Minimizing Gibbs Energy

Example 12.1 Estimate the chemical equilibrium composition of a gaseous mixture of -butane and isobutane at 298.15 K and 1 bar, based on direct minimization of Gibbs energy. Assume that -butane and isobutane form an ideal solution of ideal gases at this T and P. 

FIGURE 12.1 Calculated Gibbs energy for a mixture of normal and isobutane at 25°C = 298.15 K and 1.00 bar. Tbe pure component values are from Table A.8 and tbe intermediate values based on an assumed ideal solution. The minimum value, which corresponds to the chemical equilibrium between the two isomeric forms, occurs at 

We differentiate Eq. 12.3 with respect to Xa, remembering that for a binary mixture dxb = —dxg, finding 

Setting fliis equal to zero (to find the minimum on the g-x curve) and solving gives 

Inserting the values of the pure component Gibbs energies from Table A.8 (discussed below) for isobutane, a, and -butane, b, we find 

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In the text, we focused our attention on minimizing Gibbs energy to determine the chemical equilibrium at fixed temperature and pressure. To be fair to the other energy functions, what system function would be at an extremum (maximum or minimurn) if the following were specified. Provide a short derivation verifying your result starting with the second law, dSm > 0. 

Le Chatelier s principle is a detailed restatement of the fact that natural systems minimize their Gibbs energy. They respond to changes in T and P by moving in the direction that minimizes Gibbs energy. 

Chapter 12 discussed only single reactions in one phase, involving only nonionized species. In this chapter we apply the results of that chapter to more complex equilibria. There are no new principles or new ideas in this chapter, only applications of the principles and ideas from previous chapters to a new set of more complex problems. Again, nature minimizes Gibbs energy all we do here is estimate the concentrations at which that minimum occurs. 

Multiple reactions, in series, in parallel, or both, introduce no new concepts. Nature minimizes Gibbs energy, regardless of how many reactions are occurring. As the number or reactions increases, the mathematics and the number of variables to be accounted for increase. 

All of biochemistry obeys the general principle that nature minimizes Gibbs energy. 

This book is intended for university juniors in chemical or environmental engineering. It explains the fundamentals of physical and chemical equilibriiun and how these relate to practical problems in chemical and environmental engineering. The student will find that our understanding of equilibrium is based on thermodynamics. Nature attempts to minimize Gibbs energy this book shows some of the details of that minimization. 

Complex Clieinical-Reaction Equilibria When the composition of an equilibrium mixture is determined by a number of simultaneous reactions, calculations based on equilibrium constants become complex and tedious. A more direct procedure (and one suitable for general computer solution) is based on minimization of the total Gibbs energy G in accord with Eq. (4-271). The treatment here is... 

When the kinetics are unknown, still-useful information can be obtained by finding equilibrium compositions at fixed temperature or adiabatically, or at some specified approach to the adiabatic temperature, say within 25°C (45°F) of it. Such calculations require only an input of the components of the feed and produc ts and their thermodynamic properties, not their stoichiometric relations, and are based on Gibbs energy minimization. Computer programs appear, for instance, in Smith and Missen Chemical Reaction Equilibrium Analysis Theory and Algorithms, Wiley, 1982), but the problem often is laborious enough to warrant use of one of the several available commercial services and their data banks. Several simpler cases with specified stoichiometries are solved by Walas Phase Equilibiia in Chemical Engineering, Butterworths, 1985). 

This is a 4 2 reaction, and is thus pressure dependent. However, it is necessary to compute the equilibrium partial pressure of some alternative gaseous species, such as SiCls, and other hydrocarbons such as C2H2 and for this a Gibbs energy minimization calculation should be made. 

The values of these functions change when thermodynamic processes take place. Processes in which the Gibbs energy decreases (i.e., for which AG<0), will take place spontaneously without specific external action. The Gibbs energy is minimal in the state of equilibrium, and the condition for equilibrium are given as... 

Kulik, D. A., 2002, Gibbs energy minimization approach to model sorption equilibria at the mineral-water interface Thermodynamic relations for multi-site-surface complexation. American Journal of Science 302,227-279. ... 

At equilibrium the Gibbs energy of the system is minimal for harmonic oscillators this is the same as requesting that the potential energy is zero. By differentiation we find the equilibrium values of the solvent coordinates ... 

Minimization of the transitional Gibbs energy with respect to rgives... 

Minimization of Gibbs energy and heterogeneous phase equilibria... 

The equilibrium distribution of pairs is calculated by minimizing the Gibbs energy at a given composition. This gives... 

The equilibrium structure at applied pressure p and temperature T can be found by minimizing the Gibbs energy, G, given by... 

Gibbs energy minimization has also predicted negative isobaric expansion coefficients for certain crystalline zeolite framework structures, which subsequently were confirmed experimentally [6], Many solids show negative thermal expansion at very low temperatures, including even some alkali halides (Barron and White (Further reading)). Many other solids on heating expand in some directions and contract in others. 

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