Equations of State


Calculate thermodynamic properties using the revised Helgeson-Kirkham-Flowers (HKF) equations of state for aqueous species, or using a generic heat capacity equation for crystalline, gas, and liquid species.


  cgl(property = NULL, T = 298.15, P = 1, ghs = NULL, eos = NULL)
  hkf(property = NULL, T = 298.15, P = 1, ghs = NULL, eos = NULL,
    contrib = c("n","s","o"), H2O.PT = NULL, H2O.PrTr = NULL, 
    domega = TRUE)
  gfun(rhohat, Tc, P, alpha, daldT, beta)



character, name(s) of properties to calculate


numeric, temperature(s) at which to calculate properties (K)


numeric, pressure(s) at which to calculate properties (bar)


dataframe, values of the standard molal Gibbs energy and enthalpy of formation from the elements and entropy at 25 °C and 1 bar


dataframe, values of the equations-of-state parameters


character, which contributions to consider in the revised HKF equations equations of state: (n)onsolvation, (s)olvation (the omega terms), or (o)rigination contributions (i.e., the property itself at 25 °C and 1 bar). Default is c("n","s","o"), for all contributions


dataframe, values of the electrostatic properties of water at the temperature(s) and pressure(s) of interest


dataframe, values of the electrostatic properties of water at the reference temperature and pressure


logical, calculate the T and P derivatives of omega?


numeric, density of water (g/cm3)


numeric, temperature (°C)


numeric, coefficient of isobaric expansivity (K^-1)


numeric, temperature derivative of coefficient of isobaric expansivity (K^-2)


numeric, coefficient of isothermal compressibility (bar^-1)


The equations of state permit the calculation of the standard molal properties of species as a function of temperature and pressure. For interactive use, subcrt is usually more convenient than calling these functions directly.

The property argument is required and refers to one or more of G, H, S, Cp and V, and for aqueous species only, kT and E. The units of these properties are the first ones shown in the description for subcrt. The names of the properties are matched without regard to case.

The revised HKF equations of state (Helgeson et al., 1981; Tanger and Helgeson, 1988; Shock and Helgeson, 1988) are incorporated in hkf. The equations-of-state parameters are a1, a2, a3, a4, c1, c2, omega and Z; the units of these parameters are as indicated for thermo$obigt, sans the order of magnitude multipliers. Note that the equation-of-state parameter Z (appearing in the g-function for the temperature derivatives of the omega parameter; Shock et al., 1992) is taken from thermo$obigt and not from the makeup of the species, although in most cases the two values are coincident. H2O.PT and H2O.PrTr are optional arguments that contain the electrostatic properties of H2O required for the calculations. If either of these is NULL (the default), the required values are retrieved using water.

Unless domega, the value of which is recycled to the number of species (rows of ghs and eos), is FALSE for any of the species, the temperature and pressure derivatives of the omega parameter for charged species (where Z != 0) are calculated using the g- and f-functions described by Shock et al., 1992 and Johnson et al., 1992, and coded here in gfun. This option is currently blocked when the IAPWS-95 equations are activated (see water).

The parameters in the cgl equations of state for crystalline, gas and liquid species (except liquid water) include V, a, b, c, d, e, f and lambda. The terms denoted by a, b and c 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 gaseous or liquid organic species (Helgeson et al., 1998). The standard molal volumes (V) of species in these calculations are taken to be independent of temperature and pressure.

For both hkf and cgl, if at least one equations-of-state parameter for a species is provided, any NA values of the other parameters are reset to zero. If all equations-of-state parameters are NA, but values of Cp and/or V are available, those values are used in the integration of G, H and S as a function of temperature.

The T and P arguments (and rhohat, Tc, alpha, daldT, beta for gfun) should all be the same length; the functions perform no argument validating.


A list of length equal to the number of species (i.e., number rows of supplied ghs and eos values). Each element of the list contains a dataframe, each column of which corresponds to one of the specified properties; the number of rows is equal to the number of pressure-temperature points.


