Real gases Gibbs energy

It is tempting to view the fugacity as a sort of corrected pressure it is, however, a well-defined fiincrion related to the exponential of the difference between the real and ideal gas Gibbs energies. 

Example 1.12 Chemical potential of a real gas Similar to Eq. (1.142), the Gibbs free energy for a real gas is... 

Considering the first equality, Is measurable. It Is the Isothermal reversible work required to extract ions from phase a for electrons In a metal It represents the electronic work function. For metals, a, can also be obtained from thermo-emission or the photo-electric effect. Sometimes a is called the real (Gibbs) energy of hydration of ion i. The logic behind this last definition stems from the second equality In [3.9.61. The standard molar Gibbs energy of solvation of an Ion [1.5.3.11 equals when Is referred to the gas... 

Finally, a new class of solution properties is introduced. Known as excesspropevties, they are based on an idealization of solution behavior called tire ideal solution. Its role is like that of tire ideal gas in that it serves as a reference for real-solution belravior. Of particular interest is tire excess Gibbs energy, a property wlriclr underlies tire activity coefficient, introducedfrom a practical point of view hr tire preceding clrapter. 

Tliis equation takes on a new dimension when G f, the Gibbs energy of pure species i m the ideal-gas state, is replaced by Gj, tire Gibbs energy of pure species i as it actually exists at tire mixhire T and P and hr tire same physical state (real gas, liquid, or solid) as the mixture. It then applies to species in real solutions. We therefore define an ideal solution as one for which ... 

The functional form of Eq. (10.47) is particularly simple and convenient. It would be helpful if the molar Gibbs energy of real gases could be expressed in the same mathematical form. We therefore invent a function of the state that will express the molar Gibbs energy of a real gas by the equation... 

The function / is called the fugacity of the gas. The fugacity measures the Gibbs energy of a real gas in the same way as the pressure measures the Gibbs energy of an ideal gas. 

In order to correct for the nonideal behavior of the real gas the pressure is replaced by the fiigadty/. The Gibbs energy of a real gas is... 

The positive value indicates that work must be done on the mixture to achieve an isobaric separation. In a real isothermal-isobaric separation of ideal gases, more than this minimum amount of work would be needed, because a real process would be irreversible. Moreover, when separating real mixtures (whose components have inter-molecular forces), the total minimum work would not be given by (4.1.47). However, it could still be determined from O using (4.1.46), provided a reliable model were available for the Gibbs energy of the mixture and each pure. Expressions for G of real mixtures would be more complicated than the ideal-gas expression (4.1.47) but such expressions could be obtained from model equations of state. 

To obtain a physical interpretation for the residual Gibbs energy, we start with an ideal-gas mixture confined to a closed vessel. As the process, we consider the reversible isothermal-isobaric conversion of the ideal-gas molecules into real ones. Although this process is hypothetical, it is a mathematically well-defined operation in statistical mechanics the process amounts to a "turning on" of intermolecular forces. We first want to obtain an expression for the work, but since the process involves a change in molecular identities, we must start with the general energy balance (3.6.3). For a system with no inlets and no outlets, (3.6.3) becomes... 

This shows that the residual Gibbs energy can be interpreted physically as the reversible isothermal-isobaric shaft work involved in "turning on" intermolecular forces, thereby converting ideal-gas molecules into real molecules. In general this work may be positive or negative. For a single component (6.3.15) reduces to... 

First, the term A ix° refers to the difference in Gibbs energies of products and reactants when each product and each reactant, whether solid, liquid, gas, or solute, is in its pure reference state. This means the pure phase for solids and liquids [e.g., most minerals, H20( ), H20(/), alcohol, etc.], pure ideal gases at Ibar [e.g., 02( ), H20( ), etc.], and dissolved substances [solutes, e.g., NaCl(ag), Na+, etc.] in ideal solution at a concentration of Imolal. Although we do have at times fairly pure solid phases in our real systems (minerals such as quartz and calcite are often quite pure), we rarely have pure liquids or gases, and we never have ideal solutions as concentrated as 1 molal. 

This is the Nernst equation, after the physical chemist W. Nernst, who derived it at the end of the nineteenth century. As above, n is the number of electrons transferred in the cell reaction (2 in reaction 12.7), 5 the Faraday of charge, R the gas constant, and T the temperature (in kelvins). The constant 2.302 59 is used to convert from namral to base 10 logs. At 25 the quantity 2.30259 RT/3 has the value 0.05916, which is called the Nernst slope. The importance of (12.14) is that it allows calculation of the potentials of cells having nonstandard state concentrations (i.e., real cells) from tabulated values of standard half-cell values or tabulated standard Gibbs energies. 

Real-gas thermodynamic properties may be expressed as functions of the variables of state, according to relations which may be developed from first principles. A comprehensive list of such relations has been given by Beattie and Stockmayer. For example, the molar enthalpy H, the molar entropy S, and the molar Gibbs energy G of a real gas can be written in terms of p, T, and p (the amount density) as follows ... 

In these equations, the real-gas properties are expressed relative to the standard state values, H°, S°, and G°, which are, respectively, the molar enthalpy, the molar entropy, and the molar Gibbs energy values which the gas would have at a standard pressure p° (1.013 25 bar) if it were ideal. 

The method used to obtain a relation (6.2.7) between I/ideai and I/reai can also be used to relate the corresponding Helmholtz and Gibbs free energies. The main idea is that as p 0 or y —> oo, the thermodynamic quantities for a real gas approach those of an ideal gas. Let us consider the Helmholtz free energy F. Since (0F/0y)y = —p (5.1.6) we have the general expression... 

The chemical potential for a real gas can be derived from the expression (6.2.23) for the Gibbs free energy. Since the chemical potential of the component k is = pG/QNk)p j, by differentiating (6.2.23) with respect to Nk, we obtain... 

Molar Gibbs energy or chemical potential of a real gas... 

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