The Most Useful Equation in Thermodynamics

Consider a chemical reaction, involving any number of reactants and products and any number of phases, which may be written 

Now let s recall (from Chapter 3) what we mean by an equation such as (13.3). If there are no constraints placed on the system containing Mi, M2, and M3 other than T and P (or T and V U and V S and P etc.) then Mi, M2, and M3 react until they reach an equilibrium state characterized by a minimum in the appropriate energy potential as indicated by expressions like dGr p = 0. A corollary of this equilibrium relationship, to be fully developed in the next chapter, is that the sums of the chemical potentials of the reactants and products must be equal. In the example, this would be 

No notation is necessary for the phases involved because /ij must be the same in every phase in the system. 

However, if more than the minimum two constraints apply to the system, then any equilibrium state achieved will be in our terms a metastable state, (14.25) does not apply, and the difference in chemical potential between products and reactants is not zero. In our example, a solution might be supersaturated with H4Si04 but prevented from precipitating quartz by a nucleation constraint, so that /iH4Si04 A si02 2//H2O 0. 

As we have just noted, Arpb is not necessarily zero, and is not if the system is in a metastable state, but when the system achieves equilibrium with respect to the minimum two constraints (what we have called stable equilibrium), ArP becomes zero, the activities in the Q term take on their stable equilibrium values, and aj is called K instead of Q. Thus at stable equilibrium, 

If we added some H4Si04 to this solution it would then be supersaturated, MH4Si04 would be greater than its equilibrium value, and the reaction would tend to go to the left, precipitating quartz.  

To find out what we can say about this balanced equilibrium state when several solutes and other phases are involved, let s consider a general chemical reaction 

There may be any number of reactants and products, and so to be completely general we can write 

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. 

Therefore, A,.yu,° usually refers to quite a hypothetical situation. It is best not to try to picture what physical situation it might represent, but to think of it as just the difference in numbers that are obtained from tables. 

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Equation (1.16) is one example of the Gibbs-Duhem equation, which is one of the most useful formulae in thermodynamics. Thus, the partial molal quantity defined as the quantity satisfying the additive property is easily understandable. 

In this equation, Kp is used for gases and for reactions in solution. Note that the larger the K is, the more negative AG° is. For chemists. Equation (18.14) is one of the most important equations in thermodynamics because it enables us to find the equilibrium constant of a reaction if we know the change in standard free energy... 

This equation links the EMF of a galvanic cell to the Gibbs energy change of the overall current-producing reaction. It is one of the most important equations in the thermodynamics of electrochemical systems. It follows directly from the first law of thermodynamics, since nF% is the maximum value of useful (electrical) work of the system in which the reaction considered takes place. According to the basic laws of thermodynamics, this work is equal to -AG . 

Formal thermodynamics does not rest on KMT or other molecular assumptions (hence, their relegation to sidebar status in this book). Nevertheless, thermodynamic studies are highly valued for their ability to provide fundamental insights into the intermolecular forces that underlie chemical phenomena. Indeed, the most successful advances in thermodynamic theory and practice are often inspired by molecular insights, and the productive interplay between microscopic and macroscopic domains should be emphasized in a pedagogically useful presentation of thermodynamic principles. Accordingly, we discuss equations of state in terms of their ability to suggest improvements over the KMT ideal gas picture of intermolecular interactions. 

This is one of the most important equations in the whole of chemical thermodynamics. Its principal use is to predict the value of the equihbrium constant of any reaction from tables of thermodynamic data, hke those in the Resource section. Alternatively, we can use it to determine Afi by measuring the equilibrium constant of a reaction. 

The Gibbs-Duhem equation is one of the most extensively used relations in thermodynamics. It is written in the following equivalent forms for a binary solution at constant temperature and pressure ... 

Design of extraction processes and equipment is based on mass transfer and thermodynamic data. Among such thermodynamic data, phase equilibrium data for mixtures, that is, the distribution of components between different phases, are among the most important. Equations for the calculations of phase equilibria can be used in process simulation programs like PROCESS and ASPEN. 

The equation AG° = —RT In K is one of the most important relationships in chemical thermodynamics because it allows us to calculate the equilibrium constant for a reaction from the standard free-energy change, or vice versa. This relationship is especially useful when K is difficult to measure. Consider a reaction so slow that it takes more than an experimenter s lifetime to reach equilibrium or a reaction that goes essentially to completion, so that the equilibrium concentrations of the reactants are extremely small and hard to measure. We can t measure K directly in such cases, but we can calculate its value from AG°. 

