Читать книгу Geochemistry - William M. White - Страница 105

The tools we have introduced to deal with ideal solutions and infinitely dilute ones are based on observations of the gaseous state: Raoult's law and Henry's law. We will continue to make reference to gases in dealing with real solutions that follow neither law. While this approach has a largely historical basis, it is nevertheless a consistent one. So following this pattern, we will first introduce the concept of fugacity, and derive from it a more general parameter, activity.

In the range of intermediate concentrations, the partial pressure of the vapor of component i above a solution is generally not linearly related to the mole fraction of component i in solution. Thus, the chemical potential of i cannot be determined from equations such as 3.26, which we derived on the assumption that the partial pressure was proportional to the mole fraction. To deal with this situation, chemists invented a fictitious partial pressure, fugacity. Fugacity may be thought of as the “escaping tendency” of a real gas from a solution. It is defined to have the same relationship to chemical potential as the partial pressure of an ideal gas:

(3.43)

where ƒ° is the standard-state fugacity, which is analogous to standard-state partial pressure. We are free to choose the standard state, but the standard state for ƒ° and μ° must be the same. If we choose our standard state to be the pure substance, then ƒ° is identical to P°, but we may wish to choose some other standard state where this will not be the case. Since the behavior of real gases approaches ideal at low pressures, the fugacity will approach the partial pressure under these circumstances. Thus the second part of the definition of fugacity is:

For an ideal gas, fugacity is identical to partial pressure. Since, as we stated above, fugacity bears the same relationship to chemical potential (and other state functions) of a nonideal substance as pressure of a nonideal gas, we substitute fugacity for pressure in thermodynamic equations.

The relationship between pressure and fugacity can be expressed as:

(3.44)

where φ is the fugacity coefficient. The fugacity coefficient expresses the difference in the pressure between a real gas and an ideal gas under comparable conditions. Kerrick and Jacobs (1981) fitted the Redlich–Kwong equation (eqn. 2.15) to observations on the volume, pressure and volume of H₂O and CO₂ to obtain values for the coefficients a and b in eqn. 2.15. From these, they obtained fugacity coefficients for these gases at a series of temperatures and pressures. These are given in Table 3.1; Example 3.2 illustrates their use.

Table 3.1 H₂O and CO₂ fugacity coefficients.

H2O		T (°C)
P (MPa)	400	600	800	1000
50	0.4	0.78	0.91
200	0.2	0.52	0.79	0.94
400	0.21	0.54	0.84	1.03
600	0.28	0.67	1.01	1.22
800	0.4	0.89	1.27	1.49
CO2		T (°C)
P (MPa)	377	577	777	977
50	1.02	1.1	1.12	1.12
200	1.79	1.86	1.82	1.75
400	4.91	4.18	3.63	3.22
600	13.85	9.48	7.2	5.83
800	38.73	21.33	14.15	10.44

From Kerrick and Jacobs (1981).