"""
pyEQL activity correction library.
This file contains functions for computing molal-scale activity coefficients
of ions and salts in aqueous solution.
Individual functions for activity coefficients are defined here so that they
can be used independently of a pyEQL solution object. Normally, these functions
are called from within the get_activity_coefficient method of the Solution class.
:copyright: 2013-2024 by Ryan S. Kingsbury
:license: LGPL, see LICENSE for more details.
"""
import logging
import math
from pint import Quantity
from pyEQL import ureg
from pyEQL.utils import create_water_substance
logger = logging.getLogger(f"pyEQL.{__name__}")
[docs]
def _debye_parameter_B(temperature: str = "25 degC") -> Quantity:
r"""
Return the constant B used in the extended Debye-Huckel equation.
Args:
temperature: The temperature of the solution at which to calculate the constant.
Defaults to '25 degC'.
Returns:
The parameter B for use in extended Debye-Huckel equation (base e). For base 10,
divide the resulting value by 2.303. Note that A is often given in base 10 terms
in older textbooks and reference material (0.3281 at 25 degC).
Notes:
The parameter B is equal to:
.. math::
B = \bigg( \frac{2 N_A \rho_w e^2}{\epsilon_o \epsilon_r k T} \bigg) ^ {\frac{1}{2}}
References:
Bockris and Reddy. /Modern Electrochemistry/, vol 1. Plenum/Rosetta, 1977, p.210.
Archer, Donald G. and Wang, Peiming. "The Dielectric Constant of Water \
and Debye-Huckel Limiting Law Slopes." /J. Phys. Chem. Ref. Data/ 19(2), 1990.
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/25%3A_Solutions_II_-_Nonvolatile_Solutes/25.06%3A_The_Debye-Huckel_Theory
https://en.wikipedia.org/wiki/Debye%E2%80%93H%C3%BCckel_equation
"""
T = ureg.Quantity(temperature)
water_substance = create_water_substance(
T,
ureg.Quantity(1, "atm"),
)
param_B = (
2
* ureg.N_A
* ureg.Quantity(water_substance.rho, "g/L")
* ureg.elementary_charge**2
/ (ureg.epsilon_0 * water_substance.epsilon * ureg.boltzmann_constant * T)
) ** 0.5
return param_B.to_base_units()
[docs]
def _debye_parameter_activity(temperature: str = "25 degC") -> "Quantity":
r"""
Return the constant A for use in the Debye-Huckel limiting law (base e).
Args:
temperature: The temperature of the solution at which to calculate the constant.
Defaults to '25 degC'.
Returns:
The parameter A for use in the Debye-Huckel limiting law (base e). For base 10,
divide the resulting value by 2.303. Note that A is often given in base 10 terms
in older textbooks and reference material (0.509 at 25 degC).
Notes:
The parameter A is equal to:
.. math::
A^{\gamma} = \frac{e^3 \big( 2 \pi N_A \rho \big)^{0.5}}{(4 \pi \epsilon_o \epsilon_r k T)^{1.5}}
Note that this equation returns the parameter value that can be used to calculate
the natural logarithm of the activity coefficient. For base 10, divide the
value returned by 2.303.
References:
Archer, Donald G. and Wang, Peiming. "The Dielectric Constant of Water \
and Debye-Huckel Limiting Law Slopes." /J. Phys. Chem. Ref. Data/ 19(2), 1990.
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/25%3A_Solutions_II_-_Nonvolatile_Solutes/25.06%3A_The_Debye-Huckel_Theory
https://en.wikipedia.org/wiki/Debye%E2%80%93H%C3%BCckel_equation
See Also:
:func:`_debye_parameter_osmotic`
"""
T = ureg.Quantity(temperature)
water_substance = create_water_substance(
T,
ureg.Quantity(1, "atm"),
)
debyeparam = (
ureg.elementary_charge**3
* (2 * math.pi * ureg.N_A * ureg.Quantity(water_substance.rho, "g/L")) ** 0.5
/ (4 * math.pi * ureg.epsilon_0 * water_substance.epsilon * ureg.boltzmann_constant * T) ** 1.5
)
logger.debug(rf"Computed Debye-Huckel Limiting Law Constant A^{{\gamma}} = {debyeparam} at {temperature}")
return debyeparam.to("kg ** 0.5 / mol ** 0.5")
[docs]
def _debye_parameter_osmotic(temperature="25 degC"):
r"""
Return the constant A_phi for use in calculating the osmotic coefficient according to Debye-Huckel theory.
Args:
temperature: String representing the temperature of the solution. Defaults to '25 degC' if not specified.
Notes:
Not to be confused with the Debye-Huckel constant used for activity coefficients in the limiting law.
Takes the value 0.392 at 25 C.
This constant is calculated according to: [kim]_ [arch]_
.. math:: A^{\phi} = {1 \over 3} A^{\gamma}
References:
.. [kim] Kim, Hee-Talk and Frederick, William Jr, 1988. "Evaluation of Pitzer Ion Interaction Parameters of Aqueous Electrolytes at 25 C. 1. Single Salt Parameters,"
*J. Chemical Engineering Data* 33, pp.177-184.
