Customizing the thermodynamic database

Types of data

• Columns 1–9 have character data.
  • Column 8 is named model; here are the two models that are most frequent in the default database:
    • HKF: Revised Helgeson-Kirkham-Flowers “equation-of-state” parameters for aqueous species
    • CGL: Heat capacity coefficients for crystalline, gaseous, and liquid species. The first three terms in the CGL heat capacity equation correspond to the Maier-Kelley equation for heat capacity (Maier and Kelley, 1932); the additional terms are useful for representing heat capacities of minerals (Robie and Hemingway, 1995) and organic gases and liquids (Helgeson et al., 1998).
• Columns 10–22 have numeric data.
  • Columns 10–14 have standard-state thermodynamic properties at 25 °C and 1 bar.
  • Columns 15–21 have parameters for calculating thermodynamic properties at other temperatures and pressures.
    • The columns are named by combining the the names of the HKF and CGL coefficients, separated by a dot.
  • Column 22 has the charge used in the HKF EOS or the maximum temperature for CGL species.
    • NOTE: The value of charge used in the HKF EOS (in particular, the g function for the temperature derivatives of the ω parameter (Shock et al., 1992)) is taken from this column and not from the chemical formula of the species.

Ranges of HKF and CGL models

To a first approximation, the revised HKF equations of state are applicable within the stability region of liquid water or the supercritical fluid with a density greater than 0.35 g/cm3, and not exceeding the ranges of 0 to 1000 °C and 1 to 5000 bar (see Shock et al., 1992 for details). There are two ways in which these limits are enforced in CHNOSZ:

• The default source of water properties (H2O92D Fortran subroutine modified from SUPCRT92) yields NA values beyonds these T and P ranges. Note that the Deep Earth Water (DEW) model is available to extend the applicable range to pressures of up to 60 kbar (6 GPa) (Sverjensky et al., 2014).
• subcrt() generates NA values beyond the density limit; the exceed.rhomin argument can be used to enable calculations at lower density in the supercritical region.

The upper temperature limit for validity of the CGL heat-capacity equation is a species-dependent parameter. This value is stored as a negative value in the T column of OBIGT and is used by subcrt() to issue a warning at temperatures beyond this limit.

Required and optional data

REQUIRED:

• All species need a name and a state.
  • The state can be one of aq, gas, or cr.
  • For minerals with higher-temperature polymorphs, they are named cr2, cr3, etc.
  • NOTE: cr stands for “crystalline”; this naming convention (which was inherited from SUPCRT92 data files) refers to any solid phases including amorphous SiO₂ and other minerals.
• A chemical formula is required to do almost anything useful in CHNOSZ (e.g. check reaction balancing with subcrt() and add species with basis() or species()).
  • NOTE: The name of inorganic aqueous species and CH₄ in OBIGT is the same as the chemical formula.
  • Most minerals, gases, liquids, and organic aqueous species have a name that is a common name. This permits a shortcut to identify commonly used species in subcrt().
  • For example, info("O2") refers to dissolved oxygen, while info("oxygen") or info("O2", "gas") refers to the gas.
• E_units needs to be defined to perform any calculations of thermodynamic properties.
  • The value can be J for Joules or cal for calories.

OPTIONAL: Everything else. Really, it depends on what you need. For instance, if you just want to use subcrt() to calculate log K of a reaction from ΔG° of species at 25 °C, then G is the only parameter that is needed.

OPTIONAL but useful:

• abbrv may be an abbreviation (e.g. Qtz for quartz). It is used by info() (together with name and formula) to look up species in the database.
• date is a timestamp for the data entry (YYYY-MM-DD format in the default OBIGT database).
• ref1 and ref2 are bibliographic reference keys. They have matching entries in extdata/OBIGT/refs.csv, which is used by thermo.refs() to display references, and in vignettes/OBIGT.bib, which is used in the OBIGT thermodynamic database vignette to produce a reference list.

NOTE: Other functions in CHNOSZ do not depend on date, ref1, and ref2, so you can put anything there that is convenient for you.

