Introduction
pytzer.model executes the Pitzer model to calculate solute and solvent activity and osmotic coefficients.
Function inputs
Many of these functions have a common set of inputs: mols, ions, tempK, cflib and Izero.
The first three of these inputs can be generated from an input file by pytzer.io.getmols and their formats are described in the relevant documentation. Throughout pytzer, when we refer to a variable called ions we are including any neutral species in the solution.
The final compulsory input is a cflib (coefficient library), which defines the set of interaction coefficients to use in the model, as described on the relevant page.
Izero is an optional input with a default value of False. In this case, a full Pitzer model is executed. If Izero is instead changed to True, then only neutral-only interactions are evaluated: this is the setting to use for solutions with zero ionic strength. If you try to pass a zero-ionic-strength solution through the full model, a nan is returned along with lots of divide-by-zero warnings. You must split up your own input data and run the function twice, if you have both types of solution.
All of the usage examples below assume that you have first imported pytzer as pz:
import pytzer as pz
Excess Gibbs energy
From a physicochemical perspective, the excess Gibbs energy of a solution ($G_{ex}$) is the master variable from which many other properties are - literally - derived. Following Pitzer (1991), Eqs. (59) and (F-5), and Clegg et al. (1994), Eq. (AI1):
$$\frac{G_{ex}}{w_wRT} = f_G + 2 \sum_c \sum_a m_c m_a (B_{ca} + Z C_{ca}^{T})$$ $$+ \sum_i \sum_{i'} m_i m_{i'} \Bigl( 2 \Theta_{ii'} + \sum_j m_j \psi_{ii'j} \Bigr)$$ $$+ \sum_n m_n \Bigl(2 \sum_i m_i \lambda_{ni} + 2 \sum_{n'} m_{n'} \lambda_{nn'} + m_n \lambda_{nn} \Bigr)$$ $$+ \sum_n \sum_c \sum_a m_n m_c m_a \zeta_{nca} + \sum_n m_n^3 \mu_{nnn} $$
On the left hand side, we have the excess Gibbs energy ($G_{ex}$) divided by the universal gas constant ($R$ in J·mol·K−1; defined in pytzer.constants), temperature ($T$ in K), and mass of pure water ($w_w$ in kg, set to unity).
The first term on the right ($f_G$) is the Debye-Hückel approximation, a function of ionic strength evaluated by the function fG (see below). It includes the Debye-Hückel limiting slope $A_\phi$, which is evaluated using a function from pytzer.coeffs.
The Pitzer model then adds a series of corrections to this approximation for each dissolved cation ($c$), anion ($a$) and neutral component ($n$). In the equation above, $i$ refers to any ion and $j$ refers to an ion of the opposite sign charge. The term $i'$ refers to a different ion from $i$, but with the same sign charge. Similarly, $n'$ refers to a different neutral component from $n$.
The $m_x$ terms indicate the molality of component $x$, in mol·kg−1.
The $B_{ca}$ and $C_{ca}^T$ terms account for cation-anion interactions, and are evaluated by the functions B and CT (see below). $B_{ca}$ depends on the empirical coefficients $\beta_0$, $\beta_1$, $\beta_2$, $\alpha_1$ and $\alpha_2$, and $C_{ca}^{T}$ on $C_0$, $C_1$ and $\omega$. $Z$ is defined by Pitzer (1991), Eq. (66):
$$Z = \sum_i m_i |z_i|$$
where $z_i$ is the charge on ion $i$.
The $\Theta_{ii'}$ terms account for cation-cation and anion-anion interactions. As defined by Pitzer (1991), Eq. (B-6):
$$\Theta_{ii'} = \theta_{ii'} + ^E\theta_{ii'}$$
where the $^E\theta_{ii'}$ term is a function of ionic strength, evaluated by etheta (see below).
Finally, the "Greek letter terms" ($\beta_0$, $\beta_1$, $\beta_2$, $\alpha_1$, $\alpha_2$, $C_0$, $C_1$, $\omega$, $\theta_{ii'}$, $\psi_{ii'j}$, $\lambda_{nx}$, $\zeta_{nca}$ and $\mu_{nnn}$) are empirical coefficients, different for each combination of ions and neutral species, functions of temperature and sometimes pressure, defined in pytzer.coeffs.
In pytzer, the excess Gibbs energy is the only physicochemical equation that is actually explicitly written out. All other properties are determined by taking the appropriate differential of the excess Gibbs energy. These differentials are determined automatically, by autograd.
.Gex_nRT
Gex_nRT = pz.model.Gex_nRT(mols, ions, tempK, cflib, Izero=False)
Evaluates $G_{ex}/w_wRT$, as defined above.
Activity and osmotic coefficients
The natural logarithm of the activity coefficient ($\ln \gamma_x$) of any dissolved component of a solution is the first differential of $G_{ex}/w_wRT$ with respect to the component's molality, following Pitzer (1991), Eq. (34):
$$\ln \gamma_x = \frac{\partial (G_{ex}/w_wRT)}{\partial m_x} $$
The osmotic coefficient of a solution ($\phi$) is related to the first differential of $G_{ex}/w_wRT$ with respect to $w_w$, following Pitzer (1991), Eq. (35):
$$\phi = 1 - \frac{\partial G_{ex}/\partial w_w}{RT} \sum_x m_x$$
where $x$ includes all ions and neutral components.
The solvent (i.e. water) activity ($a_w$) is related to the osmotic coefficient by Pitzer (1991), Eq. (28):
$$\phi = - \frac{\ln a_w}{M_w \sum_x m_x}$$
where $M_w$ is the molar mass of water, in kg·mol−1 (defined in pytzer.constants).
