Description

CONDOR – Description

Computational Algorithm

The CONDOR code operates by simultaneously considering the dual constraints of mass balance and chemical equilibrium. The operation of the code is best illustrated with an example based on a simplified version of nitrogen thermochemistry in a hydrogen-rich system (e.g., the atmospheres of the gas giant planets Jupiter, Saturn, Uranus, and Neptune and the atmospheres of brown dwarfs such as Gliese 229B).

In this example, the total nitrogen elemental abundance is denoted as SN and it is assumed that the only important nitrogen gases are N₂ (g), NH₃ (g), HCN (g), and CH₃NH₂ (g) and that the system behaves ideally. The mass balance expression for nitrogen, which equates the total nitrogen elemental abundance (SN) to the sum of the abundances of all N gases, is given by

SN = 2P_N2 + P_NH3 + P_HCN + P_CH3NH2 (1)

where P_i is the partial pressure of gas i. The mass balance equation (1) can be rewritten in terms of the fugacity of molecular nitrogen (f_N2), the equilibrium constants (K_i) for forming gas i from the constituent elements in their respective reference states, and the thermodynamic activities and fugacities of the other elements combined with N in the gases considered. In this case, these other elements are carbon and hydrogen, so the thermodynamic activity of graphite (a_gr) and the fugacity of molecular hydrogen (f_H2) are used. The rewritten version of equation (1) is

SN = (f_N2)^1/2[2K_N2(f_N2)^1/2 + K_NH3(f_H2)^3/2 + K_HCNa_gr(f_H2)^1/2 + K_CH3NH2a_gr(f_H2)^5/2] (2)

One equation, like equation (2), which contains the partial pressure terms for all gases containing the element in question is written for each element in the code. This example considers only four N-bearing gases, but the actual mass balance equations in the CONDOR code are considerably more complex. For example, the mass balance equation for nitrogen contains over one hundred gases and the mass balance equation for hydrogen contains several hundred different gases. Furthermore, as evident from equation (2), the mass balance equations are coupled and nonlinear because they contain fugacities and activities for several elements.

The solution of the mass balance equations in the CONDOR code is done by making an initial guess for the activity (or fugacity) of each element. The initial guesses can be optimized and the solution will converge within fewer steps if the major gases of each element are known, but this optimization is not essential for proper operation of the code. The CONDOR code then iteratively solves the set of coupled nonlinear equations and gives the thermodynamic activity (or fugacity) for each element, the abundances of all gases in the code, and information on the quality of the solution for each element. The convergence criterion used specifies that the code reaches a solution when the calculated abundance and the input abundance for each element agree within one part in 10,000,000 or better. The database for the CONDOR code currently contains ~2300 gaseous and ~1600 solid and liquid species of all naturally occurring elements. The average time for one calculation (for all compounds at one pressure, temperature point) is less than 0.5 second on an IBM compatible PC with a Pentium III microprocessor.

Thermodynamic Data

The equilibrium constants K_i used in the mass balance equations are taken from the thermodynamic database used in the Planetary Chemistry Laboratory. This has been and continues to be compiled by careful evaluation of thermodynamic data published in compilations and in the refereed literature. This careful evaluation is necessary for two reasons: (1) errors and inconsistencies exist in thermodynamic data compilations, and (2) thermodynamic data from the refereed literature need to be analyzed and evaluated before use in computations (see 1). For example, some of the tables in the third and/or fourth edition of the JANAF Tables (2, 3) that are wrong and have to be recomputed before use are those for C₂H₂, S₂O, PH, PH₃, PN, and Mg₃P₂O₈ solid (e.g., 4; 5; 6; 7). Computations that are done by simply copying all the tabulated data from JANAF and/or other data sources (such as the different editions of the IVTAN tables) thus contain errors due to the use of incorrect thermodynamic data.