NA or 0?

If a character value (in Columns 1–9) or thermodynamic parameter (in Columns 10–14) is unknown, use NA. Note that a missing (blank) value in the file is treated as NA.

• Unknown values for character values (usually abbrv, date, ref1, or ref2) should be NA.
• If you have only two of G, H, and S, then the missing one should be NA.
  • Do NOT set a missing value of G, H, or S to 0. Zero is a numeric value that is incorrect except for very special cases.
  • NOTE: info() – and, by extension, subcrt() – “know” about the equation ΔG°_f = ΔH°_f - TΔS°_f and the entropies of the elements needed to calculate ΔS°_f from values of S in OBIGT. This equation is used to compute a missing value of G, H, or S from the other two, or to cross-check the values if all three are present for any species.
• If you don’t have Cp or V, then set it to NA.
  • If HKF or CGL parameters are present, they will be used to calculate Cp, so thermodynamic properties can be calculated at T > 25 °C.
  • If HKF or CGL parameters aren’t present, thermodynamic properties can’t be calculated at T > 25 °C (NAs will propagate to higher T).

If an “equation-of-state” parameter or heat capacity coefficient (Columns 15-21) is unknown, use 0.

• Furthermore, if you would like to assume that Cp or V is 0, then set it to 0.
  • Then, thermodynamic properties will be extrapolated to T > 25 °C and P > 1 bar assuming that Cp and V are 0.

More detail on the inner working of the functions: For both HKF and CGL, if at least one parameter for a species is provided, any NA values of the other parameters are taken to be zero. If all EOS parameters are NA, but values of Cp and/or V are present, they are assumed to be constants for extrapolating thermodynamic properties (e.g. ΔG°) as a function of temperature and pressure.

OOM scaling and info()

HKF parameters in the the CSV files and OBIGT data frame are scaled by order-of-magnitude (OOM) factors. For these parameters, OOM scaling is nearly always used in published data tables. See thermo() for details of the OOM scaling.

info() provides a simple user interface to the OBIGT database and is called by other functions in CHNOSZ to retrieve unscaled values from the database. This is a summary of its main features:

• Remove OOM scaling. This is used primarily by other functions in CHNOSZ to get a set of unscaled model parameters for calculating thermodynamic properties as a function of T and P.
• Extract the HKF or CGL parts of column names (only if all matching species have the same model).
• Calculate a missing one of G, H, or S if two of them are present.
• Cross-check G, H, and S if all of them are present, and print a message if the difference is above a threshold (see check.GHS()).
• Calculate a missing Cp or V from the model parameters, if possible.
• Cross-check Cp or V (if present) against the model parameters, if possible, and print a message if the difference is above a threshold (see check.EOS()).

NOTE: info() does NOT change the units of energy; the values it displays (including possibly calculated ones) correspond to the E_units for that species in OBIGT. On the other hand, subcrt() outputs values in the units previously selected with the function E.units().

add.OBIGT() with optional data files

The default database has parameters for many minerals from Berman (1988); a notable exception is sulfide minerals, which are from Helgeson et al. (1978). Besides the different literature sources in ref1, the model column indicates that a different model is used for these minerals (Berman equations or CGL).

info(info(c("quartz", "pyrite")))

## info.numeric: Cp° of pyrite(cr) is NA; set by EOS parameters to 14.84 cal K-1 mol-1

##        name abbrv formula state   ref1   ref2       date  model E_units       G
## 2073 quartz    Qz    SiO2    cr  Ber88   <NA> 2017-10-01 Berman       J -856288
## 2160 pyrite    Py    FeS2    cr HDNB78 RH95.7 1978-05-05    CGL     cal  -38293
##            H     S      Cp     V     a       b       c  d  e  f lambda    T
## 2073 -910700 41.46 44.7423 22.69    NA      NA      NA NA NA NA     NA   NA
## 2160  -41000 12.65 14.8425 23.94 17.88 0.00132 -305000  0  0  0      0 1015