.. [arch] Archer, Donald G. and Wang, Peiming. "The Dielectric Constant of Water \
and Debye-Huckel Limiting Law Slopes." /J. Phys. Chem. Ref. Data/ 19(2), 1990.
Examples:
>>> _debye_parameter_osmotic() #doctest: +ELLIPSIS
0.3916...
See Also:
:func:`_debye_parameter_activity`
"""
output = 1 / 3 * _debye_parameter_activity(temperature)
logger.debug(f"Computed Debye-Huckel Limiting slope for osmotic coefficient A^phi = {output} at {temperature}")
return output.to("kg ** 0.5 /mol ** 0.5")
[docs]
def _debye_parameter_volume(temperature="25 degC"):
r"""
Return the constant A_V, the Debye-Huckel limiting slope for apparent
molar volume.
Args:
temperature: String representing the temperature of the solution. Defaults to '25 degC' if not specified.
Notes:
Takes the value 1.8305 cm ** 3 * kg ** 0.5 / mol ** 1.5 at 25 C.
This constant is calculated according to: [1]_
.. math:: A_V = -2 A_{\phi} R T \big[ \frac{3}{\epsilon} \frac{\partial \epsilon}{\partial p} \
- \frac{1}{\rho}\frac{\partial \rho}{\partial p} \big]
Notes: at this time, the term in brackets (containing the partial derivatives) is approximate.
These approximations give the correct value of the slope at 25 degC and
produce estimates with less than 10% error between 0 and 60 degC.
The derivative of epsilon with respect to pressure is assumed constant (for atmospheric pressure)
at -0.01275 1/MPa. Note that the negative sign does not make sense in light
of real data, but is required to give the correct result.
The second term is equivalent to the inverse of the bulk modulus of water, which
is taken to be 2.2 GPa. [2]_
References:
.. [1] Archer, Donald G. and Wang, Peiming. "The Dielectric Constant of Water and Debye-Huckel Limiting
Law Slopes." J. Phys. Chem. Ref. Data/ 19(2), 1990.
.. [2] http://hyperphysics.phy-astr.gsu.edu/hbase/permot3.html
See Also:
:func:`_debye_parameter_osmotic`
"""
T = ureg.Quantity(temperature)
water_substance = create_water_substance(
T,
ureg.Quantity(1, "atm"),
)
# TODO - add partial derivatives to calculation
epsilon = water_substance.epsilon
dedp = ureg.Quantity(-0.01275, "1/MPa")
result = (
-2 * _debye_parameter_osmotic(temperature) * ureg.R * T * (3 / epsilon * dedp - 1 / ureg.Quantity(2.2, "GPa"))
)
# result = ureg.Quantity('1.898 cm ** 3 * kg ** 0.5 / mol ** 1.5')
if T.to("degC").magnitude != 25:
logger.warning("Debye-Huckel limiting slope for volume is approximate when T is not equal to 25 degC")
logger.debug(f"Computed Debye-Huckel Limiting Slope for volume A^V = {result} at {temperature}")
return result.to("cm ** 3 * kg ** 0.5 / mol ** 1.5")
[docs]
def get_activity_coefficient_debyehuckel(ionic_strength, z=1, temperature="25 degC"):
r"""
Return the activity coefficient of solute in the parent solution according to the Debye-Huckel limiting law.
Args:
z (int, optional): The charge on the solute, including sign. Defaults to +1 if not specified.
ionic_strength (Quantity): The ionic strength of the parent solution, mol/kg.
temperature (str, Quantity, optional): String representing the temperature of the solution. Defaults to '25 degC' if not specified.
Returns:
Quantity: The mean molal (mol/kg) scale ionic activity coefficient of solute, dimensionless.
Notes:
Activity coefficient is calculated according to:
.. math:: \ln \gamma = A^{\gamma} z_i^2 \sqrt I
Valid only for I < 0.005
See Also:
:func:`_debye_parameter_activity`
:func:`get_activity_coefficient_davies`
:func:`get_activity_coefficient_guntelberg`
References:
Stumm, Werner and Morgan, James J. Aquatic Chemistry, 3rd ed,
pp 103. Wiley Interscience, 1996.
"""
# check if this method is valid for the given ionic strength
if not ionic_strength.magnitude <= 0.005:
logger.warning("Ionic strength exceeds valid range of the Debye-Huckel limiting law")
log_f = -_debye_parameter_activity(temperature) * z**2 * ionic_strength**0.5
return math.exp(log_f) * ureg.Quantity(1, "dimensionless")
[docs]
def get_activity_coefficient_guntelberg(ionic_strength, z=1, temperature="25 degC"):
r"""
Return the activity coefficient of solute in the parent solution according to the Guntelberg approximation.
Args:
z (int, optional): The charge on the solute, including sign. Defaults to +1 if not specified.
ionic_strength (Quantity): The ionic strength of the parent solution, mol/kg.
temperature (str, Quantity, optional): String representing the temperature of the solution. Defaults to '25 degC' if not specified.