Condensate Formation

The CONDOR code also takes possible liquid and solid condensates into account. This can be illustrated using nitrogen chemistry as an example. Solid ammonium hydrosulfide (NH₄SH) is predicted to condense in the atmospheres of gas giant planets such as Jupiter. The condensation of NH₄SH (solid) occurs when the thermodynamic activity of this phase becomes equal to one. Once the CONDOR code finds that a_NH4SH is equal to (or greater than) one, it computes the temperature at which the activity first reached unity, resets the thermodynamic activity of NH₄SH (solid) to unity at all lower temperatures, and adds a new term (A_NH4SH) to the mass balance equations for nitrogen and sulfur which takes into account the amount of NH₄SH that is condensed. (The hydrogen abundance is so large relative to that of nitrogen and sulfur that a mass balance correction for NH₄SH condensation is unnecessary for a solar or near solar composition gas.) The gas phase and condensation calculations are coupled, and are done simultaneously using iterative methods.

The total abundance of each condensate is limited by the least abundant element in the condensate. Ammonium hydrosulfide condensation from a solar or a near solar composition system such as the Jovian atmosphere illustrates this point. Condensation of NH₄SH occurs via the net thermochemical reaction:

NH₃ (g) + H₂S (g) = NH₄SH (solid) (3)

at about 230 K in Jupiter’s atmosphere. The solar elemental abundances of hydrogen, nitrogen, and sulfur are 2.82 × 10¹⁰, 2.63 × 10⁶, and 4.47 × 10⁵atoms, respectively, on the cosmochemical scale where Si has an abundance of 1.00 × 10⁶ atoms (8). Thus, sulfur is the least abundant element and limits the amount of NH₄SH that condenses. At 230 K on Jupiter, NH₃ and H₂S are the dominant nitrogen-bearing and sulfur-bearing gases and contain ~100% of all nitrogen and ~100% of all sulfur in the Jovian atmosphere. The NH₃ abundance decreases by about 17% (i.e., by the S/N atomic abundance ratio) due to the condensation of NH₄SH. However, H₂S is almost completely consumed by NH₄SH condensation. The H₂S abundance at lower temperatures is very small (effectively zero) and is an exponential function of temperature because it is controlled by its vapor pressure over NH₄SH (solid). Analogous constraints control the abundances of other condensates and affect the abundances of gaseous species after condensation occurs.

References

1. Stull, D., and H. Prophet 1967. The calculation of thermodynamic properties of materials over wide temperature ranges. In The Characterization of High Temperature Vapors (J. L. Margrave, Ed.) pp. 359-424. John Wiley, New York.

2. Chase, M. W., Jr., C. A. Davies, J. R. Downey, Jr., D. J. Frurip, R. A. McDonald, and A. N. Syverud 1985. JANAF Thermochemical Tables, 3rd. ed., J. Phys. Chem. Ref. Data 14, Suppl. 1, Am. Chem. Soc. and Am. Inst. of Physics, Washington, D. C.

3. Chase, M. W. 1999. NIST-JANAF Thermochemical Tables, 4th ed., J. Phys. Chem. Ref. Data 28, Monograph no. 9, Am. Chem. Soc. and Am. Inst. of Physics, Washington, D. C.

4. Zolotov, M. Yu., and B. Fegley, Jr. 1998a. Volcanic production of sulfur monoxide (SO) on Io. Icarus 132, 431-434. reprint

5. Zolotov, M. Yu., and B. Fegley, Jr. 1998b. Volcanic origin of disulfur monoxide (S₂O) on Io. Icarus 133, 293-297. reprint

6. Heirs, R. S., III 1994. Errors in JANAF thermodynamic formation functions for acetylene. J. Thermophys. Heat Transfer 8, 181-182.

7. Lodders, K. 1999b. Revised thermochemical properties of phosphinidene (PH), phosphine (PH₃), phosphorus nitride (PN), and magnesium phosphate (Mg₃P₂O₈). J. Phys. Chem. Ref. Data 28, 1705-1712. reprint

8. Lodders, K., and B. Fegley, Jr. 1998. The Planetary Scientist’s Companion, Oxford University Press, New York.