American Mineralogist, Volume 76, pages 1547-1559, 1991

An equation of state for carbon dioxide to high pressure and temperature

Uns K. MAonn*Department of Geological Sciences, University of British Columbia, Vancouver Y6T 2B'4, Canada

Ronnnr G. Bnnrvu.NGeological Survey of Canada, 601 Booth Street, Ottawa KIA 088, Canada

Ansrucr

An equation of state for fluid CO, is presented that yields thermodynamic propertiesfor mineral equilibrium calculations reliably in the range 400-1800 K and I bar to 42kbar with good extrapolation properties to higher pressure and temperature:

P :RT A t , A ,- , T W T W, - (t, + B,r #+")

with C : B3/(Bt + 8,7) and five adjustable parameters, B,: 28.06474, Br: 1.728712.l 04 ,B . t : 8 .365341 ' l 04 ,A t :1 .094802 ' l } e ,andAr :3 .374749 ' l 0e inun i t so f cm3 /mo l ,K, and bar.

Equation-of-state parameters were optimized simultaneously to P-V-T data up to 8 kbarand phase equilibrium data up to 42kbar by a mathematical programming approach usingthe Berman (1988) data base for mineral properties. Agreement with phase equilibriumexperiments is excellent. Computed volumes compare well with measurements, althoughthey are not within uncertainties for several data sets. The enthalpy of magnesite is ad-justed to - I I12.505 kJ/mol to achieve consistency with phase equilibrium constraints atlow pressures. New experimental brackets at l2.l kbar (1173-1183 "C) and 21.5 kbar(1375-1435 "C) constrain the location of the equilibrium magneslls + periclase * CO,at high pressures. Comparisons are made with existing equations of state by Kerrick andJacobs (1981), Bottinga and Richet (1981), Saxena and Fei (1987a), and Holloway (1977).

Approximate thermophysical properties of the calcite polymorphs and aragonite werederived by mathematical programming. Results show that it is necessary to consider thehigh-temperature polymorphs, calcite-IV and calcite-V, to assess the validity of an equa-tion of state for CO, on the basis of phase equilibrium data at high temperature.

Iurnonucrrotrt

Volumetric properties of CO, at high pressures areneeded for the solution of a number of problems in pe-trology, one of the most important being the computationof phase equilibria. Because volumes have been mea-sured up to only 8 kbar (Shmonov and Shmulovich, 1974),extrapolation using an equation ofstate is required.

Equations of state for CO, suitable for computation ofgeological phase equilibria include those by Holloway(1977, l98la. l98lb), Touret and Bottinga (1979), Bot-tinga and Richet (1981), Kerrick and Jacobs (1981), Pow-ell and Holland (1985), Holland and Powell (1990), Sax-ena and Fei (1987a, 1987b), Shmulovich and Shmonov(1975), Mel'nik (1972), and Ryzhenko and Volkov (1971).These equations are based on a variety of theories andincorporate adjustable parameters to fit experimentally

measured volumes (see Ferry and Baumgartner, 1987,and Holloway, 1987, for reviews and Prausnitz et al.,1986, for theory). Although most equatiotsfrt P-V-T dataadequately, it has been pointed out that all are inconsis-tent with phase equilibrium data at pressures above l0-20 kbar (Fig. l) (Haselton et al., 1978; Berman, 1988;Chernosky and Berman, 1989). Equations that use em-pirically combined parameters, polynomial equations inparticular, may achieve excellent agreement with ob-served data, commonly at the expense of reasonable ex-trapolation. Even equations based on theory may showunconstrained behavior if they contain some empiricalelement. For example, Kerrick and Jacob's (1981) mod-ified Redlich-Kwong equation (Redlich and Kwong, 1949)has no positive volume defined at progressively highertemperatures with rising pressure, because of the math-ematical form of the a(P,T) parameter. The equation pro-posed by Bottinga and Richet (1981) shows the best ex-trapolation properties to 42kbar (Chernosky and Berman,1989, and below), but it contains discontinuities that re-

* Present address: Geological Survey of Canada, 601Street, Ottawa KIA 0E8, Canada.

0003-004x/9 1/09 1 0-l 547$02.00

Booth

r547

Hoselfon et ol, (1978)

Newlon ond Shorp (1975)

Eggler et ol, (1979)

sx

$

\ /

/.'/ tral

l 548

tn

TEMPERATURE ( 'C)Fig. l. Experimental brackets on the equilibrium Mst * En

+ Fo + CO, and reaction boundaries computed with the data-base of Berman (1988) with revised magnesite data (Chernoskyand Berman, I 989; this study) and equations of state for CO, byKerrick and Jacobs (1981) (K&J), Bottinga and Richet (1981)(B&R), Holloway (1977, 1981b) (dashed curve, HOL), and Sax-ena and Fei (1987a) (S&F). The dotted curve was computed withthe B&R equation with magnesite properties perturbed by itsestimated uncertainties as discussed in the text. Tails leading tothe symbols depict measurement uncertainty. The diagram wasproduced with Ge0-Calc (Brown et al., 1988). Solid and opensymbols indicate stability of Fo * CO, and En + Mst, respec-tively.

sult from separate parameter fitting for different volumeintervals. Powell and Holland (1985) and Holland andPowell (1990) fitted a simple polynomial function direct-ly to the logarithm offugacities derived from the equationof state of Shmonov and Shmulovich (1974) and Bottingaand Richet (1981). The function is computationally effi-cient but limited in its applications, i.e., volumes cannotbe derived reliably and extrapolation is not recommend-ed.

To circumvent the problem of inadequate P-V-T dataat high pressures, alternate approaches utilize shock wavedata, although uncertainties are large. Because measure-ments on pure fluid CO, do not seem to exist, one mustrely on approximations based on data for similar fluidsrelated through corresponding state systematics (Saxenaand Fei, 1987a; Helffrich and Wood, 1989). The equa-tions chosen by these authors to fit the data are based onformulations of the intermolecular potential (e.g., Len-nard-Jones). They have the disadvantage ofincorporatingdiscontinuities that require a diflerent function for pres-sures below several kilobars. Saxena and Fei (1987a) fit-

MADER AND BERMAN: EQUATION OF STATE FOR CO,

40

ted a modified virial equation to shock wave data validat pressures above 5 kbar.

