If the equations do not converge, a new iteration is initiated. This continues until convergence is achieved for the gaseous species or 35 iterations have been completed. If there is no convergence after 35 iterations, the program goes on to a new T or P or both.

Once convergence is achieved, the program checks the condensed species to determine if a condensed species should be added. It selects the one with the lowest or next to the lowest DELN value based on the formula:

If there are no condensed species to be added, the program goes on to print out the results, if a condensed species is to be added, a new matrix is set up to include the formula:

This formula differs from the original formula in that it now includes the contribution due to the effect of P on the activity of the solids (VΔP).

The equations are solved, changes to the mole fraction are made, a new control factor is calculated and convergence tests are made. If convergence is achieved in less than 35 iterations, the program checks to see if another condensed species should be added or removed. The program only adds or removes one condensed species at a time. If no singular matrix occurs or if there are no more condensed species with a negative G_j value, the program prints out the results and moves on to a new P-T pair, a new mole fraction ratio, or a new problem. To save

time, the program only uses the compositions from the previous points for the initial estimates of a new T point. However, it resets the initial estimates for new P or mole fractions.

The coefficients (a_i, i = 1,....,14) are derived from a set of thermodynamic data by the means of least-squares technique in LEASE (Table 8). The equations used are based on the following relations:

where a_i, a₆, and a₇ are integration constants(McBride and Gordon, 1967). Heat capacity, enthalpy, and entropy can be derived from the coefficients by use of the following equations:

There are two sets of coefficients on the THERMO cards: The first set covers the interval 1000^o-5000^o K, and the second set is for the interval 298.15^o - 1000^o K. Thermodynamic data supplied with the original program for condensed species only included data for the T range over which the species was stable at one bar. At the transition point, the program substitutes the stable phase for the unstable phase depending on the T - not the P -under consideration. When the molar volume was

included in computation of G_j, it became necessary to include data for the species beyond their normal transition points. The original program would have extrapolated the thermodynamic properties beyond the transition T if the T limits were altered on the coefficient cards. This method was tried for calculating the thermodynamic properties of calcite, brucite, and magnesite, and the results were plotted. Beyond a certain point the values became greatly exaggerated and were not reliable. Therefore another approach was used.

The heat capacity of a solid is assumed to vary linearly with T, and this relationship was used to extrapolate values for entropy and enthalpy. A program was written to compute heat capacity for a specie at any T from coefficients(a_i, i = 1,....,14) derived from available data using equation 12. Heat capacity was computed at 10^o intervals over the stable T range of the specie. Systematic trends in the plotted values were identified using Arizona State University Computer Center library program POLFIT***. This fits least-square polynominals to bivariant data, using an orthogonal polynominal method. Heat capacity for any T was obtained from the relation C_p = a + bT + cT^-1. However for brucite, magnesite, portlandite, alpha-quartz, and tremolite the relationship C_p = a₁ + a₂ T was used, where a, a₁ and a₂ are integration constants. Enthalpy and entropy were derived from heat capacity using the following equations:

where S_t1 and H_t1 are entropy and enthalpy values at the last T value input to POLFIT***. Data for brucite to 1000^o K obtained this way were found to agree closely with the JANAF tables.

A detailed explanation of the standard states of G_j , (H^o_t)_j, and (S^o_t)_j can be found in Gordon and McBride (1971). Their standard state is at 298.15^o K and one bar, and they are considered to be dimensionless in the program. The computation of G_j is derived from the gaseous species rather then the reference elements. G_j for the gases and condensed phases differs according to the μ_j value used, which is derived from the following formulae(Gordon and McBride, 1971):

The free energy values used in the program are derived from equation 3 for the gases and equation 7 for the condensed species.

(S^o_t)_j is dimensionless in the program and is derived from the formula;

and (H^o_t)_j is also dimensionless in the program. It is derived from the formula

This site built and hosted for free by FreeWebs.com. Click here to get your own free website.