Returns:
Quantity: The mean molal (mol/kg) scale ionic activity coefficient of solute, dimensionless.
Notes:
Activity coefficient is calculated according to:
.. math:: \ln \gamma = A^{\gamma} z_i^2 {\sqrt I \over (1 + \sqrt I)}
Valid for I < 0.1
See Also:
:func:`_debye_parameter_activity`
:func:`get_activity_coefficient_davies`
:func:`get_activity_coefficient_debyehuckel`
References:
Stumm, Werner and Morgan, James J. Aquatic Chemistry, 3rd ed,
pp 103. Wiley Interscience, 1996.
"""
# check if this method is valid for the given ionic strength
if not ionic_strength.magnitude <= 0.1:
logger.warning("Ionic strength exceeds valid range of the Guntelberg approximation")
log_f = -_debye_parameter_activity(temperature) * z**2 * ionic_strength**0.5 / (1 + ionic_strength.magnitude**0.5)
return math.exp(log_f) * ureg.Quantity(1, "dimensionless")
[docs]
def get_activity_coefficient_davies(ionic_strength, z=1, temperature="25 degC"):
r"""
Return the activity coefficient of solute in the parent solution according to the Davies equation.
Args:
ionic_strength (Quantity): The ionic strength of the parent solution, mol/kg.
z (int, optional): The charge on the solute, including sign. Defaults to +1 if not specified.
temperature (str, Quantity, optional): String representing the temperature of the solution. Defaults to '25 degC' if not specified.
Returns:
Quantity: The mean molal (mol/kg) scale ionic activity coefficient of solute, dimensionless.
Notes:
Activity coefficient is calculated according to:
.. math:: \ln \gamma = A^{\gamma} z_i^2 ({\sqrt I \over (1 + \sqrt I)} + 0.2 I)
Valid for 0.1 < I < 0.5
See Also:
:func:`_debye_parameter_activity`
:func:`get_activity_coefficient_debyehuckel`
:func:`get_activity_coefficient_guntelberg`
References:
Stumm, Werner and Morgan, James J. Aquatic Chemistry, 3rd ed,
pp 103. Wiley Interscience, 1996.
"""
# check if this method is valid for the given ionic strength
if not ionic_strength.magnitude <= 0.5 and ionic_strength.magnitude >= 0.1:
logger.warning("Ionic strength exceeds valid range of the Davies equation")
# the units in this empirical equation don't work out, so we must use magnitudes
log_f = (
-_debye_parameter_activity(temperature).magnitude
* z**2
* (ionic_strength.magnitude**0.5 / (1 + ionic_strength.magnitude**0.5) - 0.2 * ionic_strength.magnitude)
)
return math.exp(log_f) * ureg.Quantity(1, "dimensionless")
[docs]
def get_activity_coefficient_pitzer(
ionic_strength,
molality,
alpha1,
alpha2,
beta0,
beta1,
beta2,
C_phi,
z_cation,
z_anion,
nu_cation,
nu_anion,
temperature="25 degC",
b=1.2,
):
"""
Return the activity coefficient of solute in the parent solution according to the Pitzer model.
Args:
ionic_strength: The ionic strength of the parent solution, mol/kg
molality: The molal concentration of the parent salt, mol/kg
alpha1, alpha2: Coefficients for the Pitzer model. This function assigns the coefficients
proper units of kg ** 0.5 / mol ** 0.5 after they are entered.
beta0, beta1, beta2, C_phi: Coefficients for the Pitzer model. These ion-interaction parameters are
specific to each salt system.
z_cation, z_anion: The charge on the cation and anion, respectively
nu_cation, nu_anion: The stoichiometric coefficient of the cation and anion in the salt
temperature: String representing the temperature of the solution. Defaults to '25 degC' if not specified.
b: Coefficient. Usually set equal to 1.2 and considered independent of temperature and pressure.
If provided, this coefficient is assigned proper units of kg ** 0.5 / mol ** 0.5 after entry.
Returns:
Quantity
The mean molal (mol/kg) scale ionic activity coefficient of solute, dimensionless
Examples:
>>> get_activity_coefficient_pitzer(0.5*ureg.Quantity('mol/kg'),0.5*ureg.Quantity('mol/kg'),1,0.5,-.0181191983,-.4625822071,.4682,.000246063,1,-1,1,1,b=1.2)
0.61915...
>>> get_activity_coefficient_pitzer(5.6153*ureg.Quantity('mol/kg'),5.6153*ureg.Quantity('mol/kg'),3,0.5,0.0369993,0.354664,0.0997513,-0.00171868,1,-1,1,1,b=1.2)
0.76331...
Notes: the examples below are for comparison with experimental and modeling data presented in
the May et al reference below.
10 mol/kg ammonium nitrate. Estimated result (from graph) = 0.2725
>>> get_activity_coefficient_pitzer(10*ureg.Quantity('mol/kg'),10*ureg.Quantity('mol/kg'),2,0,-0.01709,0.09198,0,0.000419,1,-1,1,1,b=1.2)
0.22595 ...