Ihe purpose ofthe present paper is to provide petrol-ogists with an equation of state for CO, that is compatiblewith existing experimental data, thal extrapolates reliablyto upper mantle conditions, and that is mathematicallytractable. Preliminary results were reported by Miider etal. (1988). A Fortran-77 coded subroutine that computesfugacity and volume at specified pressure and tempera-ture is available from the senior author. [Requests willbe answered if they are accompanied by a formatted flop-py disk (IBM-compatible) and a self-addressed envelope.l

MnrHoo

Provided a reliable data base of thermophysical prop-erties of minerals is available, thermodynamic propertiesof CO, may be tested by comparing calculated and ex-perimentally determined phase equilibria (Haselton et al.,1978; Ferry and Baumgartner, 1987; Berman, 1988;Chernosky and Berman, 1989). Figure I indicates thatexisting equations of state for CO, calibrated on P-V-Tmeasurements only do not extrapolate well to high pres-sures. An alternate approach incorporates phase equilib-rium experiments as constraints rather than merely astests of thermodynamic parameters. This is especially im-portant in the present context because phase equilibriumdata for CO, extend to much higher pressures (42 kbar)than P-v-T dara (8 kbar).

It is important to demonstrate that the inconsistenciesillustrated in Figure I result from inadequate equationsof state for CO, rather than from errors in the thermo-dynamic properties of the minerals. This point is ad-dressed in the section on magnesite properties below.

Mathematical programming is an appropriate tool (seeWasil et al., 1989, for a survey of computer codes) tohandle this constrained optimization problem (see alsoBerman et al., 1986; Berman, 1988). Phase equilibriumconstraints are written to separate the unknown contri-butions of CO, to the Gibbs potential from the knowncontributions ofall other phases involved:

L*6r'r

solids,CO2

- r-s7,.',

o5L U ^ ^CC JU-<t>ct)Ltlctro-

20

r0

solids.co2 |

4 ,,lr,,ni'''

* I), r,,dr - r I' C.4* 5",, [,

r,, dP + v.o, [), r*,or. (r)

In Equation I A*Gar denotes the change in the Gibbs freeenergy ofa reaction; A,GP" is the apparent free energy ofa pure phase as defined in Berman (1988); AtHP"r, is the

MADER AND BERMAN: EQUATION OF STATE FOR CO,

TABLE 1. Phase equilibrium studies including pure CO2

t549

Reference Equilibrium P (kbar) rfc)s&A (1923)H&T (1955)

H&T (1956)G&H (1961)

B (1962)W (1963a)

w (1963b)J&M (1968)J (1969)s (1974)

r&w (1975)N&S (1975)H (1978)

E (1979)P (1988)M (1ee0)

0.00-0.030.o1-2.70.01-0.020.3J.1

0.01-0.081.0-5.01 .0-10

0.00-0.030.07-0.540.5-0.7

0.08-0.70.08-0.70.5-1.0

2.O0.5-0.70.5-0.720-36't94110-19374224-2513-3725-290.4-1.212-21

890-1 21 0650-900980-1 070590-800840-1 20081 0-1 01 0660-1 090900-1 21 071 0-890900-930720-930710-920700-760

560840J90790-860

1350-16001 000-1 5001 000-1 3301 1 30-1 200

1125950-1 250

1 1 00-1 250700-760

1 170-1435

C c + L m + C O 2M s t + P e + C O ,C c + L m + C O ,C c + B Q z + W o + C O ,C c + L m * C O zM s t = P e + C O ,M s t + P e + C O ,C c + L m + C O ,C c + F o + D i + M o + C O ,A k + F o + C c + M o + C O ,C c + D i + A k + C O ,C c + F o + M o + P e + C O ,M s t + P e + C O 2E n + M s t + F o + C O zC c + A n = W o + G e + C O ,C c + A n + C o = G e + C O ,M s t + P e + C O ,E n + M s t + F o + C O zC c + P O z + W o + C O zM s t + C s + E n + F o + C O ,E n + M s t + F o + C O ,M s t + R u + G k * C O zE n + M s t + F o + C O zM s t + P e + C O ,M s t = P e + C O ,

n yvvn nn yn nn nn nn yy vn (y)vvn nv yv vn (v)vvv yY Yn yy vv yn ly vv yv v

Ga, DTCSCSCSGAGAGAGAcsCScsCSCSCSPCPCPCPCPCPCPCPCPCCS, PAPC

Note: M : experimental method; CS : cold seal, GA : gas apparatus (internally heated), PC : piston-cylinder, DT : differential thermal analysis,PA:pressureanalysis.Findicatesdatasetsusedforf i t t ing(y:yes,n:no).MBindicatesconsistencywithEquat ion5ofth isstudy(y:yes,n:no, i : insufficient thermodynamic data on geikielite). See text for discussion of inconsistent data sets. Abbreviations of minerals: Cc : calcite, Lm :

lime, Mst: magnesite, Pe: periclase, BQz: P-quat1(z, Fo: forsterite, Di : diopside, Mo: monticellite, An: anorthite, Co: corundum, Wo:wollastonite, Ge: gehlenite, Cs : coesite, En : orthoenstatite, Ru : rutile, Gk : geikielite. Abbreviations of authors: S&A : Smith and Adams, H&T:Ha rke randTu t t l e ,G&H:Go ldsm i thandHea rd ,B :Bake r ,W:Wa l t e r , J&M:JohannesandMe tz , J : Johannes ,S :Shmu lov i ch , l&W: l r v i ngandWy l l i e ,N&S :NewtonandSha rp ,H :Hase l t one ta l . ,E :Egg le re ta l . ,P :Ph i l i pp ,M :Mdde r (1990and th i ss tudy ) .

enthalpy of formation from the elements; SP"r. is the thirdlaw entropy, and, VP,,r, is the molar volume at standardpressure (P.: I bar) and temperature (1n,: 298.15 K).C" and / refer to the heat capacity at constant pressureand the stoichiometric coemcient, respectively. Given areliable, internally consistent thermodynamic database forminerals, the only unknown part of Equation I is the lastterm, J V33, dP. From the relationship A.@,' : 0 atequilibrium, it follows that each experimental half brack-et leads to an inequality of the form A*G"t < 0 or A*Ger> 0 depending on whether products or reactants are sta-ble. Equation I is rearranged accordingly to impose aninequality constraint on J VP6{rdP for each halfbracket:

rPvco- | V36,dP S -ARGftI* Q)_

Jp-

where A"G*o* denotes all the terms of Equation I butthe last one, and the "less than" refers to the case whereproducts are stable. The /.6, is related to the volumeintegral by the relationship

Rrh/i{ : f' ,rr,* (3)

choosing P, : I bar and assuming approximate equalityofunit pressure and unit fugacity at I bar. R denotes thegas constant. Each inequality constraint of the form ofEquation 2 requires integration of the equation of stateand is therefore nonlinear with respect to parameters ofthe equation ofstate.