Sometimes it is useful to load mineral data from the SUPCRT92 database, corresponding largely to the compilation by Helgeson et al. (1978). This can be done with add.OBIGT(). In this example we load SUPCRT92 data for just one mineral, quartz.

add.OBIGT("SUPCRT92", "quartz")

## add.OBIGT: read 1 rows; made 1 replacements, 0 additions [energy units: cal]

info(info("quartz"))

## info.numeric: Cp° of quartz(cr) is NA; set by EOS parameters to 10.63 cal K-1 mol-1

##        name abbrv formula state   ref1 ref2       date model E_units       G
## 2073 quartz   Qtz    SiO2    cr HDNB78 <NA> 1978-05-05   CGL     cal -204646
##            H    S      Cp      V     a      b       c d e f lambda   T
## 2073 -217650 9.88 10.6275 22.688 11.22 0.0082 -270000 0 0 0      0 848

Here we load all minerals available in the optional SUPCRT92 data file and then list the names. NOTE: suppressMessages() is used to suppress messages from info() about missing parameters, and unique() is used to list each mineral only once (because each polymorph has a separate entry).

iSUPCRT92 <- add.OBIGT("SUPCRT92")

## add.OBIGT: read 177 rows; made 65 replacements, 112 additions [energy units: cal]

unique(suppressMessages(info(iSUPCRT92))$name)

##   [1] "akermanite"          "albite"              "albite,high"        
##   [4] "albite,low"          "almandine"           "andalusite"         
##   [7] "andradite"           "annite"              "anorthite"          
##  [10] "anthophyllite"       "antigorite"          "aragonite"          
##  [13] "boehmite"            "brucite"             "Ca-Al-pyroxene"     
##  [16] "calcite"             "chrysotile"          "clinozoisite"       
##  [19] "coesite"             "corundum"            "cristobalite,alpha" 
##  [22] "cristobalite,beta"   "diaspore"            "diopside"           
##  [25] "dolomite"            "enstatite"           "epidote"            
##  [28] "fayalite"            "ferrosilite"         "fluorphlogopite"    
##  [31] "fluortremolite"      "forsterite"          "gehlenite"          
##  [34] "gibbsite"            "glaucophane"         "grossular"          
##  [37] "grunerite"           "hedenbergite"        "hematite"           
##  [40] "jadeite"             "K-feldspar"          "kaolinite"          
##  [43] "kyanite"             "lawsonite"           "lime"               
##  [46] "magnesite"           "magnetite"           "margarite"          
##  [49] "merwinite"           "monticellite"        "muscovite"          
##  [52] "paragonite"          "periclase"           "phlogopite"         
##  [55] "prehnite"            "pyrope"              "pyrophyllite"       
##  [58] "quartz"              "sillimanite"         "spinel"             
##  [61] "talc"                "tremolite"           "wollastonite"       
##  [64] "zoisite"             "rutile"              "aegerine"           
##  [67] "amesite,14A"         "amesite,7A"          "amorphous silica"   
##  [70] "analcime"            "analcime,dehydrated" "Ca-phillipsite"     
##  [73] "celadonite"          "chabazite"           "chalcedony"         
##  [76] "chloritoid"          "clinochlore,14A"     "clinochlore,7A"     
##  [79] "cordierite,dry"      "cordierite,hydrous"  "cristobalite"       
##  [82] "cronstedtite,7A"     "cummingtonite"       "daphnite,14A"       
##  [85] "daphnite,7A"         "dickite"             "dolomite,disordered"
##  [88] "dolomite,ordered"    "edenite"             "epidote,ordered"    
##  [91] "epistilbite"         "ferroedenite"        "ferrogedrite"       
##  [94] "ferropargasite"      "ferrotremolite"      "fluoredenite"       
##  [97] "fluorite"            "greenalite"          "halloysite"         
## [100] "hastingsite"         "heulandite"          "K-phillipsite"      
## [103] "kalsilite"           "larnite"             "laumontite"         
## [106] "leonhardite"         "magnesiohastingsite" "magnesioriebeckite" 
## [109] "microcline,maximum"  "minnesotaite"        "Na-phillipsite"     
## [112] "natrolite"           "nepheline"           "pargasite"          
## [115] "PD-oxyannite"        "richterite"          "riebeckite"         
## [118] "sanidine,high"       "sepiolite"           "spessartine"        
## [121] "staurolite"          "stilbite"            "wairakite"          
## [124] "titanite"

