CHNOSZ (version 0.9-9)

EOSregress: Regress Equations-of-State Parameters for Aqueous Species

Description

Functions for fitting experimental volume and heat capacities using regression equations. Possible models include the Helgeson-Kirkham-Flowers (HKF) equations of state and other equations defined using any combination of terms derived from the temperature, pressure and thermodynamic and electrostatic properties of water.

Usage

EOSregress(exptdata, var = "", T.max = 9999)
  EOSvar(var, T, P)
  EOScalc(coefficients, T, P)
  EOSplot(exptdata, var = NULL, T.max = 9999, T.plot = NULL, 
    P = NULL, fun.legend = "topleft", coefficients = NULL)
  EOSlab(var, coeff = "")
  EOScoeffs(species, property)

Arguments

exptdata

dataframe, experimental data

var

character, name(s) of variables in the regression equations

T.max

numeric, maximum temperature for regression, in degrees Kelvin

T

numeric, temperature in degrees Kelvin

P

numeric, pressure in bars

T.plot

numeric, upper limit of temperature range to plot

fun.legend

character, where to place legend on plot

coefficients

dataframe, coefficients to use to make line on plot

coeff

numeric, value of equation of state parameter for plot legend

species

character, name of aqueous species

property

character, Cp or V

Value

• For EOSregress, an object of class lm. EOSvar and EOScalc both return numeric values. EOScoeffs returns a data frame.

encoding

UTF-8

Details

EOSregress uses lm to regress the experimental heat capacity or volume data in exptdata, which is a data.frame with columns T (temperature in degrees Kelvin), P (pressure in bars), and Cp or V (heat capacity in cal/mol.K or volume in cm3/mol). Only data below the temperature of T.max are included in the regression. The regression formula is specified by a vector of names in var; these names correspond to variables identified below:

ll{ T $T$ (temperature) P $P$ (pressure) TTheta $(T-\Theta)$ ($\Theta$ = 228 K) invTTheta $1/(T-\Theta)$ TTheta2 $(T-\Theta)^2$ invTTheta2 $1/(T-\Theta)^2$ invPPsi $1/(P+\Psi)$ invPPsiTTheta $1/((P+\Psi)(T-\Theta))$ V $V$ (volume of water) E $E$ (isobaric expansivity of water) kT $\kappa_T$ (isothermal compressibility of water) alpha $\alpha$ (coefficient of isobaric expansivity of water) beta $\beta$ (coefficients of isothermal compressibility of water) X $X$ (Born function $X$) Q $Q$ (Born function $Q$) TX $TX$ (temperature times $X$) drho.dT $d\rho/dT$ (temperature derivative of density of water) V.kT $V\kappa_T$ (volume times isothermal compressibility of water) }

EOSvar takes as input var (one of the names of the variables listed above), and T (temperature in degrees Kelvin), P (pressure in bars). It returns the value of the variable at the specified temperature-pressure condition(s). This function is used by EOSregress to get the values of the variables used in the regression.

EOScalc calculates the predicted heat capacities or volumes using coefficients provided by the result of EOSregress, at the temperatures and pressures specified by T and P.

EOSplot takes a table of data in exptdata, runs EOSregress and EOSpred and plots the results. The experimental data are plotted as points, and the calculated values as a smooth line. The point symbols are filled circles where the calculated value is within 10% of the experimental value; open circles otherwise.

EOSlab produces labels for the variables listed above that can be used as.expressions in plots. The value of coeff is prefixed (using substitute) to the name of the variable.

EOScoeffs retrieves coefficients in the Helgeson-Kirkham-Flowers equations from the thermodynamic database (thermo$obigt) for the given aqueous species. If the property is Cp, the resulting dataframe has column names of (Intercept), invTTheta2 and TX, respectively holding the coefficients $c_1$, $c_2$ and $\omega$ in equation $Cp^\circ = c_1 + c_2/(T-\Theta)^2 + {\omega}TX$. If the property is V, the data frame has column names of (Intercept), invTTheta and Q, respectively holding the coefficients $\sigma$, $\xi$ and $-\omega$ in $V^\circ = \sigma + \xi/(T-\Theta) - {\omega}Q$.

The motivation for writing these functions is to explore alternatives or possible modifications to the revised Helgeson-Kirkham-Flowers equations applied to aqueous nonelectrolytes. As pointed out by Schulte et al., 2001, the functional forms of the equations do not permit retrieving values of the solvation parameter ($\omega$) that closely represent the observed trends in both heat capacity and volume at high temperatures (above ca. 200 $^{\circ}$C).

References

Schulte, M. D., Shock, E. L. and Wood, R. H. (1995) The temperature dependence of the standard-state thermodynamic properties of aqueous nonelectrolytes. Geochim. Cosmochim. Acta 65, 3919--3930. http://dx.doi.org/10.1016/S0016-7037(01)00717-7

See Also

See lm for the details of the regression calculations.