The virial equation of state, first suggested by Kammerlingh-Ohnes, is probably one of the most convenient equations to use, and is used in this chapter to illustrate the development of the thermodynamic equations that are consistent with the given equation of state. The methods used here can be applied to any equation of state. 

In chemical kinetics, the key parameters are traditionally the rate con stants and the concentrations of reactants. At the same time for thermody namic analysis, the parameters such as chemical potential appear to be the most useful. That is why the most convenient way to consider chemical transformations in terms of thermodynamics of nonequihbrium processes is a thermodynamic form of kinetic equations. The main elements of the application of this form are given following. 

Arsenic oxide, AS4O10, cannot be formed by the direct combination between the elements owing to the greater thermodynamic stability of AS4O6, which promotes the back dissociation reaction illustrated in equation (20). The most useful preparative method involves the oxidation of elemental arsenic with concentrated nitric acid, followed by careful dehydration of H3ASO4 (equation 21). 

Despite its simplicity, this is probably the most widely useful equation in all of thermodynamics. It is now that we can appreciate the importance of AG° data values for different reactions may be combined to give the equilibrium constant for any other reaction of interest. 

The free volume model has been also incorporated into thermodynamic theories of liquids and solutions [Prigogine et al., 1957] and it is an integral part of theories used for the interpretation of thermodynamic properties of polymer blends [Utracki, 1989a]. In particular, it is a part of the most successful equation of state (EoS) derived for liquids and glasses [Simha and Someynsky,... 

This equation has been called, with some reason, the most useful in chemical thermodynamics, and it certainly merits the most careful attention. Several things about it need comment. 

The most critical aspect in simulation is the selection of appropriate thermodynamic models. Different models can be used on different parts of the flowsheet, or for some units. For example, equation of state model can be used for the whole flowsheet, but liquid activity models are more suitable for separations. The accuracy of model parameters should be checked systematically. Thermodynamic analysis tools should be used systematically to evaluate the accuracy of phase equilibria before detailed simulation of separations. 

Chemical equilibrium model Most reactive transport formulations use the mass action law to solve the chemical equilibrium equations. In this formulation an alternative (though thermodynamically equivalent) approach is used, based on the minimization of Gibbs Free Energy. This approach has a wider application range extending to highly non-ideal brine systems. 

The problem is to derive the equation of state and thermodynamic functions of a particular liquid crystal phase from properties of constituting molecules (a form, a polarizability, chirality, etc.). The problem we are going to discuss is one of the most difficult in physics of liquid crystals and the aim of this chapter is very modest just to introduce the reader to the basic ideas of the theory with the help of comprehensive works of the others [2, 5, 19]. To consider the problem quantitatively we need special methods of the statistical physics. In this context, the most useful function is free energy F, which is based microscopically rai the so-called partition function, see below. For the partition function, we need that energy spectrum of a molecular system, which is relevant to the problem imder cmisider-ation. The energy spectrum is related to the entropy of the system and we would like to recall the microscopic sense of the entropy. 

The concept of pH, however, is not applicable in such nonaqueous systems or in concentrated acid solutions. A new quantitative scale, therefore, was needed (5,9-11). The most useful and widely accepted method was proposed by Hammett and Deyrup in 1932 (12). They defined ho by equation 24, which can be determined experimentally by adding the neutral base B in low concentrations to an acid solution (BH+ is the acidic form of the indicator, Ksa+ is the thermodynamic equilibrium constant for BH+, and [BH+]/[B] is the ionization ratio generally determined spectrophotometrically). Equation 24 is usually written in the logarithmic form (eq. 25), where the quantity Hq is termed the Hammett acidity function. Since in dilute solutions of acids Abh+ is expressed as in equation 26, the Hammett acidity function becomes equal to pH. In concentrated solutions, however, Hq differs considerably from pH and this can be formally expressed by inserting activity coefficients in equation 24 (eq. 27). 

The most complete tabulation of thermodynamic data available for neon is given in a paper by Yendall [1]. In this paper, Yendall used the 102 experimental PVT observations of the Leiden Laboratory [2] made between 1915 and 1919 to compute an equation of state. This equation of state was a modification of the Beattie-Bridgeman and Benedict-Webb-Rubin equations and used twelve coefficients. 

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