Free energy example

Calculate an equilibrium constant from a standard Gibbs free energy (Example 9.3). 

Representative minimum free-energy examples of conformations in the different pseudophases of a 179-mer in a cavity. The substrate is shaded in light gray. From [302]. 

I.P.P.D and its relatives have become standard procedures for the characterization of the structure of both clean surfaces and those having an adsorbed layer. Somoijai and co-workers have tabulated thousands of LEED structures [75], for example. If an adsorbate is present, the substrate surface structure may be altered, or reconstructed, as illustrated in Fig. VIII-9 for the case of H atoms on a Ni(llO) surface. Beginning with the (experimentally) hypothetical case of (100) Ar surfaces. Burton and Jura [76] estimated theoretically the free energy for a surface transition from a (1 x 1) to a C(2x 1) structure as given by... 

The basic phenomenon involved is that particles of ore are carried upward and held in the froth by virtue of their being attached to an air bubble, as illustrated in the inset to Fig. XIII-4. Consider, for example, the gravity-free situation indicated in Fig. XIII-5 for the case of a spherical particle. The particle may be entirely in phase A or entirely in phase B. Alternatively, it may be located in the interface, in which case both 7sa nnd 7sb contribute to the total surface free energy of the system. Also, however, some liquid-liquid interface has been eliminated. It may be shown (see Problem XIII-12) that if there is a finite contact angle, 0sab> the stable position of the particle is at the interface, as shown in Fig. XIII-5Z>. Actual measured detachment forces are in the range of 5 to 20 dyn [60]. 

If there are other kinds of work, similar expressions apply. For example, with electromagnetic work (equation (A2.1.8)1 instead of pressure-volume work, one can write for the Helmholtz free energy... 

A second source of standard free energies comes from the measurement of the electromotive force of a galvanic cell. Electrochemistry is the subject of other articles (A2.4 and B1.28). so only the basics of a reversible chemical cell will be presented here. For example, consider the cell conventionally written as... 

PEP theory has also been applied to modelling the free energy profiles of reactions in solution. An important example is the solvent effect on the SN2 reaction... 

In 1972 Wegner [25] derived a power-series expansion for the free energy of a spin system represented by a Flamiltonian roughly equivalent to the scaled equation (A2.5.28). and from this he obtained power-series expansions of various themiodynamic quantities around the critical point. For example the compressibility... 

In prineiple, nothing more is neeessary to understand the infiuenee of the solvent on the TST rate eonstant than the modifieation of the PMF, and the resulting ehanges in the free energy barrier height should be viewed as the dominant effeet on the rate sinee tliese ehanges appear in an exponential fonn. As an example, an error... 

The applications of this simple measure of surface adsorbate coverage have been quite widespread and diverse. It has been possible, for example, to measure adsorption isothemis in many systems. From these measurements, one may obtain important infomiation such as the adsorption free energy, A G° = -RTln(K ) [21]. One can also monitor tire kinetics of adsorption and desorption to obtain rates. In conjunction with temperature-dependent data, one may frirther infer activation energies and pre-exponential factors [73, 74]. Knowledge of such kinetic parameters is useful for teclmological applications, such as semiconductor growth and synthesis of chemical compounds [75]. Second-order nonlinear optics may also play a role in the investigation of physical kinetics, such as the rates and mechanisms of transport processes across interfaces [76]. 

It is possible to calculate derivatives of the free energy directly in a simulation, and thereby detennine free energy differences by thenuodynamic integration over a range of state points between die state of interest and one for which we know A exactly (the ideal gas, or hanuonic crystal for example) ... 

We have previously calculated conformational free energy differences for a well-suited model system, the catalytic subunit of cAMP-dependent protein kinase (cAPK), which is the best characterized member of the protein kinase family. It has been crystallized in three different conformations and our main focus was on how ligand binding shifts the equilibrium among these ([Helms and McCammon 1997]). As an example using state-of-the-art computational techniques, we summarize the main conclusions of this study and discuss a variety of methods that may be used to extend this study into the dynamic regime of protein domain motion. 

Such a free energy is called a potential of mean force. Average values of Fs can be computed in dynamics simulations (which sample a Boltzmann distribution), and the integral can be estimated from a series of calculations at several values of s. A third method computes the free energy for perturbing the system by a finite step in s, for example, from si to S2, with... 

Conformational free energy simulations are being widely used in modeling of complex molecular systems [1]. Recent examples of applications include study of torsions in n-butane [2] and peptide sidechains [3, 4], as well as aggregation of methane [5] and a helix bundle protein in water [6]. Calculating free energy differences between molecular states is valuable because they are observable thermodynamic quantities, related to equilibrium constants and... 

Let us illustrate this with the example of the bromination of monosubstituted benzene derivatives. Observations on the product distributions and relative reaction rates compared with unsubstituted benzene led chemists to conceive the notion of inductive and resonance effects that made it possible to explain" the experimental observations. On an even more quantitative basis, linear free energy relationships of the form of the Hammett equation allowed the estimation of relative rates. It has to be emphasized that inductive and resonance effects were conceived, not from theoretical calculations, but as constructs to order observations. The explanation" is built on analogy, not on any theoretical method. 

As an example, experimental kinetic data on the hydrolysis of amides under basic conditions as well as under acid catalysis were correlated with quantitative data on charge distribution and the resonance effect [13]. Thus, the values on the free energy of activation, AG , for the acid catalyzed hydrolysis of amides could be modeled quite well by Eq. (5)... 

Two approaches to quantify/fQ, i.e., to establish a quantitative relationship between the structural features of a compoimd and its properties, are described in this section quantitative structure-property relationships (QSPR) and linear free energy relationships (LFER) cf. Section 3.4.2.2). The LFER approach is important for historical reasons because it contributed the first attempt to predict the property of a compound from an analysis of its structure. LFERs can be established only for congeneric series of compounds, i.e., sets of compounds that share the same skeleton and only have variations in the substituents attached to this skeleton. As examples of a QSPR approach, currently available methods for the prediction of the octanol/water partition coefficient, log P, and of aqueous solubility, log S, of organic compoimds are described in Section 10.1.4 and Section 10.15, respectively. 

A consequence of writing the partition function as a product of a real gas and an ideal g part is that thermod)mamic properties can be written in terms of an ideal gas value and excess value. The ideal gas contributions can be determined analytically by integrating o the momenta. For example, the Helmholtz free energy is related to the canonical partitii function by ... 

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