Examples

data(thermo)
## regress calculated heat capacities and volumes of CH4(aq)
## to test that regressions return known HKF parameters
# calculate the properties of CH4 on a T-P grid
T <- convert(seq(0, 350, 25), "K")
P <- seq(200, 1000, 100)
# convert=FALSE means that temperature has units of K
CH4.prop <- subcrt("CH4", T=T, P=P, grid="T", convert=FALSE)$out[[1]]
# terms in the HKF equations for Cp
Cp.var <- c("invTTheta2", "TX")
# get coefficients in Cp regression
Cp.lm <- EOSregress(CH4.prop[, c("T", "P", "Cp")], Cp.var)
Cp.coeff <- Cp.lm$coefficients
# terms in the HKF equations for V
V.var <- c("invPPsi", "invTTheta", "invPPsiTTheta", "Q")
# get coefficients in V regression
V.lm <- EOSregress(CH4.prop[, c("T", "P", "V")], V.var)
# use same units as HKF: convert from cm3.bar to calories (divide by 41.84)
V.coeff <- convert(V.lm$coefficients, "calories")
## the tests: did we get the HKF parameters that are in the database?
CH4.par <- info(info("CH4"))
# c1 and c2
stopifnot(all.equal(Cp.coeff[1], CH4.par$c1, check.attr=FALSE))
stopifnot(all.equal(Cp.coeff[2], CH4.par$c2, check.attr=FALSE))
# omega (from Cp)
stopifnot(all.equal(Cp.coeff[3], CH4.par$omega, check.attr=FALSE))
# a1, a2, a3 and a4
stopifnot(all.equal(V.coeff[1], CH4.par$a1, check.attr=FALSE))
stopifnot(all.equal(V.coeff[2], CH4.par$a2, check.attr=FALSE))
stopifnot(all.equal(V.coeff[3], CH4.par$a3, check.attr=FALSE))
stopifnot(all.equal(V.coeff[4], CH4.par$a4, check.attr=FALSE))
# omega (from V) - note negative sign
stopifnot(all.equal(-V.coeff[5], CH4.par$omega, check.attr=FALSE))

## regress experimental heat capacities of CH4
## using revised Helgeson-Kirkham-Flowers equations
# read the data from Hnedkovsky and Wood, 1997
f <- system.file("extdata/cpetc/Cp.CH4.HW97.csv", package="CHNOSZ")
d <- read.csv(f)
# have to convert J to cal and MPa to bar
d$Cp <- convert(d$Cp, "cal")
d$P <- convert(d$P, "bar")
# specify the terms in the HKF equations
var <- c("invTTheta2", "TX")
# perform regression, with a temperature limit
EOSlm <- EOSregress(d, var, T.max=600)
# the result is within 10% of the accepted
# values of c1, c2 and omega for CH4(aq)
CH4coeffs <- EOScoeffs("CH4", "Cp")
dcoeffs <- EOSlm$coefficients - CH4coeffs
stopifnot(all(abs(dcoeffs/CH4coeffs) < 0.1))
## make plots comparing the regressions
## here with the accepted EOS parameters of CH4
par(mfrow=c(2,2))
EOSplot(d, T.max=600)
title("Cp of CH4(aq), fit to 600 K")
legend("bottomleft", pch=1, legend="Hnedkovsky and Wood, 1997")
EOSplot(d, coefficients=CH4coeffs)
title("Cp from EOS parameters in database")
EOSplot(d, T.max=600, T.plot=600)
title("Cp fit to 600 K, plot to 600 K")
EOSplot(d, coefficients=CH4coeffs, T.plot=600)
title("Cp from EOS parameters in database")

## model experimental volumes of CH4
## using HKF equation and an exploratory one
f <- system.file("extdata/cpetc/V.CH4.HWM96.csv", package="CHNOSZ")
d <- read.csv(f)
d$P <- convert(d$P, "bar")
# the HKF equation
varHKF <- c("invTTheta", "Q")
# alpha is the expansivity coefficient of water
varal <- c("invTTheta", "alpha")
par(mfrow=c(2,2))
# for both HKF and the expansivity equation
# we'll fit up to a temperature limit
EOSplot(d, varHKF, T.max=663, T.plot=625)
legend("bottomright", pch=1, legend="Hnedkovsky et al., 1996")
title("V of CH4(aq), HKF equation")
EOSplot(d, varal, T.max=663, T.plot=625)
title("V of CH4(aq), expansivity equation")
EOSplot(d, varHKF, T.max=663)
title("V of CH4(aq), HKF equation")
EOSplot(d, varal, T.max=663)
title("V of CH4(aq), expansivity equation")
# note that the volume regression using the HKF gives
# a result for omega (coefficient on Q) that is
# not consistent with the high-T heat capacities