5 mol/kg ammonium nitrate. Estimated result (from graph) = 0.3011
>>> get_activity_coefficient_pitzer(5*ureg.Quantity('mol/kg'),5*ureg.Quantity('mol/kg'),2,0,-0.01709,0.09198,0,0.000419,1,-1,1,1,b=1.2)
0.30249 ...
18 mol/kg ammonium nitrate. Estimated result (from graph) = 0.1653
>>> get_activity_coefficient_pitzer(18*ureg.Quantity('mol/kg'),18*ureg.Quantity('mol/kg'),2,0,-0.01709,0.09198,0,0.000419,1,-1,1,1,b=1.2)
0.16241 ...
References:
Scharge, T., Munoz, A.G., and Moog, H.C. (2012). Activity Coefficients of Fission Products in Highly
Salinary Solutions of Na+, K+, Mg2+, Ca2+, Cl-, and SO42- : Cs+.
/Journal of Chemical& Engineering Data (57), p. 1637-1647.
Kim, H., & Jr, W. F. (1988). Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 25 degree C. 1. Single salt parameters.
Journal of Chemical and Engineering Data, (2), 177-184.
May, P. M., Rowland, D., Hefter, G., & Königsberger, E. (2011).
A Generic and Updatable Pitzer Characterization of Aqueous Binary Electrolyte Solutions at 1 bar and 25 °C.
Journal of Chemical & Engineering Data, 56(12), 5066-5077. doi:10.1021/je2009329
Beyer, R., & Steiger, M. (2010). Vapor Pressure Measurements of NaHCOO + H 2 O and KHCOO + H 2 O from 278 to 308 K
and Representation with an Ion Interaction (Pitzer) Model.
Journal of Chemical & Engineering Data, 55(2), 830-838. doi:10.1021/je900487a
See Also:
:func:`_debye_parameter_activity`
:func:`_pitzer_B_MX`
:func:`_pitzer_B_phi`
:func:`_pitzer_log_gamma`
"""
# assign proper units to alpha1, alpha2, and b
alpha1 = ureg.Quantity(alpha1, "kg ** 0.5 / mol ** 0.5")
alpha2 = ureg.Quantity(alpha2, "kg ** 0.5 / mol ** 0.5")
b = ureg.Quantity(b, "kg ** 0.5 / mol ** 0.5")
C_phi = ureg.Quantity(C_phi, "kg ** 2 /mol ** 2")
# assign units appropriate for the activity parameters
BMX = ureg.Quantity(_pitzer_B_MX(ionic_strength, alpha1, alpha2, beta0, beta1, beta2), "kg/mol")
Bphi = ureg.Quantity(_pitzer_B_phi(ionic_strength, alpha1, alpha2, beta0, beta1, beta2), "kg/mol")
loggamma = _pitzer_log_gamma(
ionic_strength,
molality,
BMX,
Bphi,
C_phi,
z_cation,
z_anion,
nu_cation,
nu_anion,
temperature,
b,
)
return math.exp(loggamma) * ureg.Quantity(1, "dimensionless")
[docs]
def get_apparent_volume_pitzer(
ionic_strength,
molality,
alpha1,
alpha2,
beta0,
beta1,
beta2,
C_phi,
V_o,
z_cation,
z_anion,
nu_cation,
nu_anion,
temperature="25 degC",
b=1.2,
):
"""
Return the apparent molar volume of solute in the parent solution according to the Pitzer model.
Args:
ionic_strength (Quantity): The ionic strength of the parent solution, mol/kg.
molality (Quantity): The molal concentration of the parent salt, mol/kg.
alpha1, alpha2 (number): Coefficients for the Pitzer model. This function assigns the coefficients proper units of kg ** 0.5 / mol ** 0.5 after they are entered.
beta0, beta1, beta2, C_phi (number): Pitzer coefficients for the apparent molar volume. These ion-interaction parameters are specific to each salt system.
V_o (number): The V^o Pitzer coefficient for the apparent molar volume.
z_cation, z_anion (int): The formal charge on the cation and anion, respectively.
nu_cation, nu_anion (int): The stoichiometric coefficient of the cation and anion in the salt.
temperature (str, Quantity): String representing the temperature of the solution. Defaults to '25 degC' if not specified.
b (number, optional): Coefficient. Usually set equal to 1.2 and considered independent of temperature and pressure. If provided, this coefficient is assigned proper units of kg ** 0.5 / mol ** 0.5 after entry.
Returns:
Quantity: The apparent molar volume of the solute, cm ** 3 / mol.
Examples:
Notes: the example below is for comparison with experimental and modeling data presented in the Krumgalz et al reference below.
0.25 mol/kg CuSO4. Expected result (from graph) = 0.5 cm ** 3 / mol
>>> get_apparent_volume_pitzer(1.0*ureg.Quantity('mol/kg'),0.25*ureg.Quantity('mol/kg'),1.4,12,0.001499,-0.008124,0.2203,-0.0002589,-6,2,-2,1,1,b=1.2)
0.404...