For this study, the internally consistent thermodynam-ic database of Berman (1988), including compressibilityand expansivity terms, is used to compute A*GftI*" forabout 120 phase-equilibrium halfbrackets (Table l). Ad-justments were made to the thermodynamic propertiesof magnesite and calcite (see below). About 440 measuredvolumes were used to optimize agreement between com-puted and observed volumes (Table 2). The nonlinearoptimization problem is solved with a general reduced-gradient strategy (GRG2, Lasden and Waren, 1982). Sta-bility problems do not occur if the mathematical form ofthe equation ofstate is chosen carefully.

Equ,l.rroN oF srATE

There are essentially four types of equations of statefrom which to choose: (l) van der Waals type (Redlichand Kwong, 1949; Holloway,1977; Touret and Bottinga,1979; Bottinga and Richet, l98l; Kerrick and Jacobs,I 98 l), (2) virial type (Saxena and Fei, 1987a, I 987b), (3)molecular potential formulations (Helfrich and Wood,1989), and (4) empirical functions to fit directly the log-arithm of the fugacity (Powell and Holland, 1985; Hol-land and Powell, 1990). A van der Waals type ofequationwas chosen for the following reasons: simple mathemat-ical form, favorable behavior in the limits (P - o, P -0), potentially small number of adjustable parameters,and the simplest imitation of subcritical behavior. Juza(1961) presented a successfully modified van der Waalsequation for HrO applicable to pressures up to 100 kbar.

1 5 5 0

TABLE 2. Experimentally measured P-V-f properties of CO,

MADER AND BERMAN: EQUATIoN oF STATE FoR co,

Average percent deviation

Referen@ rcc) P (kbar)

Shmonov and Shmulovich (1974)Tsiklis et at. (1971)Juza et al. (1965)Michels et al. (1935)Kennedy (1954)Amagat (1891)vukalovich et al. (1962)Vukalovich et al. (1963a)vukalovich er al. (1963b)Michels and Michels (1935)

vvvvvvvnnn

400-700100-4001 50-4751 25-1 50200-1000137-258200-7501 25-1 50650-8001 00-1 50

1.0-8.02.O-7.O0.74.O

0.07J.10.03-1.40.05-1.00.01-0.60.02-0.6o.o2-o.20.03-0.07

cb4440467260

104432235

2.070.951 . 1 72.140.792.200.681.540 . 1 00.58

1.620.800.691.330.551.770.611.240.080.66

0.890.680.95

1 . 1 7

0.39

o.241.33

1 . 1 11.420.861 . 1 4o.47o.870.440.830.230.24

Note.-Average percent deviations of calculated volumes trom measured volumes are quoted for several equations of state: M&B : Equation 5 ofthis study, PVT: Equation 5 with parameters based on P-y-fdata only (see text); K&J : Kerrick and Jacobs (1981), B&R : Bottinga and Richet(1981). F indicates data sets that were used to constrain Equation 5 (y) and data that were used tor comparison only (n). No. indicates the number ofdata points.

An equation discussed but not adopted by Bottinga andRichet (1981) served as a starting point for our devel-opment:

RT

presgibility at high pressures. The D parameter, a measureof the "incompressible volume," is commonly assumedto be a constant, an approximation valid up to 50 kbaraccording to Holloway (1987). Corresponding states the-ory applied to the van der Waals equation predicts a bparameter of 43 cm3lmol. De Santis et al. (1974) deriveda value for b of 29.7 cm3lmol for their modified Redlich-Kwong equation based on P-V-T data up to 1400 bar;the same value was adopted by Holloway (1977). Thesmallest measured volume is 3l cm3/mol at 7 .l kbar and100'C (Tsiklis et al., l97l), and shock wave dala (Zu-barev and Telegin, 1962) on solid CO, indicate a volumeof 18 cm3/mol (with large uncertainties) at 180 kbar and900 'C (as revised by Ross and Ree, 1980). It seems ob-vious that an equation successful at high pressures re-quires a compressible "incompressible volume," whichmay be achieved mathematically by making D dependenton volume (see also Kerrick and Jacobs, l98l; Bottingaand Richet, 198 l). An inverse power of the volume de-pendency of the b-term was determined empirically sub-ject to the restriction ofobtaining an integrable equation.A small temperature dependence of b also proved to beadvantageous, resulting in the final equation:

RZ

P - -A ,

+ -v4

A l

V - b T V '(4)

:oE

E{f[l

=-l

o

where b, Ar, and l, are adjustable parameters, and R isthe gas constant. The last term of Equation 4 becomesimportant at small volumes because of increased com-

PRESSURE (kbar )

Fig.2. Mathematical behavior of Equation 5 at 1200 K. Thevolume at infinite pressure is labeled 4"r. Inset figure depictsvan der Waals behavior near the critical point with volumes ofthe liquid, V,ro, and the vapor, I/".0, connected by tie lines.

with C : B3/(Bt + BrT) and five independently adjust-able parameters: -B,, Br, Br, Ar, Ar. The Ar/TVZ termyielded a b€tter fit to the data than the Redlich-Kwongterm a/y/-TV(V + b). The function approximates the ide-al gas law at very low pressures, and at infinite pressureit has a limiting volume that is dependent on temperatureand parameter values (Fig. 2). The volume at specifiedpressure and temperature is determined iteratively, andthe integratiol I V dP may be performed analytically(Appendix 1). At subcritical pressures and temperatures,Equation 5 behaves like the van der Waals equation (Fig.2, inset). The equation is continuous between 0 bar and

P - -

, - (",+ B2r --t-r. )

A , A ,..- _+_ - ,\ lTV2 V4 \ " /

500 600 700 800TEMPERATURE (K)

Fig. 3. Isobaric volume-temperature diagram of high-pres-srure P-V-T measurements. Linear trends within data sets ofShmonov and Shmulovich (solid lines) and Tsiklis et al. (dashedlines) are delineated for claritv. See text for discussion.

infinite pressure and at temperatures above 0 K for vol-umes larger than the discontinuity at V,n in Figure 2.

