Gibbs-Duhem integrations with negative mole fractions
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C studies of metal-metal-sulfide and metal-metal-oxide systems at high temperature begin with the equilibration of solid or liquid samples with a gas phase containing the active species, H2S/H2 mixtures when working with sulfides, and CO2/CO or H2O/H2 mixtures with oxides. Having established a pattern of equilibrium gas pressures as a function of the composition of the condensed phase, the chemical activities of the metal component are calculated through integration of the Gibbs–Duhem equation. This is given as Eq. [3] and rearranged in Eq. [4], where x is the mole fraction of the indicated species and d log is the differential change in the logarithm of activity or partial pressure. The ratio of the partial pressures of H2S and H2 can be used to replace the square root of the sulfur pressure; this ratio is usually expressed in the technical literature simply as H2S/H2. Me ðlÞ 1 0:5S2 ðgÞ 5 MeS ðlÞ
[1]
K 5 aðMeSÞ=aðMeÞ pðS2 Þ0:5
[2]
xðMeÞ d log aðMeÞ 1 xðSÞ d log pðS2 Þ0:5 5 0
[3]
d log aðMeÞ 5 xðSÞ=xðMeÞ d log pðS2 Þ0:5
[4]
Taking the activity of Me at the stoichiometric composition of MeS, a value of the equilibrium constant is calculated so as to make the activity of MeS unity at this same composition. The activities of MeS at all compositions may then be calculated for any combination of temperature, metal activity, and sulfur pressure. Alternatively, the Gibbs–Duhem equation can be back integrated taking the activity of MeS to be a function of metal activity in an
J.W. MATOUSEK is a Consulting Metallurgical Engineer, Englewood, CO 80111-5415. Contact e-mail: [email protected] Manuscript submitted June 11, 2006. METALLURGICAL AND MATERIALS TRANSACTIONS B
expression such as Eq. [4]. There are, however, potential shortcomings with each method. In the vicinity of composition of the stoichiometric compound, the equilibrium pressure of sulfur increases rapidly, as seen particularly in the systems iron-sulfur[1,2,3] and nickel-sulfur.[3,4,5] The specification of the sulfur pressure at this composition can, therefore, be challenging and perhaps somewhat arbitrary. On the opposite side of the binary, sulfur pressures decline toward zero as the melt composition approaches pure metal. These characteristics can lead to difficulties in the integration of the Gibbs–Duhem equation. The problem is well covered in standard texts on thermodynamics,[6,7,8] which describe the use of functions of activity coefficients (not straightforward in the case of gaseous sulfur) in place of activities. To simplify the procedure, the concept of the ‘‘negative’’ mole fraction is introduced in the discussions that follow. The integration technique is illustrated taking the system silver-sulfur as investigated by Professor Terkel Rosenqvist.[9] It is stressed, however, that the intent of the current note is to introduce the concept of negative mole fractions, not to critique the previous work. The experimental results of Rosenqvist were reworked to convert H2S/H2 gas ratios to sulfur pressures, as shown in Figure 1. At 1000 °C and a
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