add.OBIGT() with other CSV files

add.OBIGT() can also be used to add data from a user-specified file to the OBIGT database. The file must be a CSV (comma separated value) file with column headers that match those in the default database (i.e., thermo()$OBIGT). As an example, here are the contents of BZA10.csv, which has parameters taken from Bazarkina et al. (2010). Missing values are indicated by NA:

file <- system.file("extdata/adds/BZA10.csv", package = "CHNOSZ")
read.csv(file, as.is = TRUE)

##      name abbrv formula state  ref1 ref2       date model E_units       G  H
## 1   CdCl+    NA   CdCl+    aq BZA10   NA 2010-07-03   HKF     cal  -52629 NA
## 2   CdCl2    NA   CdCl2    aq BZA10   NA 2010-07-03   HKF     cal  -84883 NA
## 3  CdCl3-    NA  CdCl3-    aq BZA10   NA 2010-07-03   HKF     cal -115399 NA
## 4 CdCl4-2    NA CdCl4-2    aq BZA10   NA 2010-07-03   HKF     cal -145583 NA
##       S     Cp     V    a1.a    a2.b    a3.c    a4.d    c1.e    c2.f
## 1  7.06  11.12  2.20  2.2303 -2.3357  6.6681 -2.6824 16.6723 -0.7693
## 2 25.72 116.01 42.21  7.5221 10.5852  1.5895 -3.2166 73.7023 20.5956
## 3 45.15  97.78 63.47 10.8045 18.5994 -1.5605 -3.5479 72.0244 16.8832
## 4 50.61  42.52 81.35 13.8329 25.9938 -4.4669 -3.8536 53.6766  5.6267
##   omega.lambda z.T
## 1       0.4372   1
## 2      -0.0495   0
## 3       0.9378  -1
## 4       2.4766  -2

Loading the data with add.OBIGT() produces a message that the new data replace existing species. We can then use subcrt() to calculate the equilibrium constant for a reaction involving the new species. Note the decrease in the stepwise stability constant for the second cadmium chloride complex with increasing pressure (Bazarkina et al., 2010, Fig. 4).

iCd <- add.OBIGT(file)

## add.OBIGT: read 4 rows; made 4 replacements, 0 additions [energy units: cal]

subcrt(c("CdCl+", "Cl-", "CdCl2"), c(-1, -1, 1), T = 25, P = c(1, 2000))

## subcrt: 3 species at 2 values of T (ºC) and P (bar) (wet) [energy units: J]

## $reaction
##     coeff  name formula state ispecies model
## 377    -1 CdCl+   CdCl+    aq      377   HKF
## 29     -1   Cl-     Cl-    aq       29   HKF
## 378     1 CdCl2   CdCl2    aq      378   HKF
## 
## $out
##    T    P      rho     logK         G       H       S       V      Cp
## 1 25    1 0.997061 0.641379 -3661.000 2669.61 21.3384 22.6816 561.306
## 2 25 2000 1.071783 0.061853  -353.058 2732.39 10.4541 12.2632 539.264

After running reset() we can look up the source of data in the default OBIGT database (Sverjensky et al., 1997). Running the reaction with thermodynamic parameters from the default database, we now see that the equilibrium constant is not as sensitive to pressure:

reset()

## reset: resetting "thermo" object

## OBIGT: loading default database with 2018 aqueous, 3570 total species

thermo.refs(iCd)[, 1:3]

##      key                                           author year
## 87 SSH97 D. A. Sverjensky, E. L. Shock and H. C. Helgeson 1997

subcrt(c("CdCl+", "Cl-", "CdCl2"), c(-1, -1, 1), T = 25, P = c(1, 2000))