P-V-T ntrs

Experimentally measured volumes (Table 2) were usedto minimize the difference between computed and mea-sured volumes during optimization. The accuracy of theP-V-T data is almost impossible to establish. Precisionat pressures below I kbar is generally better than 0.2010,and at higher pressures it is about 0.50/o depending on thetype of equipment used. A complete review of equipmentand quality of data through 1972 is provided by Anguset al. (1973). At lower pressures and high temperaturesproblems may arise from the presence of species otherthan CO, such as CO, Or, from the precipitation of graph-ite, or from oxidation of the pressure vessel walls. Graph-ite coatings were reported by Vukalovich et al. (1963b)at temperatures above 800'C. High-pressure equipmentis prone to uncertainties because of sealing problems, cal-ibration, thermal gradients, pressure and temperaturemeasurement, and deformation under pressure. The non-existent or small areas of overlap between individual high-pressure data sets make the detection of inconsistenciesdifficult. If, however, one extrapolates measurements fromone data set into an adjacent one by some conservativemethod such as isobaric sections, inconsistencies of theorder of 2o/o or more are common (Fig. 3). One of the

t 5 5 l

TEMPERATURE (K)Fig. 4. Isobaric volume-temperature diaglam showing com-

parison of different equations of state and experimentally mea-sured volumes. Equation 5 in this figure is constrainedby P-V-Tdata only. The discontinuities in the slopes at 47.22 cm3/mol ofthe Bottinga and Richet equation are a result of using separatefit parameters for different volume intervals.

most disturbing discrepancies is found between the l-kbardata of Shmonov and Shmulovich (1974) and those ofKennedy (1954) (Fie. a). This suggests that parts ofentiredata sets are systematically in error by more than 2ol0.

PTUSN EQUILIBRIUM DATA

About 120 phase equilibrium experiments from 25 datasets of 16 laboratory studies (Table l) were used to putbounds on J V.o,dPaccording to Equation 2. All phaseequilibria used include magnesite or calcite as CO'-bear-ing phases and only stoichiometric phases. High-pressuredata sets (>25 kbar) involve magnesite only. Studies in-cluding dolomite were avoided because of uncertaintiesstemming from its ordering state (see also Berman, 1988).Similarly, phase equilibria including geikielite, meionite,spurrite, and tilleyite were not considered. A thermody-namic analysis of the system CaO-SiOr-CO, at high tem-peratures is provided by Treiman and Essene (1983) in-cluding larnite, rankinite, spurrite, and tilleyite. Reactions

MADER AND BERMAN: EQUATION OF STATE FOR CO,

-oE

EO

LU

DJ

o

;E

Esl,u 40==J

O>

r 0009004003 0 f

300

O Shmonov & Shnulovich (1974)

QTsiktis et al. (1971)

. Juzaetal. (1965)

n Anagat(1891)

- Kerick & Jacobs.---" Bottinga & Richet- equation (5)PW

1552

Tnale 3. Constraints on1 97s)

MADER AND BERMAN: EQUATION OF STATE FOR CO,

f"o, imposed by experimental brackets on the equilibrium Mst + En + Fo + CO2 (Newton and Sharp,

Rf ln f@.

B&R K&J S&F HOI(kJ) (kJ) (kJ) (kJ)

Obs. M&B(kJ) (kJ)

P T P TExp. Exp. Adl. Adj.

St. (kbar) (K) (kbar) (K)

RePrRePrRePI

19.019.032.O32.041.041.0

127312981523155317231773

20.517.533.530.s42.539.5

12481 3 1 31498156816981 788

> 154.6< 158.2>213.6<219.4>260.0<268.9

158.9154.7220.6217.9265.2264.7

161 .7157.8224.6222.4270.1270.4

163.2159.0227.6225.1274.4274.3

156.6153.3213.021 ' t .2253.0253.2

162.5157.4228.3224.3275.9274,O

/Votej St. indicates whether reactants (Re) or products (Pr) are stable. Adj. shows pressures and temperatures adjusted for experimental uncertainties,Obs. (observed) was computed with Equation 2 and the data base of Berman (1988) with revised magnesite properties (see text). The remainder otthe columns were @mputed with various equations of state: M&B : Equation 5 ot this study, B&R : Bottinga and Richet (1981), K&J : Kerrick andJacobs (1981), S&F: Saxena and Fei (1987a), Hol : Holloway (1977). Numbers printed in bold face are inconsistent with experimental brackets (Obs.).

including calcite were only used below the transition ofcalcite-I to calcite-IV (ca.790 'C) because ofpoorly con-strained properties ofcalcite-IV and calcite-V (see sectionbelow). The equilibrium En * Mst + Fo + COr, con-strained between 19 and 4l kbar by three studies (New-ton and Sharp, 1975; Haselton et al., 1978;- Eggler et al.,1979), forms one of the cornerstones of the present work.Equally important are tight constraints on the equilibri-um magnesite + periclase + CO, at low pressures (Phil-ipp, 1988) and at high pressures (new experiments, thisstudy).

Experimental uncertainties were accounted for by dis-placing positions ofthe halfbrackets away from the equi-librium based on best estimates of uncertainties in pres-sure and temperature. Table 3 shows an example of howone data set (Newton and Sharp, 1975) is treated to im-pose constraints on the CO, between 19 and 4l kbar and1373 and 1773 K. Many of the experimental studies usedas constraints in this paper do not properly demonstratereversibility of the phase equilibria studied. This is ac-counted for by increasing the range ofuncertainty on theside of the equilibrium boundary approximated by sta-bility or synthesis experiments rather than reversals. Thisis in most cases the reactant-stable side.

NBw nxpnnrMENTs oNMAGNESITE = PERICLASE + CO2

Existing experiments by Irving and Wyllie (1975) onthe equilibrium magnesite = periclase + CO, do not con-strain the equilibrium position at high pressure. Theequilibrium was therefore reversed at l2.l kbar betweenll73 and ll83 "C and at 21.5 kbar between 1375 and1435 'C in a piston-cylinder apparatus using %-in. talc-pyrex assemblies. Friction corrections were calibratedagainst the melting curve of Au and Ag (Mirwald et al.,1975); they amounted to 2.9 kbar at I 5 kbar and 3.5 kbarat 25 kbar nominal pressure. Thermal gradients and theeffect of pressure on the electromotive force of the Pt-PtlOo/oRh thermocouples (Getting and Kennedy, 1970)were accounted for. Uncertainties are estimated at + 1.0kbar and + l0 K. Details are reported elsewhere (Miider,1990, and in preparation).

