Cation exchange competition among Na+, K+ and Ca2+

Cation exchange competition among Na+, K+ and Ca2+#

In this tutorial, we show how Reaktoro can be used to model ion exchange processes. We use the PHREEQC database phreeqc.dat for this purpose. We assume that there is an ion exchanger X- on which the aqueous cations Na+, K+ and Ca2+ can bind and form the exchange phases NaX, KX and CaX2 respectively. We are interested in calculating the amounts of these phases as the concentrations of K+ and Ca2+ vary in the aqueous solution.

First, we define a chemical system representative of the process we want to model.

import numpy as np
from reaktoro import *
import pandas as pd

# Initialize the PHREEQC database
db = PhreeqcDatabase("phreeqc.dat")

# Define an aqueous phase
solution = AqueousPhase("H2O Na+ Cl- H+ OH- K+ Ca+2 Mg+2")
solution.set(ActivityModelPhreeqc(db))

# Define an ion exchange phase
exchange = IonExchangePhase("NaX KX CaX2")
exchange.set(ActivityModelIonExchangeGainesThomas())

# Create the chemical system
system = ChemicalSystem(db, solution, exchange)

As we are going to perform a sequence of chemical equilibrium calculations, we define an object of EquilibriumSolver as well as IonExchangeProps and AqueousProps using the chemical system. These objects will be updated after every equilibrium calculation.

# Define the equilibrium solver
solver = EquilibriumSolver(system)

# Define exchange, aqueous, and chemical properties
exprops = IonExchangeProps(system)
aqprops = AqueousProps(system)

Note

Note that at the end of each equilibrium calculation, state.props() can be used to access a ChemicalProps object with the chemical properties of the system evaluated in the last iteration of the algorithm.

Next, we define the range of Ca2+ and K+ initial amounts as well as auxiliary pandas.DataFrame to collect the results of the ion exchange simulations.

# Output dataframe
df = pd.DataFrame(columns=[
    "amount_K", 
    "amount_Ca",
    "amount_K_ion", 
    "amount_Na_ion", 
    "amount_Ca_ion",
    "amount_KX", 
    "amount_NaX", 
    "amount_CaX2",
    "pH"
])

# Sampling arrays storing the amounts of ions
steps = 21

mols_K  = np.linspace(0.1, 0.0, num=steps)
mols_Ca = np.linspace(0.0, 0.5, num=steps)

We’ll now perform a sequence of equilibrium calculations by varying the initial amounts of ions Ca2+ and K+ ions (taken from the arrays mols_Ca and mols_K, respectively). At the end of each calculation, we’ll extract the following properties from the computed chemical state:

  • amount of K,

  • amount of Ca,

  • amount of K+,

  • amount of Na+,

  • amount of Ca2+,

  • amount of KX,

  • amount of NaX,

  • amount of CaX2,

  • pH.

for mol_K, mol_Ca in zip(mols_K, mols_Ca):

    # Define initial chemical state
    state = ChemicalState(system)
    state.setTemperature(25, "celsius")
    state.setPressure(1, "atm")
    state.set("H2O" , 1.0   , "kg")
    state.set("NaX" , 0.4   , "mol")
    state.set("K+"  , mol_K , "mol")
    state.set("Ca+2", mol_Ca, "mol")

    # Equilibrate the chemical state
    res = solver.solve(state)
    
    # Stop if the equilibration did not converge or failed
    assert res.succeeded()

    # Update exchange and aqueous properties
    chemprops = state.props()

    exprops.update(chemprops)
    aqprops.update(chemprops)

    # Update output arrays:
    df.loc[len(df)] = [
        chemprops.elementAmount('K'),   # amount_K
        chemprops.elementAmount('Ca'),  # amount_Ca
        state.speciesAmount('K+'),      # amount_K+
        state.speciesAmount('Na+'),     # amount_Na+
        state.speciesAmount('Ca+2'),    # amount_Ca+2
        state.speciesAmount('KX'),      # amount_KX
        state.speciesAmount('NaX'),     # amount_NaX
        state.speciesAmount('CaX2'),    # amount_CaX2
        aqprops.pH()                    # pH
    ]