## subcrt: 3 species at 2 values of T (ºC) and P (bar) (wet) [energy units: J]

## $reaction
##     coeff  name formula state ispecies model
## 377    -1 CdCl+   CdCl+    aq      377   HKF
## 29     -1   Cl-     Cl-    aq       29   HKF
## 378     1 CdCl2   CdCl2    aq      378   HKF
## 
## $out
##    T    P      rho     logK        G        H        S        V      Cp
## 1 25    1 0.997061 0.619389 -3535.48 -8522.81 -16.7360 10.04043 214.488
## 2 25 2000 1.071783 0.425896 -2431.02 -9603.90 -24.0664  2.34196 191.416

mod.OBIGT() for aqueous species

Let’s add data for CoCl₄^-2 from Liu et al. (2011). The values are taken from Table 5 of that paper; note that they are reported in caloric units, which is rather common for the HKF model. The entry includes the date in ISO 8601 extended format (e.g. 2020-08-16); Sys.Date() is used in this example to get the current date.

mod.OBIGT("CoCl4-2", formula = "CoCl4-2", state = "aq", ref1 = "LBT+11", E_units = "cal",
  date = as.character(Sys.Date()), G = -134150, H = -171558, S = 19.55, Cp = 72.09, V = 27.74)

## mod.OBIGT: updated CoCl4-2(aq)

## [1] 867

The function prints a message saying that the species was added and returns the species index of the new species. Now let’s modify the new species by adding the HKF coefficients including the OOM multipliers, as they are usually given in publications. The z at the end refers to the charge of the species, and is used only for calculating the “g function” in the revised HKF model, not for balancing reactions.

mod.OBIGT("CoCl4-2", a1 = 6.5467, a2 = 8.2069, a3 = 2.0130, a4 = -3.1183,
  c1 = 76.3357, c2 = 11.6389, omega = 2.9159, z = -2)

## mod.OBIGT: no change for CoCl4-2(aq)

## [1] 867

Let us now calculate the equilibrium constant for the formation of CoCl₄^-2 from Co⁺² and Cl^-.

T <- c(25, seq(50, 350, 50))
sres <- subcrt(c("Co+2", "Cl-", "CoCl4-2"), c(-1, -4, 1), T = T)
round(sres$out$logK, 2)

## [1] -3.20 -2.96 -2.02 -0.74  0.77  2.50  4.57  7.29

The calculated values of logK are identical to those in Table 9 of Liu et al. (2011), which provides a good indication that the thermodynamic parameters were entered correctly. Nevertheless, this isn’t a guarantee that the thermodynamic parameters are consistent with the provided values of C_P° and V°. We can see this by running info() to cross-check the parameters for the new CoCl₄^-2 species:

inew <- info("CoCl4-2")
info(inew)

## check.EOS: calculated Cp° of CoCl4-2(aq) differs by 1.33 cal K-1 mol-1 from database value

## check.EOS: calculated V° of CoCl4-2(aq) differs by -1.19 cm3 mol-1 from database value

The messages indicate that the given values of C_P° and V° differ slightly from those calculated using the HKF parameters.

mod.OBIGT() for minerals

Let’s add data for magnesiochromite from Klemme et al. (2000). The parameters in this paper are reported in Joules, so we set the E.units() to J. The value for volume, in cm³ mol^-1, is from Robie and Hemingway (1995).

H <- -1762000
S <- 119.6
V <- 43.56
mod.OBIGT("magnesiochromite", formula = "MgCr2O4", state = "cr", ref1 = "KOSG00",
          date = as.character(Sys.Date()), E_units = "J", H = H, S = S, V = V)

## mod.OBIGT: added magnesiochromite(cr) with CGL model and energy units of J

## [1] 3571

Here are the heat capacity parameters for the Haas-Fisher polynomial equation (Cp = a + bT + cT^− 2 + dT^− 0.5 + eT²). As of CHNOSZ 2.0.0, OOM multipliers are not used for these coefficients. 1500 K is a generic value for the high-temperature limit; experimental heat capacities were only reported up to 340 K (Klemme et al., 2000).

