation is based on Bethke,
     * Appendix 2.
     *
     * @param is Ionic strength
     */
    void calc_lambdas(double is) const;
    mutable double m_last_is = -1.0;
 
    /**
     * 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[out] etheta return value of etheta
     * @param[out] etheta_prime return value of 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;
 
    //! Precalculate the IMS Cutoff parameters for typeCutoff = 2
    void calcIMSCutoffParams_();
 
    //! Calculate molality cut-off parameters
    void calcMCCutoffParams_();
};
 
}
 
#endif