Fugacity with Pressure

The dependence of fugacity on pressure can be derived by differentiating Equation (10.29)  

This equation can be integrated to find a fugacity at one pressure from that at another  

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In Section I, we indicated that significant progress in understanding high-pressure thermodynamics of mixtures requires a quantitative description of the variation of fugacity with pressure as given by Eq. (3). To obtain the effect of pressure on activity coefficient we substitute as follows ... 

All of the information presented so far in this chapter is practically independent of the system pressure, as long as the pressure is above the boiling-point pressure of the mixture. The reason, as shown in Chapter 7, is that change in fugacity with pressure is given by Eqs. 7.6 and 7.14, which show that... 

We have a considerable body of knowledge to help us to say something about the third coefficient, the variation of fugacity with composition. Many empirical and semiempirical expressions (e.g., Margules, Van Laar, Scat-chard-Hildebrand) have been investigated toward that end. Most of our experience in this regard is limited to liquid mixture at low pressures, where... 

With the standard state expressed in this manner, the activity of the gas becomes the fugacity expressed in bars. We will usually follow this convention as we work with activities of gases. An added convenience comes from being able to relate fugacity to pressure through the fugacity coefficient [Pg.284]

By now we should be convinced that thermodynamics is a science of immense power. But it also has serious limitations. Our fifty million equations predict what — but they tell us nothing about why or how. For example, we can predict for water, the change in melting temperature with pressure, and the change of vapor fugacity with temperature or determine the point of equilibrium in a chemical reaction but we cannot use thermodynamic arguments to understand why we end up at a particular equilibrium condition. 

Of the problems presented, correlation of the NH3-CO2-H2S-H2O system is most important. Data that might be used for direct empirical correlation of partial pressures or fugacities with total concentrations of ammonia, carbon dioxide, and hydrogen sulfide in the liquid are available for relatively limited ranges... 

The equation shows that the fugacity (the activity ) of the dissolved gas molecules increases exponentially with pressure. With... 

MPa). Also, a pressure, usually near 1 bar, will exist at which the real gas has a fugacity of unity. bar also real gas at zero pressure. (See Exercise 1, this chapter.) (0.1 MPa). Also, a pressure of the real gas will exist, not zero and not that of unit fugacity, with an entropy equal to that in the standard state. (0.1 Mpa). V =(RT/P°). 

For a solid component taking part in a reaction, fugacity variations with pressure are small and can usually be ignored. Hence... 

The first term on the right-hand side is the idea gas limit, and the remaining -logarithmic terms express the successive virial corrections for the real gas behavior. It is evidently most convenient for this problem to choose the standard state pressure as P° = 0, where all gases are ideal. With this choice, we can write the relationship between fugacity and pressure as... 

One method of calculating fugacity and hence y is based on the measured deviation of the volume of a real gas from that of an ideal gas. Considet the case of a pute gas The ftee energy F and chemical potential /i changes with pressure according to the equation... 

The first two terms in the product on the right side of Equation (4C-14) give the fugacity at the saturation pressure. The exponential term, the Poynting correction, is a correction to the fugacity for compressing the condensed phase from the saturation pressure to pressure P. Assuming that the condensed phase molar volume does not vary with pressure this relation reduces to... 

The comparison highlights the difference between the nonideal hydrogen/steam/water case and the ideal carbonmonox-ide/carbondioxide case. The difference can be detected only if fugacity-based calculations as displayed in the introduction to this book are made using the JANAF tables, (Chase etah, 1998). The equilibrium concentrations, the equilibrium constant and the Nernst potential difference V, in the hydrogen case, are a function of both pressure and temperature. declines with pressure. In the carbon monoxide perfect gas case, the same variables are a function of temperature only. The pressure coefficient is zero. 

The equilibrium constant K is a function of temperature only. However, Eq. (13.25) relates K to fugacities of the reacting species as tliey exist in the real equilibrium mixture. These fugacities reflect the nonidealities of the equilibrium mixture and are functions of temperature, pressure, and composition. Tliis means that for a fixed temperature the composition at equilibrium must change with pressure in such a way that W ifilP ") remains constant. 

As a first approximation, Cp may be treated as independent of the pressure, and if MJ.T. is expressed as a function of the pressure, it is posnble to carry out the integration in equation (29.24) alternatively, the integral may be evaluated graphically. It is thus posdble to determine the variation of the fugacity with temperature. 

By following the procedure given in 29f, with / representing the fugacity of pure liquid or solid, an equation exactly analogous to (29.22) is obtained for the variation of the fugacity with temperature at constant pressure. As before, H is the molar heat content of the gas, i.e., vapor, at low pressure, but H is now the molar heat content of the pure liquid or solid at the pressure P. The difference — H has been called the ideal heat of vaporization, for it is the heat absorbed, per mole, when a very small quantity of liquid or solid vaporizes into a vacuum. The pressure of the vapor is not the equilibrium value, but rather an extremely small pressure where it behaves as an ideal gas. 

The equations derived in 30c, 30d thus also give the variation with pressure and temperature of the fugacity of a constituent of a liquid (or solid) solution. In equation (30.17), Vi is now the partial molar volume of the particular constituent in the solution, and in (30.21), i is the corresponding partial molar heat content. The numerator — fti thus represents the change in heat content, per mole, when the constituent is vaporized from the solution into a vacuum (cf. 29g), and so it is the ideal" heat of vaporization of the constituent i from the given solution, at the specified temperature and total pressure. 

T and pressure P. It should be noted that equation (33.26) is the exact form of the Clausius-Clapeyron equation (27.12). If the vapor is assumed to be leal, so that the fugacity may be replaced by the vapor pressure, and the total pressuic is taken as equal to the equilibrium pressure, the two equations become identical. In this simplification the assumption is made that the activity of the liquid or solid does not vary with pressure this is exactly equivalent to the approximation used in deriving the Clausius-Clapeyron equation, that the volume of the liquid or solid is negligible. 

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