a <- 221.4
b <- -0.00102030
c <- -1757210
d <- -1247.9
mod.OBIGT("magnesiochromite", E_units = "J", a = a, b = b, c = c, d = d,
          e = 0, f = 0, lambda = 0, T = 1500)

## mod.OBIGT: updated magnesiochromite(cr)

## [1] 3571

NOTE: An additional f term is available, which can have any exponent given in lambda. This offers some flexibility for using heat capacity equations that are different from the Haas-Fisher polynomial.

Now we can use subcrt() to calculate the heat capacity of magnesiochromite. For this calculation, we set the temperature units to Kelvin. We also specify a pressure of 1 bar because the default setting of P_sat (liquid-vapor saturation) causes an error below the freezing temperature of water.

T.units("K")

## changed temperature units to K

Tref <- c(250, 300, 340)
(sres <- subcrt("magnesiochromite", property = "Cp", T = Tref, P = 1))

## subcrt: 1 species at 3 values of T (K) and P (bar) [energy units: J]

## $species
##                  name formula state ispecies model
## 3571 magnesiochromite MgCr2O4    cr     3571   CGL
## 
## $out
## $out$magnesiochromite
##     T P      Cp
## 1 250 1 114.105
## 2 300 1 129.522
## 3 340 1 138.175

Next we check that the calculated values are within 0.3 J K^-1 mol^-1 of reference values taken from Fig. 1 of Klemme et al. (2000).

Cpref <- c(114.3, 129.8, 138.4)
stopifnot(max(abs(sres$out[[1]]$Cp - Cpref)) < 0.3)

Finally, let’s restore the units setting for later calculations with subcrt(). (Another way would be to run reset(), which also resets the OBIGT database.)

T.units("C")

## changed temperature units to C

Fitting formation constants

logB.to.OBIGT() requires three things:

• Experimental decimal logarithms of formation constants (log β) as a function of temperature;
• The stoichiometry of the formation reaction in terms of known species (the new species must be last);
• The experimental temperature and pressure.

logB.to.OBIGT() does three things:

• Combines the formation constants with standard Gibbs energies (ΔG°) of the known species to calculate ΔG° of the new species;
• Fits ΔG° of the new species using 25 °C thermodynamic properties and selected HKF model parameters (i.e., G, S, c1, c2, and omega parameters in OBIGT);
• Adds the parameters to OBIGT for use by other functions in CHNOSZ.

First we set the pressure for all log β data.

P <- "Psat"

Add first species: HWO₄^- (Wang et al., 2019).

T <- c(250, 300, 350)
logB <- c(5.58, 6.51, 7.99)
species <- c("WO4-2", "H+", "HWO4-")
coeff <- c(-1, -1, 1)
logB.to.OBIGT(logB, species, coeff, T, P)

## mod.OBIGT: updated HWO4-(aq)

## logB.to.OBIGT: mean difference between logB (experimental) and logK (calculated) is 0

## [1] 427

Add second species: H₃WO₄F₂^- (Wang et al., 2021).

T <- seq(100, 250, 25)
logB <- c(17.00, 17.11, 17.46, 17.75, 18.17, 18.71, 19.23)
# Species and coefficients in the formation reaction
species <- c("H+", "WO4-2", "F-", "H3WO4F2-")
coeff <- c(-3, -1, -2, 1)
logB.to.OBIGT(logB, species, coeff, T, P)

## mod.OBIGT: updated H3WO4F2-(aq)

## logB.to.OBIGT: mean difference between logB (experimental) and logK (calculated) is 0.0307

## [1] 3572

Add third species: H₂WO₄ (Wang et al., 2021). Here we increase the tolerance because there is considerable scatter in the experimental values.

logB <- c(7.12, 7.82, 7.07, 7.76, 7.59, 7.98, 8.28)
species <- c("H+", "WO4-2", "H2WO4")
coeff <- c(-2, -1, 1)
logB.to.OBIGT(logB, species, coeff, T, P, tolerance = 0.3)

## mod.OBIGT: updated H2WO4(aq)

## logB.to.OBIGT: mean difference between logB (experimental) and logK (calculated) is 0.2035

## [1] 906

After running, logB.to.OBIGT() returns the species indices; the low values for HWO₄^- (427) and H₂WO₄ (906) indicate that the function replaced parameters for these species that were already present in OBIGT.

