"""
pyEQL functions that take Solution objects as inputs or return Solution objects.
:copyright: 2013-2024 by Ryan S. Kingsbury
:license: LGPL, see LICENSE for more details.
"""
import logging
import math
from pyEQL import Solution, ureg
logger = logging.getLogger(__name__)
[docs]
def gibbs_mix(solution1: Solution, solution2: Solution):
r"""
Return the Gibbs energy change associated with mixing two solutions.
Args:
solution1: a solution to be mixed.
solution2: a solution to be mixed.
Returns:
The change in Gibbs energy associated with complete mixing of the
Solutions, in Joules.
Notes:
The Gibbs energy of mixing is calculated as follows
.. math::
\Delta_{mix} G = \sum_i {(n_c + n_d) R T \ln a_b} - \sum_i {n_c R T \ln a_c} - \sum_i {n_d R T \ln a_d}
Where :math:`n` is the number of moles of substance, :math:`T` is the temperature in kelvin,
and subscripts :math:`b`, :math:`c`, and :math:`d` refer to the concentrated, dilute, and blended
Solutions, respectively.
Note that dissociated ions must be counted as separate components,
so a simple salt dissolved in water is a three component solution (cation,
anion, and water).
References:
Koga, Yoshikata, 2007. Solution Thermodynamics and its Application to Aqueous Solutions:
A differential approach. Elsevier, 2007, pp. 23-37.
"""
concentrate = solution1
dilute = solution2
blend = solution1 + solution2
term_list = {concentrate: 0, dilute: 0, blend: 0}
# calculate the entropy change and number of moles solute for each solution
for solution in term_list:
for solute in solution.components:
if solution.get_amount(solute, "fraction") != 0:
term_list[solution] += solution.get_amount(solute, "mol") * math.log(solution.get_activity(solute))
return (ureg.R * blend.temperature.to("K") * (term_list[blend] - term_list[concentrate] - term_list[dilute])).to(
"J"
)
[docs]
def entropy_mix(solution1: Solution, solution2: Solution):
r"""
Return the ideal mixing entropy associated with mixing two solutions.
Parameters:
solution1, solution2: The two solutions to be mixed.
Returns:
The ideal mixing entropy associated with complete mixing of the
Solutions, in Joules.
Notes:
The ideal entropy of mixing is calculated as follows
.. math::
\Delta_{mix} S = \sum_i {(n_c + n_d) R T \ln x_b} - \sum_i {n_c R T \ln x_c} - \sum_i {n_d R T \ln x_d}
Where :math:`n` is the number of moles of substance, :math:`T` is the temperature in kelvin,
and subscripts :math:`b`, :math:`c`, and :math:`d` refer to the concentrated, dilute, and blended
Solutions, respectively.
Note that dissociated ions must be counted as separate components,
so a simple salt dissolved in water is a three component solution (cation,
anion, and water).
References:
Koga, Yoshikata, 2007. *Solution Thermodynamics and its Application to Aqueous Solutions:
A differential approach.* Elsevier, 2007, pp. 23-37.
"""
concentrate = solution1
dilute = solution2
blend = solution1 + solution2
term_list = {concentrate: 0, dilute: 0, blend: 0}
# calculate the entropy change and number of moles solute for each solution
for solution in term_list:
for solute in solution.components:
if solution.get_amount(solute, "fraction") != 0:
term_list[solution] += solution.get_amount(solute, "mol") * math.log(
solution.get_amount(solute, "fraction")
)
return (ureg.R * blend.temperature.to("K") * (term_list[blend] - term_list[concentrate] - term_list[dilute])).to(
"J"
)
[docs]
def donnan_eql(solution: Solution, fixed_charge: str):
r"""
Return a solution object in equilibrium with fixed_charge.
Args:
solution : Solution object
The external solution to be brought into equilibrium with the fixed
charges
fixed_charge : str quantity
String representing the concentration of fixed charges, including sign.
May be specified in mol/L or mol/kg units. e.g. '1 mol/kg'
Returns:
A Solution that has established Donnan equilibrium with the external
(input) Solution
Notes:
The general equation representing the equilibrium between an external
electrolyte solution and an ion-exchange medium containing fixed charges
is
.. math::
\big(\frac{a_{-}}{\bar a_{-}} \big)^(\frac{1}{z_{-})
\big(\frac{\bar a_{+}}{a_{+}}\big)^(\frac{1}{z_{+})
\exp(\frac{\Delta \pi \bar V}{RT z_{+} \nu_{+}})
Where subscripts :math:`+` and :math:`-` indicate the cation and anion, respectively,
the overbar indicates the membrane phase,
:math:`a` represents activity, :math:`z` represents charge, :math:`\nu` represents the stoichiometric
coefficient, :math:`V` represents the partial molar volume of the salt, and
:math:`\Delta \pi` is the difference in osmotic pressure between the membrane and the
solution phase.