1.0 mol/kg CuSO4. Expected result (from graph) = 4 cm ** 3 / mol
>>> get_apparent_volume_pitzer(4.0*ureg.Quantity('mol/kg'),1.0*ureg.Quantity('mol/kg'),1.4,12,0.001499,-0.008124,0.2203,-0.0002589,-6,2,-2,1,1,b=1.2)
4.424...
10.0 mol/kg ammonium nitrate. Expected result (from graph) = 50.3 cm ** 3 / mol
>>> get_apparent_volume_pitzer(10.0*ureg.Quantity('mol/kg'),10.0*ureg.Quantity('mol/kg'),2,0,0.000001742,0.0002926,0,0.000000424,46.9,1,-1,1,1,b=1.2)
50.286...
20.0 mol/kg ammonium nitrate. Expected result (from graph) = 51.2 cm ** 3 / mol
>>> get_apparent_volume_pitzer(20.0*ureg.Quantity('mol/kg'),20.0*ureg.Quantity('mol/kg'),2,0,0.000001742,0.0002926,0,0.000000424,46.9,1,-1,1,1,b=1.2)
51.145...
Notes: the examples below are for comparison with experimental and modeling data presented in the Krumgalz et al reference below.
0.8 mol/kg NaF. Expected result = 0.03
>>> get_apparent_volume_pitzer(0.8*ureg.Quantity('mol/kg'),0.8*ureg.Quantity('mol/kg'),2,0,0.000024693,0.00003169,0,-0.000004068,-2.426,1,-1,1,1,b=1.2)
0.22595 ...
References:
May, P. M., Rowland, D., Hefter, G., & Königsberger, E. (2011).
A Generic and Updatable Pitzer Characterization of Aqueous Binary Electrolyte Solutions at 1 bar and 25 °C.
Journal of Chemical & Engineering Data, 56(12), 5066-5077. doi:10.1021/je2009329
Krumgalz, Boris S., Pogorelsky, Rita (1996).
Volumetric Properties of Single Aqueous Electrolytes from Zero to Saturation Concentration at 298.15 K
Represented by Pitzer's Ion-Interaction Equations.
Journal of Physical Chemical Reference Data, 25(2), 663-689.
See Also:
:func:`_debye_parameter_volume`
:func:`_pitzer_B_MX`
"""
# TODO - find a cleaner way to make sure coefficients are assigned the proper units
# if they aren't, the calculation gives very wrong results
alpha1 = ureg.Quantity(alpha1, "kg ** 0.5 / mol ** 0.5")
alpha2 = ureg.Quantity(alpha2, "kg ** 0.5 / mol ** 0.5")
b = ureg.Quantity(b, "kg ** 0.5 / mol ** 0.5")
C_phi = ureg.Quantity(C_phi, "kg ** 2 /mol ** 2 / dabar")
V_o = ureg.Quantity(V_o, "cm ** 3 / mol")
# assign units appropriate for the volume parameter
BMX = ureg.Quantity(_pitzer_B_MX(ionic_strength, alpha1, alpha2, beta0, beta1, beta2), "kg /mol/dabar")
second_term = (
(nu_cation + nu_anion)
* abs(z_cation * z_anion)
* (_debye_parameter_volume(temperature) / 2 / b)
* math.log(1 + b * ionic_strength**0.5)
)
third_term = (
nu_cation
* nu_anion
* ureg.R
* ureg.Quantity(temperature)
* (2 * molality * BMX + molality**2 * C_phi * (nu_cation * nu_anion) ** 0.5)
)
volume = V_o + second_term + third_term
return volume.to("cm ** 3 / mol")
[docs]
def _pitzer_f1(x):
r"""
The function of ionic strength used to calculate \beta_MX in the Pitzer ion interaction model.
.. math:: f(x) = 2 [ 1- (1+x) \exp(-x)] / x ^ 2
References:
Scharge, T., Munoz, A.G., and Moog, H.C. (2012). Activity Coefficients of Fission Products in Highly
Salinary Solutions of Na+, K+, Mg2+, Ca2+, Cl-, and SO42- : Cs+.
/Journal of Chemical& Engineering Data (57), p. 1637-1647.
Kim, H., & Jr, W. F. (1988). Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 25 degree C. 1. Single salt parameters.
Journal of Chemical and Engineering Data, (2), 177-184.
"""
# return 0 if the input is 0
if x == 0:
return 0
return 2 * (1 - (1 + x) * math.exp(-x)) / x**2
[docs]
def _pitzer_f2(x):
r"""
The function of ionic strength used to calculate \beta_\gamma in the Pitzer ion interaction model.
.. math:: f(x) = -\frac{2}{x ^ 2} [ 1 - (\frac{1+x+ x^2}{2}) \exp(-x)]
References:
Scharge, T., Munoz, A.G., and Moog, H.C. (2012). Activity Coefficients of Fission Products in Highly
Salinary Solutions of Na+, K+, Mg2+, Ca2+, Cl-, and SO42- : Cs+.