Diagram 1: Constant molality of F^-

Now we’re ready to make a speciation diagram. Our aim is to reproduce Fig. 7b of Wang et al. (2021), which is made for 300 °C. A constant molality of F^- is based on the assumption of complete dissociation of 0.1 m NaF (we’ll change this later). An ionic strength of 0.9 mol/kg is estimated for a solution with 1.8 m NaCl (use NaCl(1.8, T = 300)). NOTE: because the ionic strength is non-zero, the calculations here refer to molality instead of activity of species (see An Introduction to CHNOSZ).

basis(c("H+", "WO4-2", "F-", "H2O", "O2"))
basis("F-", log10(0.1))
iaq <- retrieve("W", c("O", "H", "F"), "aq")
species(iaq)
a <- affinity(pH = c(2, 7), T = 300, IS = 0.9)
e <- equilibrate(a)
col <- c(1, 4, 5, 2)
diagram(e, alpha = TRUE, col = col, lty = 1, lwd = 2, ylab = "Fraction total W")

This isn’t quite the diagram we were looking for. The published diagram shows a broad region of coexistence of H₃WO₄F₂^- and HWO₄^- at pH < 5 and increasing abundance of H₂WO₄ at lower pH.

Diagram 2: Variable molality of F^-

In reality, the molality of F^- depends strongly on pH according to the reaction H⁺ + F^- = HF. With a little algebra, we can calculate the molality of F^- (a_F in the code below) from the equilbrium constant of this reaction for a given total F concentration (F_tot). NOTE: It is important to call subcrt() with a non-zero IS so that it returns effective equilibrium constants corrected for ionic strength (try setting IS = 0 yourself and look at what happens to the diagram).

T <- 300
pH <- seq(2, 7, 0.1)
logK_HF <- subcrt(c("H+", "F-", "HF"), c(-1, -1, 1), T = T, IS = 0.9)$out$logK

## info.character: found HF(aq); also available in gas

## subcrt: 3 species at 300 ºC and 85.84 bar (wet) [energy units: J]

## nonideal: calculations for F- (Bdot equation)

## nonideal: calculations for HF (Setchenow equation)

F_tot <- 0.1
a_F <- F_tot / (1 + 10^(logK_HF - pH))

Now that we have the molality of F^- as a function of pH, we can provide it in the call to affinity().

basis(c("H+", "WO4-2", "F-", "H2O", "O2"))
iaq <- retrieve("W", c("O", "H", "F"), "aq")
species(iaq)
a <- affinity(pH = pH, "F-" = log10(a_F), T = T, IS = 0.9)
e <- equilibrate(a)
diagram(e, alpha = TRUE, col = col, lty = 1, lwd = 2, ylab = "Fraction total W")

That’s more like it. We have captured the basic geometry of Fig. 7b in Wang et al. (2021). For instance, in accord with the published diagram, HWO₄^- plateaus at around 40% of total W, and H₂WO₄ and H₃WO₄F₂^- are nearly equally abundant at pH = 2.

The highest experimental temperature for the formation constants of H₂WO₄ and H₃WO₄F₂^- is 250 °C, but this diagram is drawn for 300 °C. Wang et al. (2021) used the modified Ryzhenko-Bryzgalin (MRB) model to extrapolate to 300 °C. In contrast, we used a different model but obtained quite similar results.

NOTE: The coefficients in the model used by logB.to.OBIGT() include 25 °C values of G and S. These should be conservatively treated only as fitting parameters and should not be used to compute thermodynamic properties close to 25 °C unless they were fit to experimental data in that temperature range.