Fugacity with Temperature

Let us consider an isothermal process in which a gas is transformed form one state A at a pressure P to another A at a different pressure P. Such a transformation can be represented as follows  

The change in the Gibbs function for such a transformation is given by the expression AG = p. p, 

The partial derivative of the fugacity with respect to temperature is given by 

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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. 

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. 

Equation (9-177) describes the variation of vapor fugacity with temperature when the liquid composition is held constant and the total pressure is adjusted so as to preserve equilibrium. The quantity dpjdT) can be obtained from the slope of a pressure-temperature diagram. When p is small,... 

In their correlation, Chao and Seader use the original Redlich-Kwong equation of state for vapor-phase fugacities. For the liquid phase, they use the symmetric convention of normalization for y and partial molar volumes which are independent of composition, depending only on temperature. For the variation of y with temperature and composition, Chao and Seader use the equation of Scatchard and Hildebrand for a multicomponent solution ... 

In summary, the composition of Fe-Ti oxides in magmatic rocks provides the pet-rologist with important information about the oxygen fugacity and temperature and also the silicon activity of the magma. It also has a strong effect on the magnetic properties of these phases (see Chap. 6 7). 

According to eq. (2.3-30), the temperature mainly influences the sublimation pressure and the fugacity coefficient in the supercritical phase. The sublimation pressure always increases with increasing temperature, and if this is the main effect the solubility must always increases with temperature. To the contrary, at relatively low pressures (close to the critical pressure of the supercritical fluid) the fugacity coefficient in the supercritical phase plays the most important and preponderant role. 

This equation also gives the dependence of the activity Op on the temperature, since this activity is equal to the fugacity. Further, since Op is equal to ypP, where yp is the activity coefficient, the same equation represents the variation of the activity coefficient with temperature, p being constant. 

Heterogeneous physical equilibria, e.g., between a pure solid and its vapor or a pure liquid and its vapor, can be treated in a manner similar to that just described. If the total pressure of the system is 1 atm., the fugacity of the vapor is here also equivalent to the equilibrium constant. The variation of In/ with temperature is again given by equation (33.16), where MP is now the ideal molar heat of vaporization of the liquid (or of sublimation of the solid) at the temperature T and a pressure of 1 atm. If the total pressure is not 1 atm., but is maintained constant at some other value, the dependence of the fugacity on the temperature can be expressed by equation (29.22), since the solid or liquid is in the pure state thus,... 

The use of the foregoing definition of an ideal solution implies certain properties of such a solution. The variation of the fugacity / of a pure liquid i with temperature, at constant pressure and composition, is given by equation (29.22), viz.. 

Calcnlation of vapor-liqnid eqnilibrinm states nsing K values is particularly convenient for ideal mixtures for which the K values are independent of composition, changing only with temperature or pressure or both. While convenient, the A -value method does satisfy all the equilibrium conditions the component fugacities are equal in both phases for all components in the mixture, and temperature and pressure are equal in both phases. It follows that the A-value method can be used in general for noifideal mixtures as well. The composition dependence of the A values of nonideal mixtures is addressed next. 

Most approximations of this class involve the relative magnitudes of the partial derivatives of the activity coefficients, fugacities, and the departure function Q with respect to temperature. If, for example, the Q is independent of temperature or its variation with temperature is small, then the approximation dQ/dT = 0 may be made. 

Normally of course the expression for the variation of K with P is simpler than this, perhaps because all three states of matter may not be present, but also because it is quite unusual to use a variable pressure standard state for constituents whose fugacities are known or sought, (because this adds complexities rather than simplifying matters), and the In Qig) term is therefore essentially never required. To take a real example, let s consider the brucite-periclase reaction again. We have discussed the variation of the equilibrium constant for the brucite-periclase-water reaction with temperature at one bar, and showed that the equilibrium temperature for the reaction at one bar is about 265°C. Calculation of the equilibrium temperature of dehydration reactions such as this one at higher pressures was discussed briefly in 13.2.2. Here we will discuss the reaction in different terms to demonstrate the relationships between activities, standard states and equilibrium constants. 

The variation of fugacity with the temperature can be found by examining the derivative of the equation (5.93) ... 

In this section, we consider a binary solution in equilibrium with vapor. We shall investigate in some detail the dependence of the vapor fugacities on temperature, pressure, and composition. 

The derivative of the fugacity with respect to composition at constant temperature can be obtained from Eq. (9-168) in the form... 

Equation (9-168) can be utilized to determine how the vapor composition varies with temperature at constant fugacity. We find... 

Determine the equilibrium composition that is achieved at 300 bar and 700 K when the initial mole ratio of hydrogen to carbon monoxide is 2. You may use standard enthalpy and Gibbs free energy of formation data. For purposes of this problem you should not neglect the variation of the standard heat of reaction with temperature. You may assume ideal solution behavior but not ideal gas behavior. You may also use a generalized fugacity coefficient chart based on the principle of corresponding states as well as the heat capacity data listed below. 

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