/Journal of Chemical& Engineering Data (57), p. 1637-1647.
Kim, H., & Jr, W. F. (1988). Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 25 degree C. 1. Single salt parameters.
Journal of Chemical and Engineering Data, (2), 177-184.
"""
# return 0 if the input is 0
if x == 0:
return 0
return -2 * (1 - (1 + x + x**2 / 2) * math.exp(-x)) / x**2
[docs]
def _pitzer_B_MX(ionic_strength, alpha1, alpha2, beta0, beta1, beta2):
r"""
Return the B_MX coefficient for the Pitzer ion interaction model.
.. math:: B_MX = \beta_0 + \beta_1 f1(\alpha_1 I ^ {0.5}) + \beta_2 f2(\alpha_2 I ^ {0.5})
Args:
ionic_strength: The ionic strength of the parent solution, mol/kg
alpha1, alpha2: Coefficients for the Pitzer model, kg ** 0.5 / mol ** 0.5
beta0, beta1, beta2: Coefficients for the Pitzer model. These ion-interaction parameters are
specific to each salt system.
Returns:
The B_MX parameter for the Pitzer ion interaction model.
References:
Scharge, T., Munoz, A.G., and Moog, H.C. (2012). Activity Coefficients of Fission Products in Highly
Salinary Solutions of Na+, K+, Mg2+, Ca2+, Cl-, and SO42- : Cs+.
/Journal of Chemical& Engineering Data (57), p. 1637-1647.
Kim, H., & Jr, W. F. (1988). Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 25 degree C. 1. Single salt parameters.
Journal of Chemical and Engineering Data, (2), 177-184.
See Also:
:func:`_pitzer_f1`
"""
coeff = beta0 + beta1 * _pitzer_f1(alpha1 * ionic_strength**0.5) + beta2 * _pitzer_f1(alpha2 * ionic_strength**0.5)
return coeff.magnitude
# def _pitzer_B_gamma(ionic_strength,alpha1,alpha2,beta1,beta2):
# '''
# Return the B^\gamma coefficient for the Pitzer ion interaction model.
#
# .. math:: B_\gamma = [ \beta_1 f2(\alpha_1 I ^ 0.5) + beta_2 f2(\alpha_2 I^0.5) ] / I
#
# Parameters
# ----------
# ionic_strength: number
# The ionic strength of the parent solution, mol/kg
# alpha1, alpha2: number
# Coefficients for the Pitzer model, kg ** 0.5 / mol ** 0.5.
# beta1, beta2: number
# Coefficients for the Pitzer model. These ion-interaction parameters are
# specific to each salt system.
#
# Returns
# -------
# float
# The B^gamma parameter for the Pitzer ion interaction model.
#
# References
# ----------
# Scharge, T., Munoz, A.G., and Moog, H.C. (2012). Activity Coefficients of Fission Products in Highly
# Salinary Solutions of Na+, K+, Mg2+, Ca2+, Cl-, and SO42- : Cs+.
# /Journal of Chemical& Engineering Data (57), p. 1637-1647.
#
# Kim, H., & Jr, W. F. (1988). Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 25 degree C. 1. Single salt parameters.
# Journal of Chemical and Engineering Data, (2), 177-184.
#
# See Also
# --------
# _pitzer_f2
#
# '''
# coeff = (beta1 * _pitzer_f2(alpha1 * ionic_strength ** 0.5) + beta2 * _pitzer_f2(alpha2 * ionic_strength ** 0.5)) / ionic_strength
# return coeff * ureg.Quantity('kg/mol')
[docs]
def _pitzer_B_phi(ionic_strength, alpha1, alpha2, beta0, beta1, beta2):
r"""
Returns the B^\Phi coefficient for the Pitzer ion interaction model.
This function calculates the B^\Phi coefficient using the formula:
.. math:: B^\Phi = \beta_0 + \beta1 \exp(-\alpha_1 I ^{0.5}) + \beta_2 \exp(-\alpha_2 I ^ {0.5})
or
.. math:: B^\Phi = B^\gamma - B_{MX}
Args:
ionic_strength: The ionic strength of the parent solution, mol/kg.
alpha1, alpha2: Coefficients for the Pitzer model, kg ** 0.5 / mol ** 0.5.
beta0, beta1, beta2: Coefficients for the Pitzer model. These ion-interaction parameters are
specific to each salt system.
Returns:
float: The B^Phi parameter for the Pitzer ion interaction model.
References:
Scharge, T., Munoz, A.G., and Moog, H.C. (2012). Activity Coefficients of Fission Products in Highly
Salinary Solutions of Na+, K+, Mg2+, Ca2+, Cl-, and SO42- : Cs+.
/Journal of Chemical& Engineering Data (57), p. 1637-1647.
Kim, H., & Jr, W. F. (1988). Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 25 degree C. 1. Single salt parameters.
Journal of Chemical and Engineering Data, (2), 177-184.
