Free-energy minimization

The above calculation used free energies and the equilibrium constants for specific chemical reactions. For this reason the calculations apply only to the system, species, and reactions considered. It is possible to generalize free energy based equilibrium calculations so that they can be used with systems and species chosen arbitrarily, in a manner analogous to the program EQBRM described above. 

We observed in Chapter 3 that a chemical system is at thermodynamic equilibrium when the Gibbs free energy of the system is at a minimum. For a given pressure, temperature, and bulk composition, at equilibrium there will be one or more phases in which the concentrations of all species are fixed. Many different methods have been developed to compute the equilibrium state, as outlined by Van Zeggeren and Storey (1970) and Smith and Missen (1982). As an illustration, we will summarize an algorithm derived by Harvie et al. (1987), which has been applied with great success to geochemical systems. 

All generalized equilibrium-solving algorithms based on free energies use some numerical technique to locate the minimum on the free energy-composition surface. These are called optimization methods, and there are many from which to choose (see the summary by Van Zeggeren and Storey, 1970, Ch. 3, for example). The procedure developed by Harvie et al. (1987) uses Lagrangian multipliers. 

Let s state the problem in non-mathematical language first. We need to find the unique concentrations of species in a system that minimize the total free energy subject to the following constraints or conditions. 

To phrase the problem mathematically, the same notation used to describe the EQBRM algorithm above will be used as much as possible. Consider a system containing c components, N chemical species, and f) phases. At equilibrium, the total free energy is minimized  

---

Reactor types modeled A, stoichiometric conversion B, equiUbrium/free-energy minimization, continuous stirred tank, and plug flow C, reactive distillation. Some vendors have special models for special reactions also, private company simulators usually have reactors of specific interest to their company. 

Flame Temperature Theoretical temperature achieved based on chemical equilihrium with the assumption of Gihhs free energy minimization... 

R.C. Oliver et al, USDeptCom, Office Tech-Serv ..AD 265822,(1961) CA 60, 10466 (1969) Metal additives for solid proplnts formulas for calculating specific impulse and other proplnt performance parameters are given. A mathematical treatment of the free-energy minimization procedure for equilibrium compn calcns is provided. The treatment is extended to include ionized species and mixing of condensed phases. Sources and techniques for thermodynamic-property calcns are also discussed... 

Daoud and Cotton [10] pioneered this geometrical analysis of tethered layers with spherical symmetry, which was later extended by Zhulina et al. [36] and Wang et al. [37] to cylindrical layers. The subsequent analysis is purely geometrical and requires no free energy minimization. The tethered layer consists of a stratified array of blobs such that all blobs in a given sublayer are of equal size, E , but blobs in different layers differ in size. This corresponds to the uniform stretching assumption of the Alexander model. 

Smit et al. [19] used the partition function given by (10.4) and a free energy minimization procedure to show that, for a system with a first-order phase transition, the two regions in a Gibbs ensemble simulation are expected to reach the correct equilibrium densities. 

Harvie, C.E., J.P. Greenberg and J. H. Weare, 1987, A chemical equilibrium algorithm for highly non-ideal multiphase systems free energy minimization. Geochimica et Cosmochimica Acta 51, 1045-1057. 

Typically, solving (5.151) to find fc(oo ) is not the best approach. For example, in combusting systems Srp(0 4)1 < 1 so that convergence to the equilibrium state will be very slow. Thus, equilibrium thermodynamic methods based on Gibbs free-energy minimization are preferable for most applications. 

The change in state from liquid to solid as a material crystallizes, is driven by thermodynamics and the principle of free energy minimization, in turn this results from a trade between the total enthalpy and entropy of a system. 

Thermodynamic calculations presented here are based on Gibbs free energy minimization and were carried out using HSC Chemistry. The equilibrium amount of each species that is formed is normalized on the basis of one mole of n-Ci6, a model compound for diesel fuel, fed to the reactor. Carbon formation is a function of both the S/C ratio and reforming temperature. Figure 17 shows the minimum amount of S/C ratio thermodynamically required for carbon-free SR of n-Ci6 at a given temperature. Carbon-free operation of n-Cig is thermodynamically possible above the curve. Higher temperatures and S/C ratios inhibit carbon formation. 

At first, the superconducting state was not thought to be a thermodynamic equilibrium state. But, as we know from Chapter 2, any equilibrium state must be the result of a free energy minimization. It can be shown that the superconducting and normal states will be in equilibrium when their free energies are equal, and that the free energy difference between the normal state at zero field, G H = 0), and the superconducting state at zero field, Gs H = 0), is... 

Van der Waals acknowledges the unexpected difficulty in extending his equation to binary mixtures, and the key role of Gibbs theory of free energy minimization in handling both binary mixtures and association complexes. 

The coefficients in Eq. 19.7 may be taken as constants independent of the form of the distribution of cluster sizes. The relationship between the two coefficients fa and qaa+i may then be obtained by imposing an artificial constraint on the system no clusters are allowed to grow beyond a limiting size, Af m, which is considerably larger than the critical size Afc. At the same time, all clusters of size below this limit are allowed to equilibrate with respect to one another so that detailed balance is achieved between them and all fluxes in cluster space go to zero. A new distribution of cluster sizes, N q, will be produced in this constrained system. It is assumed that the same free-energy minimization procedure used previously to find the size distribution under true equilibrium conditions, and which led to Eq. 19.6, may be used for the constrained system, resulting in... 

Carlo simulations [16] or free energy minimizations [18], in u, yield twin domain walls oriented along a diagonal. 

Mathews, D. H., and Turner, D. H. (2006). Prediction of RNA secondary structure by free energy minimization. Cun. Opin. Struct. Biol. 16, 270-278. 

Wittenberg, G.K., Haun, D.V., and Parsons, M.L., The use of free-energy minimization for calculating beta factors and equilibrium compositions in flame spectroscopy, Appl. Spectrosc., 33, 626, 1979. 

The second reason for a reaction not going to completion is that it proceeds to a state of material equilibrium in which both reactants and products exist. This is the case for all gas-phase reactions, because free energy minimizes in gas mixtures when some of each component is present. (See Problem 16 in Chapter 4.) In fact, even in the case of the spark-initiated H2-02 reaction discussed earlier, some H2 and 02 will remain after the reaction. Equilibrium is more evident with the gaseous molecule N02, a fraction of which exists as the dimer, N204, near ambient conditions. This fraction depends on the temperature and pressure and rapidly adjusts to changes in these variables, indicating that there is... 

Thermodynamic software packages may be used to find equilibrium compositions at prescribed temperatures and pressures. Such calculations require knowledge of feed components and products and their thermodynamic properties and are based on Gibbs free energy minimization techniques. Examples of thermodynamic packages may be found in Smith and Missen (Chemical Reaction Equilibrium Analysis Theory and Algorithms, Wiley, 1982) and in Walas (Phase Equilibria in Chemical Engineering, Butterworths, 1985). 

In one approach, T is assumed and Gibbs free energy minimized to give compositions. Then T is adjusted to satisfy the specification e.g., H or S. Naphtali (10) suggests this procedure, but does not present an algorithm for adjusting T. 

---