M^c,cNnsrrn PRoPERTTES

The standard state properties, ArHP"r,, SP"r,, and VP,'r, ,of all minerals involved in the equilibria of Table I aretightly constrained by calorimetry and low-pressure phaseequilibrium data (see Berman, 1988, for details). The mostpoorly constrained properties are those for magnesite,particularly the heat capacity and thermal expansivity,for which the data extend only ro 750 K (Kelly, 1960)and 773 K (Markgraf and Reeder, 1985), respectively.The sensitivity of the calculated phase equilibria to pos-sible errors in extrapolation ofthese magnesite propertiescan be examined by adjusting the functions given by Ber-man (1988) and Chernosky and Berman (1989) withinexperimental uncertainties so that the inconsistencies be-tween existing equations of state and phase equilibriumdata are minimized. The heat capacity function of Ber-man (1988), adjusted from Berman and Brown (1985), isalready aimed at minimizing inconsistencies with theKerrick and Jacobs (1981) equation ofstate at high pres-sure, i.e., to render magnesite less stable. To decrease thestability of magnesite, the thermal expansion was maxi-mized while maintaining average percent deviations frommeasured values within l00o/o of those computed withthe best fit of Berman (1988). This results in an increaseof 25o/o of the vo term and a decrease of 2lo/o of the v,term (to adjust for increased curvature) compared to Ber-man (1988) using his Equation 5, W'r/W,r, : | * vz(T- T,) + vo(T - Z)'z. Phase equilibria computed with thisperturbed volume function for magnesite are still incon-sistent with all equations of state for CO, (Fig. l). It isconcluded that inconsistencies cannot be attributed en-tirely to errors in the properties of minerals and that,instead, phase equilibrium data may be used to constrainthe high-pressure properties of COr.

At pressures below l0 kbar and temperatures below800'C, magnesite properties are well constrained and anyconsistent set analysis (e.g., Berman, 1988) is not ham-pered by uncertainties in CO, properties because existingequations of state do not differ significantly at pressuresbelow l0 kbar. It is therefore possible to derive standardstate thermodynamic properties of magnesite prior to theoptimization of CO, properties at higher pressures. Cher-

MADER AND BERMAN: EQUATION OF STATE FOR CO,

TABLe 4. Thermodynamic properties of calcite polymorphs and magnesite

1 553

Std. state prop.L,ly,r,

(kJ/mol)s4L

(J/mol K) (J/bar) Reference

AragoniteCalcitecl)CalciteJVCalcite-VMagnesite

-1207.597- 1 206.970-1204.580- 1 1 99.768-1112.505

87.49091.89394.1 0097.89565.090

3.4153.6903.6833.6892.803

f ) , () ,0)(),(),(B)(), c), f)(),(-),(.)(.), (B), (B)

c" coefficients Ko & ( x 1 0 , ) k , (x 10 5) k " ( x 1 0 J

Calcite(-l)Ar, CcJV, Cc-VMagnesite

178 .19178 .19194.08

-16.577-16.577-21.210

-4.827-4.827-1.533

16.66016.66017.491

(2 ,3 ,4 , B)(a)(BB)

Y coetficients 4 (x 106) vz(x 10'21 v. (x 10') vo (x 10 'o )

AragoniteCalcitecl)Cc-lV, Cc-VMagnesite

- 1 .620-1.400-1.400-0.890

0.00800.00600.00600.0221

3.6700.89070.89071.844

227.4227.4227.4416.0

(s, 6, 7), (5,8)(B), (a), (B), (B)(a)(cB), (B)

Note: Standard state thermodynamic properties of calcite polymorphs at 1 bar and 298.15 K, isobaric heat capacity function coefficients (ko-k3), andvofume function coetficients (4-vo). The heat capacity function of Berman and Brown (1985) is used: C" : I<o + k1T 05 + k2T-2 + k3f-3; the volumef u n c t i o n o f B e r m a n ( 1 9 8 8 ) i s u s e d : y e ' l y 4 r : 1 + v , ( P - P , l + v z l P - P , 1 2 + v d T - T , ) + v 4 ( T - l i Y . C " i s i n J / m o l ' K a n d y i n J i b a r , f i n K , P i nba r .Re fe rences :B :Be rman (1988 ) ,BB :Be rmanandBrown(1985 ) ,CB :Che rnoskyandBerman (1989 ) , . : va l uesde r i ved in th i ss tudy ,a :assumed ,see tex t f o rd i scuss ion , l :Rob iee ta l . 1979 ,2 :S tave l yandL in fo rd (1969 ) ,3 : Jacobse ta l . ( 1981 ) , 4 :Ke l l y (1960 ) ,5 :Sa l i eandViswanathan (1976), 6: Madelung and Fuchs (1921),7: Singh and Kennedy (1974), 8: Kozu and Kani (1934).

nosky and Berman (1989) revised the enthalpy of for-mation for magnesite (- I I14.505 kJ/mol) based on ex-perimental constraints on phase equilibria in systems withan HrO-CO, fluid phase. Trommsdorff and Connolly(1990) propose an increase of the Gibbs free energy offormation for magnesite (an enthalpy of about - I I I I kJ/mol) based on field evidence of phase diagram topologies(CaO-MgO-SiOr-COr-HrO) and new phase equilibriumexperiments by Philipp (1988) on the equilibrium mag-nesite + periclase + COr.

A value of - I I 12.505 kJlmol for the enthalpy of for-mation from the elements for magrresite was derived inthis study by linear programming analysis. This value isconsistent with Philipp's (1988) accurate pressure analy-sis experiments and with the brackets on the same equi-librium determined by Harker and Tuttle (1955) and Jo-hannes and Metz (1968) using conventional cold-sealtechniques. This value used in conjunction with the database ofBerman (1988) produces the essential features ofthe phase diagram topologies suggested from naturalmineral assemblages as outlined by Trommsdorff andConnolly (1990). The destabilization of magnesite wasminimized in order not to deviate more than necessaryfrom experimental constraints in systems including mag-nesite and an HrO-CO, fluid mixture (e.g., magnesite +talc + forsterite * HrO + CO2, Greenwood, 1967). Anysuch discrepancy has to be counterbalanced by more non-ideal mixing of H,O-CO, (larger excess volume on mix-ing) compared with the mixing model of Kerrick and Ja-cobs (1981) if all other thermodynamic parameters arewell constrained.

The uncertainty on extrapolating heat capacity andthermal expansion of magnesite is the largest single con-tribution to the uncertainty of the CO, properties derived

in this study. The preferred thermophysical properties ofmagnesite are summarized in Table 4.

Rnsur,rsThe ability of Equation 5 to fit P-V-T data in the ab-

sence of additional constraints from phase equilibriumexperiments is demonstrated in Figure 4. The followingequation-of-state parameters are derived: Bt : 29.57 13,B z : 3 .16418' l0 4, B 3 : 10.2554' l0o, A, : l . I 0002. I 0 ' ,Az: 2.54456. l0e, in units of cm3lmol, K, and bar. Theresults are difficult to compare with other equations ofstate because each was calibrated with different weightsgiven to various sets of data. Figure 4 shows only thehigh-pressure subset of all constraining P-V-T measure-ments, with the overall quality of fit of the three equa-tions being comparable (Table 2). The equations shareone feature: volumes extrapolated toward high pressuresare larger than those inferred from the high-pressureP-V-T data of Shmonov and Shmulovich (1974).