In addition, electroneutrality must prevail within the membrane phase:
.. math:: \bar C_{+} z_{+} + \bar X + \bar C_{-} z_{-} = 0
Where :math:`C` represents concentration and :math:`X` is the fixed charge concentration
in the membrane or ion exchange phase.
This function solves these two equations simultaneously to arrive at the
concentrations of the cation and anion in the membrane phase. It returns
a solution equal to the input solution except that the concentrations of
the predominant cation and anion have been adjusted according to this
equilibrium.
NOTE that this treatment is only capable of equilibrating a single salt.
This salt is identified by the get_salt() method.
References:
Strathmann, Heiner, ed. *Membrane Science and Technology* vol. 9, 2004. Chapter 2, p. 51.
http://dx.doi.org/10.1016/S0927-5193(04)80033-0
See Also:
get_salt()
"""
# identify the salt
salt = solution.get_salt()
# convert fixed_charge in to a quantity
fixed_charge = ureg.Quantity(fixed_charge)
# identify variables from the external solution
conc_cation_soln = solution.get_amount(salt.cation, str(fixed_charge.units))
conc_anion_soln = solution.get_amount(salt.anion, str(fixed_charge.units))
act_cation_soln = solution.get_activity(salt.cation)
act_anion_soln = solution.get_activity(salt.anion)
z_cation = salt.z_cation
z_anion = salt.z_anion
nu_cation = salt.nu_cation
# get the partial molar volume for the salt, or calculate it from the ions
# TODO - consider how to incorporate pitzer parameters
molar_volume = solution.get_property(salt.formula, "size.molar_volume")
if molar_volume is None:
cation_vol = solution.get_property(salt.cation, "size.molar_volume")
anion_vol = solution.get_property(salt.anion, "size.molar_volume")
if cation_vol is not None and anion_vol is not None:
molar_volume = cation_vol + anion_vol
else:
logger.error("Required partial molar volume information not available. Aborting.")
return None
# initialize the equilibrated solution - start with a direct copy of the
# input / external solution
donnan_soln = solution.copy()
# do nothing if either of the ion concentrations is zero
if conc_cation_soln.magnitude == 0 or conc_anion_soln.magnitude == 0:
return donnan_soln
# define a function representing the donnan equilibrium as a function
# of the two unknown activities to feed to the nonlinear solver
# the stuff in the term below doesn't change on iteration, so calculate it up-front
# assign it the correct units and extract the magnitude for a performance gain
exp_term = (molar_volume / (ureg.R * solution.temperature * z_cation * nu_cation)).to("1/Pa").magnitude
def donnan_solve(x):
"""Where x is the magnitude of co-ion concentration."""
# solve for the counter-ion concentration by enforcing electroneutrality
# using only floats / ints here instead of quantities helps performance
if fixed_charge.magnitude >= 0:
# counter-ion is the anion
conc_cation_mem = x / abs(z_cation)
conc_anion_mem = -(conc_cation_mem * z_cation + fixed_charge.magnitude) / z_anion
elif fixed_charge.magnitude < 0:
# counter-ion is the cation
conc_anion_mem = x / abs(z_anion)
conc_cation_mem = -(conc_anion_mem * z_anion + fixed_charge.magnitude) / z_cation
# match the units given for fixed_charge
units = str(fixed_charge.units)
# set the cation and anion concentrations in the membrane phase equal
# to the current guess
donnan_soln.set_amount(salt.cation, str(conc_cation_mem) + units)
donnan_soln.set_amount(salt.anion, str(conc_anion_mem) + units)
# get the new concentrations and activities
act_cation_mem = donnan_soln.get_activity(salt.cation)
act_anion_mem = donnan_soln.get_activity(salt.anion)
# compute the difference in osmotic pressure
# using the magnitudes here helps performance
delta_pi = donnan_soln.osmotic_pressure.magnitude - solution.osmotic_pressure.magnitude
return (act_cation_mem / act_cation_soln) ** (1 / z_cation) * (act_anion_soln / act_anion_mem) ** (
1 / z_anion
) - math.exp(delta_pi * exp_term)
# solve the function above using one of scipy's nonlinear solvers
from scipy.optimize import brentq
# determine which ion concentration represents the co-ion
# call a nonlinear solver to adjust the concentrations per the donnan
# equilibrium, unless the membrane is uncharged
# the initial guess is to set the co-ion concentration in the membrane
# equal to that in the solution
if fixed_charge.magnitude > 0:
x = conc_cation_soln.magnitude
brentq(donnan_solve, 1e-10, x, xtol=0.001)
elif fixed_charge.magnitude < 0:
x = conc_anion_soln.magnitude
brentq(donnan_solve, 1e-10, x, xtol=0.001)
else:
pass
# return the equilibrated solution
return donnan_soln