Gibbs functions

Gibbs function (see Gibbs free energy, free energy) gram... 

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. 

Arguments have been presented that this difference in changes in Gibbs function, rather than the similar difference in enthalpies of activation, AHl — AH, better represents the quantity with which... 

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. 

The heat capacity of thiazole was determined by adiabatic calorimetry from 5 to 340 K by Goursot and Westrum (295,296). A glass-type transition occurs between 145 and 175°K. Melting occurs at 239.53°K (-33-62°C) with an enthalpy increment of 2292 cal mole and an entropy increment of 9-57 cal mole °K . Table 1-44 summarizes the variations as a function of temperature of the most important thermodynamic properties of thiazole molar heat capacity Cp, standard entropy S°, and Gibbs function - G°-H" )IT. 

Fuel Cell Efficiency The theoretical energy conversion efficiency of a fuel cell ° is given by the ratio of the free energy (Gibbs function) of the cell reaction at the cell s operating temperature AG to the enthalpv of reaction at the standara state AH°, both quantities being based on a mole of fuel ... 

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... 

Let us apply Equation (6.8) to the two-phase liquid-vapor equilibrium requirement for a pure substance, namely p = p T) only. This applies to the mixed-phase region under the dome in Figure 6.5. In that region along a p-constant line, we must also have T constant. Then for all state changes along this horizontal line, under the p—v dome, dg = 0 from Equation (6.8b). The pure end states must then have equal Gibbs functions ... 

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...

In the following sections, we introduce the concept of a thermodynamic universe (i.e. a system plus its surroundings). For a reaction to occur spontaneously in a system, we require the change in Gibbs function G to be negative. We then explore the thermodynamic behaviour of G as a function of pressure, temperature and reaction composition. 

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. 

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