Phase equilibrium constraints at pressures below 8 kbarare firlly compatible with P-V-T data and the equationof state. The high-pressure phase equilibrium constraints(Table l), however, are not compatible. The prominentfeature observed is, as would be expected from Figures Iand 4, that the RZ ln f,o, terms and thus the volumesare forced to become smaller (more stable COr) com-pared with predictions based on P-V-T data alone (Fig.5). The final equation-of-state parameters, incorporatingphase equilibrium constraints, are given in Table 5. Table3 lists the magnitudes of mismatch for several equationsof state compared with experimental data by Newton andSharp (1975).

Table I summarizes the consistency of Equation 5 withphase equilibrium experiments involving pure CO, and

1554

TEMPERATURE (K)

Fig. 5. Isobaric volume-temperature diagtam comparingEquation 5 constrained by P-V-T measurements only (dashedlines) with Equation 5 constrained additionally by phase equi-librium experiments (solid lines). Filled hexagons depict vol-umes extrapolated by Shmonov and Shmulovich based on theirown experimental data (hexagons with dots). The 9-kbar and7-kbar data points and curves are omitted for clariry.

stoichiometric phases. The two data sets that are incon-sistent with the equation include some experiments byGoldsmith and Heard (1961) with Pco, likely less thanP,., and one set by Walter (1963b) that is inconsistent(> 100 K) with any other data. Berman (1988) attributesthe latter inconsistency to Walter's failure to recognizepericlase in any experimental products. Figure 6 com-pares computed high-pressure phase equilibria includingmagnesite with experimental data. Reactions, includinghigh-temperature polymorphs of calcite, calcite-IV, andcalcite-Y, are discussed in a separate section below. Theequation of state thus performs reliably up to at leasl 42kbar.

Figure 5 and Table 2 document the comparison ofmeasured and calculated volumes. Agreement at lowpressures and supercritical temperatures is excellent be-

TaBLE 5. Parameter values for Equation 5

Br 82

28.0847 1.72871.104 8.36534 10' 1.09480.1o'g 3.37475'1o'g

MADER AND BERMAN: EQUATION OF STATE FOR CO,

6c=LUE=<t><J)LIJctro-

EC

O

LU

l-J

A2B3

800 1000 1200 1400 1600 1800TEMPERATURE ("C)

Fig. 6. Pressure-temperature diagtam with phase equilibriainvolving magnesite computed with the equation of state for CO,of this study compared with phase equilibrium experiments. Afriction correction of -3 kbar was applied to the brackets byIrving and Wyllie (1975) based on calibrations by Huang andWyllie (1975). Abbreviations of minerals: Cs : coesite, Mst :

magnesite, En : orthoenstatite, Fo : forsterite, ps : periclase,

Qz : quattz. Tails leading to symbols depict measurement un-certainties. The diagram was produced with Ge0-Calc (Brown etal., 1988). Solid and open symbols indicate stability of En +CO, and Mst + SiO', resPectivelY.

cause the equation approximates the ideal gas law. Theequation was not constrained by data below 373 K, andtherefore the shape ofthe subcritical area is only approx-imate and solely a result of its van der Waals mathemat-ical form. Between 100'C and -10 "C the volumes ofthe fluid, liquid, or vapor deviate not more than 0'50/ofrom measurements except very close to the critical point.At -50 "C (5-300 bar), computed volumes are too smallby 2.5o/o compared with measurements. The critical curvecannot be adequately represented with only five param-eters because of the severe mathematical constraints im-posed by equating fio-a to .4"oo.. The calculated criticalpoint is al332.74 (+ 0.01) K, 88.754 (+ 0.001) bar, andI 15.3 (+ 0.3) cm3lmol. Experimentally determined val-ues (Angus et al., 1973) are 304.20 (+ 0.05) K, 73.858(+ 0.05) bar, and 94.07 (+ 0.1) cm3/mol.

Rigorous comparisons of computed CO, properties withexperimental phase equilibria have previously been ham-pered by insufficient thermophysical data of solid phases,magnesite in particular (Haselton et al., 1978; Bottingaand Richet, 198 l). The properties of solids chosen bySaxena and Fei (1987a) lead to good agreement with CO,properties obtained with their equation ofstate and ren-

O Hotket & luttle 0955)Alrving & Wyllie (1975)o Hoselton et ol. (1978)OEggler et ol. (1979)ENewlon & Shorp (1975)#Mdden fhis studY

O Shmonov & Shmulovich (1974)

a Shmonov & Shmulovich(1974), ertnpolated

fr Michels et al. (1935)

o Tsi4is et al. (1971)

o Juzaetal.(1965)

I Kennsdy(1954)

400 500 600 700 800 900 1000 I 100 I

Nofe.'Units are cm3/mol, K, and bar. R : 83.147.

der the Bottinga and Richet (1981) equation grossly in-consistent at 40 kbar pressure. This is in contrast withour computations (cf. Fig. l), which render CO, far toostable with the Saxena and Fei (1987a) equation. Thisdiscrepancy results, at least in part, from Saxena and Fei'schoice ofheat capacity function (Robie et al., 1979) notsuitable for extrapolation beyond the highest temperaturedata for magnesite (750 K).

Equation 5 is able to fit measured volumes significantlybetter without the additional constraints from phaseequilibria (Fig. 5); the notable exceptions are volumesmeasured at the highest pressures, which show excellentagreement. This may indicate too little flexibility of theequation, some systematic problems with high-pressureexperimental P-V-T equipment, or inaccurate mineralproperties used to constrain the parameters. It seems thata more complex equation is not justified with the amount,extent, and quality of P-V-T data available.

Additional volumetric constraints imposed by shockwave data could possibly improve the extrapolationproperties ofEquation 5 toward pressures above 100 kbar.Shock wave measurements (Zubarev and Telegin, 1962)on solid CO, (or more likely a mixture of solid and liquidat initial conditions) are difficult to apply, and shock mea-surements on liquid CO, appear to be nonexistent. Theequation of state based on shock data of similar fluidsand corresponding state systematics of Saxena and Fei(1987a) is inconsistent with phase equilibrium experi-ments (Table 3, Fig. l). A new approach taken by Helf-frich and Wood (1989) appears to bridge shock wave dataand phase equilibrium data more successfully.

