The Gibbs function

To implement the reconstruction of the initial image, using denoised and/or noisy data given by simulated projections The algorithm (1) and the Gibbs functional in the form (12) were used for the reconstruction. The coefficients a and P were optimized every time. 

The results of the estimation of the distortion criterion s are presented in fig. 4. It is seen that the distortion increases with the noise. But the main conclusion is that the Gibbs functional (12) provides a satisfactory reconstruction having quite large noise up to 0.15, what can not be provided by other known reconstruction techniques. 

As has been noted above, there is no gross change in the mechanism of nitration of PhNH3+ down to 82 % sulphuric acid. The increase in o- andp-substitution at lower acidities has been attributed differential salt effects upon nitration at the individual positions. The two sets of partial rate factors quoted for PhNH3+ in table 9.3 show the effect of the substituent on the Gibbs function of activation at the m- and -positions to be roughly equal for reaction in 98 % sulphuric acid, and about 28 % greater at the -position in 82 % sulphuric acid. ... 

If this electrostatic treatment of the substituent effect of poles is sound, the effect of a pole upon the Gibbs function of activation at a particular position should be inversely proportional to the effective dielectric constant, and the longer the methylene chain the more closely should the effective dielectric constant approach the dielectric constant of the medium. Surprisingly, competitive nitrations of phenpropyl trimethyl ammonium perchlorate and benzene in acetic anhydride and tri-fluoroacetic acid showed the relative rate not to decrease markedly with the dielectric constant of the solvent. It was suggested that the expected decrease in reactivity of the cation was obscured by the faster nitration of ion pairs. 

For large departures from T, we have to fall back on eqn. (5.21) in order to work out Wf. Thermodynamics people soon got fed up with writing H - TS all the time and invented a new term, the Gibbs function G, defined by... 

Fig. 5.6. Plot of the Gibbs functions for ice and water os functions of temperature. Below the melting point Tn, > Gjjj and ice is the stable state of HjO above T , G...

Among many other valuable results in his memoir on heterogeneous equilibrium is a formulation of the Gibbs free energy, also called the Gibbs function, which is defined by the equation... 

Note that eq. (3.16) involves the Helmholtz function, A, rather than the Gibbs function G, but the difference between A A and AG is negligible in solutions and we will use AG in this book. 

The Landau theory applies in the vicinity of a critical point where the order parameter is small and assumed continuous. The Gibbs function... 

If the continuous order parameter is written in terms of an order parameter density m(x), = f rn(x)dx and the Gibbs function... 

The Clausius-Clapeyron equation provides a relationship between the thermodynamic properties for the relationship psat = psat(T) for a pure substance involving two-phase equilibrium. In its derivation it incorporates the Gibbs function (G), named after the nineteenth century scientist, Willard Gibbs. The Gibbs function per unit mass is defined... 

Figure 3.4 If geographical position were a thermodynamic variable, it would be a state function because it would not matter if we travelled from London to New York via Athens or simply flew direct. The net difference in position would be identical. Similarly, internal energy, enthalpy, entropy and the Gibbs function (see Chapter 4) are all state functions...

The Gibbs function G is named after Josiah Willard Gibbs (1839-1903), a humble American who contributed to most areas of physical chemistry. He also had a delightful sense of humour A mathematician may say anything he pleases, but a physicist must be at least partially sane. ... 

As well as calling G the Gibbs function, it is often called the Gibbs energy or (incorrectly) free energy. ... 

The Gibbs function is the energy available for reaction after adjusting for the entropy changes of the surroundings. 

This compound variable occurs so often in chemistry that we will give it a symbol of its own G, which we call the Gibbs function. Accordingly, a spontaneous process in a system is characterized by saying,... 

The Gibbs function is a function of state, so values of AG obtained with the van t Hoff isotherm (see p. 162) and routes such as Hess s law cycles are identical. 

In a similar way, we say that the value of the Gibbs function changes in response to changes in pressure and temperature. We write this as... 

Figure 4.6 The value of the Gibbs function AG decreases as the extent of reaction until, at (eq), there is no longer any energy available for reaction, and AG = 0. =0 represents no reaction and = 1 mol represents complete reaction...

The sign of the Gibbs function determines reaction spontaneity, so a reaction will occur if AG is negative and will not occur if AG is positive. When the reaction is poised at 7)cntlcai between spontaneity and non-spontaneity, the value of AG = 0. 

The temperature dependence of the Gibbs function change is described quantitatively by the Gibbs-Helmholtz equation. 

What is the value of the Gibbs function for this reaction when the temperature is increased by a further 34 K ... 

Writing the equation in this way tells us that if we know the enthalpy of the system, we also know the temperature dependence of G -i-T. Separating the variables and defining Gj as the Gibbs function change at Ti and similarly as the value of G2 at T2, yields... 

The chapter also outlines the criteria for equilibrium in terms of the Gibbs function and chemical potential, together with the criteria for spontaneity. 

Earlier, on p. 181, we looked at the phase changes of a single-component system (our examples included the melting of an ice cube) in terms of changes in the molar Gibbs function AGm. In a similar manner, we now look at changes in the Gibbs function for each component within the mixture and because several components participate, we need to consider more variables, to describe both the host and the contaminant. 

We look once more at Figure 5.18, but this time we concentrate on the thinner lines. These lines are seen to be parallel to the bold lines, but have been displaced down the page. These thin lines represent the values of Gm of the host within the mixture (i.e. the once pure material following contamination). The line for the solid mixture has been displaced to a lesser extent than the line for the liquid, simply because the Gibbs function for liquid phases is more sensitive to contamination. 

The chemical potential // can be thought of as the constant of proportionality between a change in the amount of a species and the resultant change in the Gibbs function of a system. 

The way we wrote 3G in Equation (5.13) suggests the chemical potential // is the Gibbs function of 1 mol of species i mixed into an infinite amount of host material. For example, if we dissolve 1 mol of sugar in a roomful of tea then the increase in Gibbs function is /x,sugar> - An alternative way to think of the chemical potential // is to consider dissolving an infinitesimal amount of chemical i in 1 mol of host. 

The protons and oxide ions combine to form water. Again, the value of AGr for Equation (7.36) is negative, because the reaction is spontaneous. AG would be positive if we wrote Equation (7.36) in reverse. The change in sign follows because the Gibbs function is a function of state (see p. 83). 

Each reaction has a unique value of rate constant k. For example, the value of k in Worked Example 8.2 would have been different if we had chosen ethyl formate, or ethyl propanoate, or ethyl butanoate, etc., rather than ethyl ethanoate. The value of k depends ultimately on the Gibbs function of forming reaction intermediates, as discussed below. 

Figure 6.1 Search for the minimum of the Gibbs function in a two-component space (nn and ni2 are mole numbers) with the mass conservation constraints Bn = q. The search direction is the projection of the gradient onto the constraint subspace. Minimum is attained when the gradient is orthogonal to the constraint direction, which is the geometrical expression of the Lagrange multiplier methods.

Figure 8.10 According to Darken s theory, the sign of the diffusion coefficient changes where the second derivative of the Gibbs function relative to the molar fraction Xt vanishes (spinodal).

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