Beyer, R., & Steiger, M. (2010). Vapor Pressure Measurements of NaHCOO + H 2 O and KHCOO + H 2 O from 278 to 308 K
and Representation with an Ion Interaction (Pitzer) Model.
Journal of Chemical & Engineering Data, 55(2), 830-838. doi:10.1021/je900487a
"""
return beta0 + beta1 * math.exp(-alpha1 * ionic_strength**0.5) + beta2 * math.exp(-alpha2 * ionic_strength**0.5)
# def _pitzer_C_MX(C_phi,z_cation,z_anion):
# '''
# Return the C^\Phi coefficient for the Pitzer ion interaction model.
#
# .. math:: C_MX = C^\Phi / 2 \sqrt( \abs(z_+ z_-))
#
# Parameters
# ----------
# C_phi: number
# The C_phi parameter for the Pitzer ion interaction model.
# z_cation, z_anion: int
# The formal charge on the cation and anion, respectively
#
# Returns
# -------
# float
# The C_MX parameter for the Pitzer ion interaction model.
#
# References
# ----------
# Kim, H., & Jr, W. F. (1988).
# Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 25 degree C. 1. Single salt parameters.
# Journal of Chemical and Engineering Data, (2), 177-184.
#
# May, P. M., Rowland, D., Hefter, G., & Königsberger, E. (2011).
# A Generic and Updatable Pitzer Characterization of Aqueous Binary Electrolyte Solutions at 1 bar and 25 °C.
# Journal of Chemical & Engineering Data, 56(12), 5066-5077. doi:10.1021/je2009329
# '''
#
# coeff = C_phi / ( 2 * abs(z_cation * z_anion) ** 0.5 )
# return coeff * ureg.Quantity('kg ** 2 /mol ** 2')
[docs]
def _pitzer_log_gamma(
ionic_strength,
molality,
B_MX,
B_phi,
C_phi,
z_cation,
z_anion,
nu_cation,
nu_anion,
temperature="25 degC",
b=ureg.Quantity(1.2, "kg**0.5/mol**0.5"),
):
r"""
Returns the natural logarithm of the binary activity coefficient calculated by the Pitzer
ion interaction model.
.. math::
\ln \gamma_{MX} = -|z_+ z_-| A^{Phi} ( \frac{I ^ {0.5}}{(1 + b I ^ {0.5})} + \frac{2}{b} \ln{(1 + b I ^ {0.5})} )+ \frac{m (2 \nu_+ \nu_-)}{(\nu_+ + \nu_-)} (B_{MX} + B_{MX}^\Phi) + \frac{m^2(3 (\nu_+ \nu_-)^{1.5}}{(\nu_+ + \nu_-))} C_{MX}^\Phi
Args:
ionic_strength (Quantity): The ionic strength of the parent solution, mol/kg.
molality (Quantity): The concentration of the salt, mol/kg.
B_MX, B_phi, C_phi (Quantity): Calculated parameters for the Pitzer ion interaction model.
z_cation, z_anion (int): The formal charge on the cation and anion, respectively.
nu_cation, nu_anion (int): The stoichiometric coefficient of the cation and anion in the salt.
temperature (str, Quantity): String representing the temperature of the solution. Defaults to '25 degC' if not specified.
b (number, optional): Coefficient. Usually set equal to 1.2 kg ** 0.5 / mol ** 0.5 and considered independent of temperature and pressure.
Returns:
float: The natural logarithm of the binary activity coefficient calculated by the Pitzer ion interaction model.
References:
Kim, H., & Jr, W. F. (1988). Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 25 degree C. 1. Single salt parameters.
Journal of Chemical and Engineering Data, (2), 177-184.
May, P. M., Rowland, D., Hefter, G., & Königsberger, E. (2011). A Generic and Updatable Pitzer Characterization of Aqueous Binary Electrolyte Solutions at 1 bar and 25 °C.
Journal of Chemical & Engineering Data, 56(12), 5066-5077. doi:10.1021/je2009329
"""
first_term = (
-1
* abs(z_cation * z_anion)
* _debye_parameter_osmotic(temperature)
* (ionic_strength**0.5 / (1 + b * ionic_strength**0.5) + 2 / b * math.log(1 + b * ionic_strength**0.5))
)
second_term = 2 * molality * nu_cation * nu_anion / (nu_cation + nu_anion) * (B_MX + B_phi)
third_term = 3 * molality**2 * (nu_cation * nu_anion) ** 1.5 / (nu_cation + nu_anion) * C_phi
return first_term + second_term + third_term
[docs]
def get_osmotic_coefficient_pitzer(
ionic_strength,
molality,
alpha1,
alpha2,
beta0,
beta1,
beta2,
C_phi,
z_cation,
z_anion,
nu_cation,
nu_anion,
temperature="25 degC",
b=1.2,
):
"""
Return the osmotic coefficient of water in an electrolyte solution according to the Pitzer model.