The temperature and pressure range of validity of the revised HKF equations of state for aqueous species corresponds to the stability region of liquid water or the supercritical fluid at conditions between 0 to 1000 °C and 1 to 5000 bar (Tanger and Helgeson, 1988; Shock and Helgeson, 1988). The hkf function does not check these limits and will compute properties as long as the requisite electrostatic properties of water are available. There are conceptually no temperature limits (other than 0 Kelvin) for the validity of the cgl equations of state. However, the actual working upper temperature limits correspond to the temperatures of phase transitions of minerals or to those temperatures beyond which extrapolations from experimental data become highly uncertain. These temperature limits are stored in the thermodynamic database for some minerals, but cgl ignores them; however, subcrt warns if they are exceeded.


Helgeson, H. C., Kirkham, D. H. and Flowers, G. C. (1981) Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures. IV. Calculation of activity coefficients, osmotic coefficients, and apparent molal and standard and relative partial molal properties to 600°C and 5 Kb. Am. J. Sci. 281, 1249–1516. http://www.ajsonline.org/cgi/content/abstract/281/10/1249

Helgeson, H. C., Owens, C. E., Knox, A. M. and Richard, L. (1998) Calculation of the standard molal thermodynamic properties of crystalline, liquid, and gas organic molecules at high temperatures and pressures. Geochim. Cosmochim. Acta 62, 985–1081. http://dx.doi.org/10.1016/S0016-7037(97)00219-6

Maier, C. G. and Kelley, K. K. (1932) An equation for the representation of high-temperature heat content data. J. Am. Chem. Soc. 54, 3243–3246. http://dx.doi.org/10.1021/ja01347a029

Robie, R. A. and Hemingway, B. S. (1995) Thermodynamic Properties of Minerals and Related Substances at 298.15 K and 1 Bar (10^5 Pascals) Pressure and at Higher Temperatures. U. S. Geol. Surv., Bull. 2131, 461 p. http://www.worldcat.org/oclc/32590140

Shock, E. L. and Helgeson, H. C. (1988) Calculation of the thermodynamic and transport properties of aqueous species at high pressures and temperatures: Correlation algorithms for ionic species and equation of state predictions to 5 kb and 1000°C. Geochim. Cosmochim. Acta 52, 2009–2036. http://dx.doi.org/10.1016/0016-7037(88)90181-0

Shock, E. L., Oelkers, E. H., Johnson, J. W., Sverjensky, D. A. and Helgeson, H. C. (1992) Calculation of the thermodynamic properties of aqueous species at high pressures and temperatures: Effective electrostatic radii, dissociation constants and standard partial molal properties to 1000 °C and 5 kbar. J. Chem. Soc. Faraday Trans. 88, 803–826. http://dx.doi.org/10.1039/FT9928800803

Tanger, J. C. IV and Helgeson, H. C. (1988) Calculation of the thermodynamic and transport properties of aqueous species at high pressures and temperatures: Revised equations of state for the standard partial molal properties of ions and electrolytes. Am. J. Sci. 288, 19–98. http://www.ajsonline.org/cgi/content/abstract/288/1/19

See Also

info for retrieving equations of state parameters from the thermodynamic database, water for equations of state of water, subcrt for calculations that use these equations.


## aqueous species
a <- info(info("methane","aq"))	
# the non-solvation heat capacity
# at different temperature and pressure

## crystalline, gas, liquid species
a <- info(info("methane","gas"))	
# melting and vaporization of n-octane
a <- info(info(rep("n-octane",3),c("cr","liq","gas")))
b <- cgl(property="G",ghs=a,eos=a,T=seq(200,420,10),P=1)
which.pmax(b,pmin=TRUE)  # 1 = cr, 2 = liq, 3 = gas
# compare that result with the tabulated transition temperatures

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