The data collected in the previous calculations can be inspected by printing the pandas.DataFrame object df:

df
amount_K amount_Ca amount_K_ion amount_Na_ion amount_Ca_ion amount_KX amount_NaX amount_CaX2 pH
0 0.1 2e-16 0.00579097 0.094209 1e-16 0.094209 0.305791 1e-16 6.98671
1 0.095 0.025 0.00873692 0.134683 0.000790228 0.0862631 0.265317 0.0242098 6.98361
2 0.09 0.05 0.011732 0.171596 0.00333599 0.078268 0.228404 0.046664 6.98079
3 0.085 0.075 0.0144722 0.202807 0.00886037 0.0705278 0.197193 0.0661396 6.97806
4 0.08 0.1 0.0166446 0.227346 0.0180048 0.0633554 0.172654 0.0819952 6.97538
5 0.075 0.125 0.0181137 0.245914 0.0304859 0.0568863 0.154086 0.0945141 6.97277
6 0.07 0.15 0.0189176 0.259953 0.0455647 0.0510824 0.140047 0.104435 6.97029
7 0.065 0.175 0.0191628 0.270776 0.0625305 0.0458372 0.129224 0.112469 6.96797
8 0.06 0.2 0.0189568 0.279337 0.0808531 0.0410432 0.120663 0.119147 6.96583
9 0.055 0.225 0.0183882 0.286279 0.100166 0.0366118 0.113721 0.124834 6.96384
10 0.05 0.25 0.0175251 0.292034 0.120221 0.0324749 0.107966 0.129779 6.96199
11 0.045 0.275 0.0164195 0.296895 0.140843 0.0285805 0.103105 0.134157 6.96028
12 0.04 0.3 0.0151111 0.301068 0.161911 0.0248889 0.0989323 0.138089 6.95869
13 0.035 0.325 0.0136307 0.304698 0.183336 0.0213693 0.0953023 0.141664 6.95721
14 0.03 0.35 0.0120026 0.307892 0.205053 0.0179974 0.0921077 0.144947 6.95582
15 0.025 0.375 0.010246 0.310732 0.227011 0.014754 0.0892683 0.147989 6.95452
16 0.02 0.4 0.00837679 0.313277 0.249173 0.0116232 0.0867231 0.150827 6.9533
17 0.015 0.425 0.00640769 0.315575 0.271508 0.00859231 0.0844246 0.153492 6.95214
18 0.01 0.45 0.00434943 0.317665 0.293993 0.00565057 0.0823352 0.156007 6.95105
19 0.005 0.475 0.00221099 0.319575 0.316607 0.00278901 0.0804249 0.158393 6.95002
20 2e-16 0.5 1e-16 0.32133 0.339335 1e-16 0.0786695 0.160665 6.94904

We can now construct an interactive plot from this data:

from reaktplot import *

fig = Figure()
fig.title("AMOUNTS OF EXCHANGE SPECIES KX, CaX<sub>2</sub>, NaX")
fig.xaxisTitle('Ca<sup>2+</sup> [molal]')
fig.yaxisTitle('Amount [mol]')

fig.drawLineWithMarkers(df["amount_Ca"], df["amount_KX"], name="KX")
fig.drawLineWithMarkers(df["amount_Ca"], df["amount_CaX2"], name="CaX<sub>2</sub>")
fig.drawLineWithMarkers(df["amount_Ca"], df["amount_NaX"], name="NaX")

fig.show()

This figure shows that as the concentration of Ca2+ increases from 0.0 to 0.5 molal, while the concentration of K+ decreases from 0.1 to 0.0 molal, there is a tendency for the cations Na+ and K+ to unbind from their respective NaX and KX exchange phases. In the process, the cations Ca2+ tend to bind to the available ion exchange species X-, with the exchange phase CaX2 becoming predominant among the others.

The next figure shows how the concentrations of Na+, K+ and Ca2+ behave in the process.

fig = Figure()
fig.title("CONCENTRATIONS OF IONS K<sup>+</sup>, Ca<sup>2+</sup>, Na<sup>+</sup>")
fig.xaxisTitle('Ca<sup>2+</sup> [molal]')
fig.yaxisTitle('Concentration [molal]')

fig.drawLineWithMarkers(df["amount_Ca"], df["amount_K_ion"], name="K<sup>+</sup>")
fig.drawLineWithMarkers(df["amount_Ca"], df["amount_Ca_ion"], name="Ca<sup>2+</sup>")
fig.drawLineWithMarkers(df["amount_Ca"], df["amount_Na_ion"], name="Na<sup>+</sup>")

fig.show()

Finally, we present how the pH of the solution changes in the process, whose changes are not significant.

fig = Figure()
fig.title("AQUEOUS SOLUTION pH")
fig.xaxisTitle('Ca<sup>2+</sup> [molal]')
fig.yaxisTitle('pH')

fig.drawLineWithMarkers(df["amount_Ca"], df["pH"], name="")

fig.show()