Args:
ionic_strength (Quantity): The ionic strength of the parent solution, mol/kg.
molality (Quantity): The molal concentration of the parent salt, mol/kg.
alpha1, alpha2 (number): Coefficients for the Pitzer model. This function assigns the coefficients proper units of kg ** 0.5 / mol ** 0.5 after they are entered.
beta0, beta1, beta2, C_phi: Coefficients for the Pitzer model. These ion-interaction parameters are specific to each salt system.
z_cation, z_anion (int): The formal charge on the cation and anion, respectively.
nu_cation, nu_anion (int): The stoichiometric coefficient of the cation and anion in the salt.
temperature (str, Quantity): String representing the temperature of the solution. Defaults to '25 degC' if not specified.
b (number, optional): Coefficient. Usually set equal to 1.2 and considered independent of temperature and pressure. If provided, this coefficient is assigned proper units of kg ** 0.5 / mol ** 0.5 after entry.
Returns:
Quantity: The osmotic coefficient of water, dimensionless.
Examples:
Experimental value according to Beyer and Stieger reference is 1.3550
>>> get_osmotic_coefficient_pitzer(10.175*ureg.Quantity('mol/kg'),10.175*ureg.Quantity('mol/kg'),1,0.5,-.0181191983,-.4625822071,.4682,.000246063,1,-1,1,1,b=1.2)
1.3552 ...
Experimental value according to Beyer and Stieger reference is 1.084
>>> get_osmotic_coefficient_pitzer(5.6153*ureg.Quantity('mol/kg'),5.6153*ureg.Quantity('mol/kg'),3,0.5,0.0369993,0.354664,0.0997513,-0.00171868,1,-1,1,1,b=1.2)
1.0850 ...
Notes: the examples below are for comparison with experimental and modeling data presented in the May et al reference below.
10 mol/kg ammonium nitrate. Estimated result (from graph) = 0.62
>>> get_osmotic_coefficient_pitzer(10*ureg.Quantity('mol/kg'),10*ureg.Quantity('mol/kg'),2,0,-0.01709,0.09198,0,0.000419,1,-1,1,1,b=1.2)
0.6143 ...
5 mol/kg ammonium nitrate. Estimated result (from graph) = 0.7
>>> get_osmotic_coefficient_pitzer(5*ureg.Quantity('mol/kg'),5*ureg.Quantity('mol/kg'),2,0,-0.01709,0.09198,0,0.000419,1,-1,1,1,b=1.2)
0.6925 ...
18 mol/kg ammonium nitrate. Estimated result (from graph) = 0.555
>>> get_osmotic_coefficient_pitzer(18*ureg.Quantity('mol/kg'),18*ureg.Quantity('mol/kg'),2,0,-0.01709,0.09198,0,0.000419,1,-1,1,1,b=1.2)
0.5556 ...
References:
Scharge, T., Munoz, A.G., and Moog, H.C. (2012). Activity Coefficients of Fission Products in Highly Salinary Solutions of Na+, K+, Mg2+, Ca2+, Cl-, and SO42- : Cs+.
/Journal of Chemical& Engineering Data (57), p. 1637-1647.
Kim, H., & Jr, W. F. (1988). Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 25 degree C. 1. Single salt parameters.
Journal of Chemical and Engineering Data, (2), 177-184.
May, P. M., Rowland, D., Hefter, G., & Königsberger, E. (2011). A Generic and Updatable Pitzer Characterization of Aqueous Binary Electrolyte Solutions at 1 bar and 25 °C.
Journal of Chemical & Engineering Data, 56(12), 5066-5077. doi:10.1021/je2009329
Beyer, R., & Steiger, M. (2010). Vapor Pressure Measurements of NaHCOO + H 2 O and KHCOO + H 2 O from 278 to 308 K and Representation with an Ion Interaction (Pitzer) Model.
Journal of Chemical & Engineering Data, 55(2), 830-838. doi:10.1021/je900487a
See Also:
:func:`_debye_parameter_activity`
:func:`_pitzer_B_MX`
:func:`_pitzer_B_phi`
:func:`_pitzer_log_gamma`
"""
# assign proper units to alpha1, alpha2, and b
alpha1 = ureg.Quantity(alpha1, "kg ** 0.5 / mol ** 0.5")
alpha2 = ureg.Quantity(alpha2, "kg ** 0.5 / mol ** 0.5")
b = ureg.Quantity(b, "kg ** 0.5 / mol ** 0.5")
C_phi = ureg.Quantity(C_phi, "kg ** 2 /mol ** 2")
B_phi = ureg.Quantity(_pitzer_B_phi(ionic_strength, alpha1, alpha2, beta0, beta1, beta2), "kg/mol")
first_term = 1 - _debye_parameter_osmotic(temperature) * abs(z_cation * z_anion) * ionic_strength**0.5 / (
1 + b * ionic_strength**0.5
)
second_term = molality * 2 * nu_cation * nu_anion / (nu_cation + nu_anion) * B_phi
third_term = molality**2 * (2 * (nu_cation * nu_anion) ** 1.5 / (nu_cation + nu_anion)) * C_phi
return first_term + second_term + third_term