In summary, the region where our proposed equationperforms reliably spans 400-1773 K and I bar to 42kbar.Extrapolation to higher pressures and temperatures is ex-pected to yield useful results, possibly to 80 kbar and2300 K.

Car-crrn (I-IV-V) -,q.RAGoNrrE

In calibrating the equation ofstate, inconsistencies be-came evident between constraints on .fcoz imposed bymagrresite phase equilibria and those by calcite reactions.The well-constrained high-pressure brackets on En + Mst+ Fo * CO, and those on Mst + Pe * CO, (Fig. 6)demand smaller.6o, (more stable COr) than high-pres-sure brackets on Cc + 6Qz - Wo * CO, (Fig. 7). Onelikely explanation is the presence of the more stable cal-cite polymorphs calcite-IV and calcite-V (e.g., Mirwald,1976, for nomenclature and summary) at high tempera-tures, which would tend to shift the above equilibrium tohigher temperatures.

As a first step, all experimental brackets including cal-cite at high temperatures were not considered as con-straints on the properties of CO, in order to remove thepossibly significant effects of calcite-Iv and calcite-V. Totest the above hypothesis, thermophysical properties ofthe calcite polymorphs were derived from constraints im-posed by the aragonite-calcite-I-calcite-IV-calcite-Vphase diagram.

I 555

400 600 800 1000 1200 1400 1600TEMPERATURE ( 'C)

Fig. 7. Pressure-temperature diagram with phase equilibriainvolving calcite polymorphs computed with thermodynamicproperties ofTable 4 and Berman (1988) and the equation ofstate for CO, of this study. Solid symbols indicate the stabilityof the low-pressure, high-temperature assemblages. Open sym-bols indicate the stability of the high-pressure, low-temperatureassemblages. The dotted curve shows the position of the wollas-tonite equilibrium calculated with calcite-I (omitting Cc-IV andCc-V). The dashed curve was computed with the Kerrick andJacobs (1981) equation ofstate not considering any ofthe high-temperature calcite polymorphs. Abbreviations of minerals: Arag: aragonite, Cc : calcite, BQz : B-qruartz, Wo : wollastonite.Tails leading to symbols depict measurement uncertainties. Thediagram was produced with Ge0-Calc (Brown et al., 1988).

Unfortunately, few quantitative data are available onthe unquenchable polymorphs calcite-IV and calcite-V.The transition boundaries have been the subject of nu-merous studies (Boeke, l9l21' Eitel, 1923; Clark, 1957:Bell and England, 1964; Crawford and Fyfe, 1964;Boettcher and Wyllie, 1968; Johannes and Puhan, l97l;Crawford and Hoersch, 19721, Cohenand Klement,1973:Irving and Wyllie, 1975; Mirwald,, 1976, 1979a,1979b).The phase transitions between calcite-I-IV-V are mostlikely related to order-disorder within the anion sublat-tice (Lander, 1949; Salje and Viswanathan, 1976; Mir-wald, 1979a, 1979b Dove and Powell, 1989). There is,however, still some debate about the existence and exactnature of discrete phase transitions in calcite (Markgrafand Reeder, 1985). To minimize the number of ther-mophysical parameters to be derived, all polymorphictransitions were treated as first-order phase transitionswith small volume discontinuities at the phase bound-aries. This should lead to a reasonable first approxima-tion of the energetics of these phase transitions in the

MADER AND BERMAN: EQUATION OF STATE FOR CO,

<l'aY

LIJctr-ct><J>UJEo-

E MiNVold (1976,1979o)

@ lrving & Wyllie (1975)

O Cohen & Klement (I

OHorker & Tultle (

AHosefton et ol. (

I 556

absence of sufficient constraints on the order-disorderprocesses.

In view of the lack of data on thermal and volumetricproperties, the following assumptions were made: (l) Allpolymorphs share the same heat capacity function. Ex-perimental data by Kobayashi (1950b) that suggest a low-er heat capacity for aragonite than calcite were disregard-ed because his aragonite and calcite (Kobayashi, 1950a)dara are imprecise and inconsistent with data of otherstudies (Stavely and Linford, 1969; Jacobs et al., l98l;Kelly, 1960). (2) Calcite-Iv and calcite-V share the sameexpansivity and compressibility as calcite-I. X-ray databy Mirwald (1979a) suggest a larger thermal expansionfor Cc-IV than for Cc-I, which is however not observedin Markgraf and Reeder's (1985) measurements. The ex-tremely large thermal expansion of aragonite relative tocalcite suggested by measurements up to 450 "C (Saljeand Viswanathan, 1976; Kozu and Kani, 1934) was re-duced by l5olo during fitting in order to constrain thecurvature of the Cc = Ar equilibrium to the phase equi-librium data. The difference in compressibility betweencalcite and aragonite was derived from studies that in-clude measurements on both minerals by the same tech-nique (Salje and Viswanathan, 1976; Madelung and Fuchs,l92l). All other thermodynamic properties were fitted tomeasurements as summarized in Table 4. Calcite-I prop-erties are nearly identical to those in Berman (1988) andare compatible with the phase equilibrium experimentsof Table 2.

The standard state properties (A,rHr,'r,, 5r,,r,, p/r"r,) ofCc-IV and Cc-V were refined by mathematical program-ming analysis using phase equilibrium constraints (Cc-I+ Ar, Cc-IV + Ar, Cc-V + Ar, Cc-I + Cc-IV, Cc-IV +Cc-V; Fig. 7). The derived thermophysical properties areconsistent with most phase equilibrium studies with theexception of the Ar + Cc-I transition at low temperatures(<200 "C) and the Ar + Cc-V transition. At 100 "C, thecomputed Arag + Cc-I boundary is located 0.5 kbar toolow compared with experiments by Crawford and Fyfe(1964). Data of Mirwald(1979a) on the Ar + Cc-V equi-librium are inconsistent and not compatible with thebrackets obtained by Irving and Wyllie (1975) (Fig. 7).This boundary is therefore poorly constrained and allowssome latitude in the properties of Cc-V.