To evaluate $\gamma_x$ and $\phi$, pytzer evaluates the appropriate differentials of $G_{ex}/w_wRT$ by using autograd to differentiate the Gex_nRT function.
.acfs / .ln_acfs
ln_acfs = pz.model.ln_acfs(mols, ions, tempK, cflib, Izero=False)
acfs = pz.model.acfs (mols, ions, tempK, cflib, Izero=False)
Returns a matrix of activity coefficients ($\gamma_x$, acfs) or their natural logarithm ($\ln \gamma_x$, ln_acfs) of the same size and shape as input mols. Each activity coefficient is for the same ion and solution composition as the corresponding input molality.
.ln_acf2ln_acf_MX
ln_acf_MX = pz.model.ln_acf2ln_acf_MX(ln_acfM, ln_acfX, nM, nX)
Combines the natural logarithms of the activity coefficients of a cation ($\ln \gamma_c$, ln_acfM) and anion ($\ln \gamma_a$, ln_acfX) into a mean activity coefficient of an electrolyte ($\ln \gamma_\pm$, ln_acf_MX) with stoichiometric ratio between cation and anion of nM:nX.
.osm
osm = pz.model.osm(mols, ions, tempK, cflib, Izero=False)
Calculates the osmotic coefficient ($\phi$) for each input solution composition.
.aw / .lnaw
lnaw = pz.model.lnaw(mols, ions, tempK, cflib, Izero=False)
aw = pz.model.aw (mols, ions, tempK, cflib, Izero=False)
Calculates the water activity ($a_w$) or its natural logarithm for each input solution composition.
.osm2aw
aw = pz.model.osm2aw(mols, osm)
Converts an osmotic coefficient ($\phi$, osm) into a water activity ($a_w$, aw).
.aw2osm
aw = pz.model.aw2osm(mols, aw)
Converts a water activity ($a_w$, aw) into an osmotic coefficient ($\phi$, osm).
Pitzer model subfunctions
pytzer.model breaks down the full Pitzer model equation into some component subfunctions for clarity.
.Istr
Calculates the ionic strength of the solution ($I$, I), following Pitzer (1991), Eq. (11):
$$I = \frac{1}{2} \sum_i m_i z_i^2$$
I = pz.model.Istr(mols, zs)
Input zs is a list of the charge on each ion, which can be generated from ions using pytzer.props.charges.
.fG
The Debye-Hückel approximation of the excess Gibbs energy. From Pitzer (1991), Eq. (48):
$$f_G = \frac{4 I A_\phi}{-b} \ln (1 + b\sqrt{I})$$
where $b$, here and hereafter, is equal to 1.2 (mol·K−1)1/2 (as defined in pytzer.constants).
fG = pz.model.fG(tempK, I, cflib)
.g
The function $g$, following Pitzer (1991), Eq. (50):
$$g = 2[1 - (1 + x) \exp(-x)] / x^2$$
g = pz.model.g(x)
.h
The function $h$, following Clegg et al. (1994), Eq. (AI15):
$$h = \{ 6 - [6 + x (6 + 3x + x^2)] \exp(-x) \} / x^4$$
h = pz.model.h(x)
.B
The function $B_{ca}$, following Pitzer (1991), Eq. (49):
$$B_{ca} = \beta_0 + \beta_1 g(\alpha_1 \sqrt{I}) + \beta_2 g(\alpha_2 \sqrt{I})$$
where $\beta_0$, $\beta_1$, $\beta_2$, $\alpha_1$ and $\alpha_2$ take different values for each $ca$ combination, as defined by the functions in pytzer.coeffs.
B = pz.model.B(tempK, I, cflib, iset)
where input iset is a string that references the appropriate entry in cflib.bC for the ion pair (e.g. 'Na-Cl' for the Na+-Cl− interaction; see how a CoefficientDictionary works for details).
.CT
The function $C_{ca}^T$, following Clegg et al. (1994) Eq. (AI10):
$$C_{ca}^T = C_0 + 4 C_1 h(\omega \sqrt{I})$$
where $C_0$, $C_1$ and $\omega$ take different values for each $ca$ combination, as defined by the functions in pytzer.coeffs.
CT = pz.model.CT(tempK, I, cflib, iset)
where input iset is a string that references the appropriate entry in cflib.bC for the ion pair (e.g. 'Na-Cl' for the Na+-Cl− interaction; see how a CoefficientDictionary works for details).
.xij
The variable $x_{ii'}$, following Pitzer (1991), Eq. (B-14):
$$x_{ii'} = 6 z_i z_{i'} A_\phi \sqrt{I}$$
xij = pz.model.xij(tempK, I, z0, z1, cflib)
where z0 and z1 are the charges on ions $i$ and $i'$ respectively (i.e. $z_i$ and $z_{i'}$).
.etheta
The function $^E\theta_{ii'}$, following Pitzer (1991), Eq. (B-15):
$$^E\theta_{ii'} = \frac{z_i z_i'}{4 I} \Bigl[J(x_{ii'}) - \frac{1}{2} J(x_{ii}) - \frac{1}{2} J(x_{i'i'})\Bigr]$$
This is only evaluated when $z_i \neq z_{i'}$. Otherwise, it is equal to zero.
References
Clegg, S. L., Rard, J. A., and Pitzer, K. S. (1994). Thermodynamic properties of 0–6 mol kg–1 aqueous sulfuric acid from 273.15 to 328.15 K. J. Chem. Soc., Faraday Trans. 90, 1875–1894. doi:10.1039/FT9949001875.
Pitzer, K. S. (1991). “Ion Interaction Approach: Theory and Data Correlation,” in Activity Coefficients in Electrolyte Solutions, 2nd Edition, ed. K. S. Pitzer (CRC Press, Florida, USA), 75–153.