Computed phase equilibria involving Cc-I, Cc-IV, andCc-V (Fig. 7) are consistent with all phase equilibriumexperiments (Table l) with the exception of small incon-sistencies (<4 K) with one half bracket on the reactionAk + Cc-IV + Fo = Mtc * CO, (Walter, 1963a) and An+ Cc-IV = Wo * Ge + CO, (Shmulovich, 1974). Mostimportantly, however, the computed equilibria are com-patible with the high-pressure brackets on the reactionCc-IV/V + BQz + Wo + CO, (Haselton et al., 1978) andthe high-temperature brackets on the equilibrium Cc-(IV/V) = Lime + CO, (Smith and Adams, 1923;Baker,1962).It appears that only the experimental data of Smith andAdams (1923) and Baker (1962) for the calcite decarbon-ation can be reconciled with the calorimetric data of lime

MADER AND BERMAN: EQUATION OF STATE FOR CO,

and the above treatment. The data of Harker and Tuttle(1955) and Goldsmith and Heard (1961) are inconsistent.Smith and Adams' (1923) differential thermal analysisexperiments are, however, not reversed and constrain onlythe product-stable assemblage.

CoNcr-usroNs

The following points are emphasized: (l) phase equi-librium brackets put important constraints on the f,o, atpressures not accessible by conventional P-V-T measure-ments, (2) mathematical programming analysis of P-V-Tand phase equilibrium data allows significant improve-ments of the equation of state for CO', and (3) someexperimentally measured volumes of CO, at high pres-sures may have much larger uncertainties than previouslyestimated"

The data needed for further improvements include (l)more high-quality phase equilibrium brackets at highpressures on reactions involving COr, (2) a set of carefullymeasured CO, volumes up to 12 kbar, (3) shock wavedata on fluid COr, (4) measurements on the expansivityof magnesite to high temperatures under pressure, and (5)quantitative evaluation of the complex phase transitionsundergone by calcite. Progress is required in providingmathematically practical equations of state with a theo-retical basis.

AcxNowr,nncMENTs

The senior author is indebted to H.J. Greenwood for support and ad-vice during U.M.'s Ph.D. thesis research. The manuscript was improvedby H.J. Greenwood and T.H. Brown- Helpful reviews were provided byN. Chatterjee, M. Ghiorso, and an anonyrnous reviewer. J.R. Hollowaykindly supplied computer codes for the evaluation ofhis equation ofstate.We gratefully acknowledge financial support by NSERC to R.G.B. (OGP0037234) and H.J. Greenwood (J.M.)(67-4222)and funds to H.J. Green-wood (U.M.) administered by the British Columbia Ministry of Energy,Mines and Petroleum Resources under the Canada-British ColumbiaMineral Development Agreement.

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Berman, R.G., and Brown, T.H. (1985) The heat capacity of minerals inthe system KrO-NarO-CaO-MgO-FeO-Fe'Or-AlrOr-SiOr-TiOr-HrO-COr: Representation, estimation, and high temperature extrapolation.Contributions to Mineralogy and Petrology, 89, 168-183.

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MADER AND BERMAN: EQUATION OF STATE FOR CO,

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AppnNorx 1. In-rrcnnuoN oF THE EeuATroNOF STATE

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Salje, E., and Viswanathan, K. (1976) The phase diagram calcite-aragonite fPas derived from the crysrallographic properties. Contributions to Min- | V dPeralogy and Petrology, 55, 55-67. r Po

: I:

P dv + v"(P - P) - Po(vo - vP)

n-('--t'A- t#o,.I::#un+ vP(P - PJ - Po(Vo - V,)

with B : B, * BrT and C: Bt/B.The attractive terms (A, and A,

rntegratron

RTd V

(Al)

A,( | _ t \_+( tT\V" V,l 3 \r'a

terms) become upon

t; (A2)

MADER AND BERMAN: EQUATIoN oF STATE FoR Co, I 559

The repulsive term (RZ term) requires integration of arational function ofthe form

Case 2. Polynomial aV' + bW + cV + d yields onereal root X,. Integral ,A'3 may be split into partial fractionsby solving

m V t l n

V(aV3 + bV'z + cV +d)

I J K V + L: V + v - x , + y , * " y a ( A l 4 )

with a : b/a + X, and B : c/a * X,a, which yields fourequations in four unknowns (1, J, K, L\:

N: - IOX , (Al5)

0: t(0 - aX') + J0 - LX, (416)

(A3)

t ' /rth m: RZ, n : RZC, a : l, b: -B, c : 0, d : C :B,/8, and B: B, + B2T.

Integral A3 may be solved after splitting into partialfractions provided the real roots of the polynomial aI/3+ bW + cV + d are known.

Case 1. Polynomial aW + bV2 + cV + dyields threereal roots X,, Xr, X.. Integral A3 may be split into partialfractions by solving

m V t + n

f ^ m V 3 r nJ , , v (ov ' + bv , + tv + ^- dv

d )

V ( a V 3 + b V 2 + c V + d )

I J K L: _ +v -x r - v -x r - v -x ,

0 : I ( a - X , ) + J a - K X , + Lm : I + J + K .

Equations Al5 through Al8 are solved simultaneously(44) to obrain

I: - pxJ-,Cross multiplication and collecting terms in powers of Z

yield four equations in four unknowns (1, J, K, L) bycomparing coefficients:

(Al7)(Al8)

(A1e)

(A20)

(A2l)

(422)

(424)

rntegrals by appropriate substitutions:Equations A'5 through A8 are solved simultaneously to f KV + Lob ta in

J Va "n

* Udv

n -- I(-X,XrXr)0: I(x,x, + xtx3 + x2x3)

+ J(X,X) + K(XI) + L(XX,)0: I(xt + x2 + x3) + J(x' + L)

+ K(X.+ X) + L(Xt + X,)m : I + J + K + L .

I :xtx2x3

r - m X l + n

K :

X,(-Xi+ XtX2- XrXr+ XX3)

m X ) + nxr(x} - x,Xr- x2x3+ xtx3)

m X l + n&(x1 + X,X,- XrXr- X,Xr) '

m X l t n

K :

X , ( a X , + p + X ? )

a B m X , * a n * B 2 m * n X ,B@X,+p+X l )

L :a 2 n * a n X , + p 2 m X - 0 n

B@X,+p+X1)(A8) The last term of Equation Al4 is expanded to standard

(A5)

(A6)

(A7)

(Ae)

(Al0)

(Al l )

(Al2)

Integral ,A,3 may now be integrated by parts to obtain

,"(t).,,"(#)* K h(m) . ","(m) (A, 3) #+;

f Kz f r - x t .: t , . d _ - - | _ d _ - ( A 2 3 )J z 2 i t r - - J 2 2 + 1 2 - -

with z : V * s, s : ot/2, t : VF-_-p, and dV : dz.Integration of ,A'3 yields

,^(?).r'"(m).{,"(ff+#v).'#l*""'(^,)

_ "."""(,!#)l

L :

b/a * Xr , B : c /a * aXr, s : a/2, and l :