Second Derivatives of the Free Energy

Among the second-order derivatives of the free energy, we distinguish between two subgroups those that are definable only in the frozen-in system and those that may be defined in either the frozen-in or the equilibrated system. We start with quantities belonging to the first group. The partial molar entropies of L and H are obtained by differentation, with respect to temperature, of the corresponding chemical potentials in (6.47) and (6.48)  

Note that in our model, the quantities E, E, and are assumed to be temperature independent hence, the partial molar entropies are 

One should refer strictly to partial molecular, rather than partial molar quantities. The distinction between the two must be made clear by the context in which they are used. 

Other partial molar quantities are obtainable from (6.47) and (6.48) from simple thermodynamic relations  

Note that on the rhs of (6.59)-(6.61), we have molecular quantities, such as El, Eb, and V, whereas on the Ihs, we have partial molecular quantities. 

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In a second-order transition, the second derivatives of the free energy show changes ... 

Second-order transitions are treated similarly. Properties that are second derivatives of the free energy show a discontinuity with changes in temperature. For example, the second derivative with respect to temperature and pressure is... 

For a molecule in solution described by the PCM model, the nuclear shielding constant and the indirect spin-spin coupling constants are determined as second derivatives of the free energy functional G of the solute-solvent system [34] ... 

First and Second Derivatives of the Free Energy in Solution... 

To account for indirect solvent effects, solvation models must allow for geometry optimizations and frequency calculations including the solute-solvent interactions. Indeed, many ab initio continuum solvation models and in particular those belonging to the family of the PCM [3] provide analytical first and second derivatives of the free energy with respect to the nuclear coordinates [4,5], In the following we shall present in detail the formalism for the derivatives in the PCM and Conductor-PCM (CPCM) [6] models. 

Another important extension of the theory concerns NMR chemical shift. Yamazaki et al. proposed a theory for computing the chemical shift of solvated molecules [17]. The nuclear magnetic shielding tensor o-x of a nucleus X can be represented as a mixed second derivative of the free energy A with respect to the magnetic field B and the nuclear magnetic moment mx ... 

Because of the complexity of the processes involved and the various methods of measuring them, phase diagrams of organogels can be envisioned by different theoretical approaches. In one option, network formation can be considered according to a spinodal decomposition mechanism. Briefly, the spinodal curve in a phase diagram represents the limit of metastability defined by the second derivative of the free energy with respect to concentration Important features of such a mechanism are (1) the phase separation in solute-rich and solute-poor... 

Similarly, if a quantity such as the volume exhibits an abrupt change in slope, which occurs at the T, then there is a discontinuity in quantities associated with first derivatives of this parameter, or second derivatives of the free energy (with respect to appropriate thermodynamic variables), such as the specific heat (Figure 10-19). Accordingly, the Tg may be related to a second-order phase transition, but this remains in dispute. The experimentally observed transition is clearly governed by kinetics and the standard method of measuring this transition is by differential scanning calorimetry (DSC), which measures the specific heat. 

The term in the second derivative of the free energy with respect to composition (or first derivative of the chemical potential or osmotic pressure), brings in a virial equation, usually expressed in a slightly different form than we used in our discussion of osmotic pressure (Equation 12-35) ... 

Within the miscibility gap there are unstable and metastable regions, separated by inflection points at which the second derivative of the free energy is zero (5 AFtiiix/ 0 = 0)- Between the inflection points, the second derivative of the free energy is negative and the homogeneously mixed state is unstable. Even the smallest fluctuations in composition... 

Further cooling of the polymer below results in a further decrease in V, which again follows the coefficient of thermal expansion of the solid polymer, until there is a change in slope of the plot at the glass-transition temperature, Tg. This is a second-order transition (in contrast to melting, which is a first-order transition) since there is a discontinuity in the second derivative of the free energy with respect to temperature and pressure, i.e. 

Fig. 3. Schematic of second-order (first line) and first-order (second line) phase transitions. The first and second columns are variables related to the first and second derivatives of the free energy, respectively. V = volume H = enthalpy S = entropy C = heat capacity a = coefficient of thermal expansion p = order parameter and x l = the inverse of the susceptibility x l df/dp.

The question is now whether the phase separation will occur spontaneously. Consider the second derivative of the free energy G" = (d2G/dm2)Tp. The points where G" = 0 denote inflection points, in... 

The Fm3m—>74/m transition is second order since it is characterized by discontinuities in the second derivatives of the free energy, such as heat capacity [37]. Figure 2b shows that for some Ln3+, a linear relation exists between Tc and the lattice constant extrapolated to the phase transition temperature. Actually, the Tc=0 intercept from linear regression in Fig. 2b for the Cs2NaLnCl6 series is 1070.4 pm, which suggests that these compounds... 

Evidently, Dj depends on the second derivative of the free energy, which in turn can be expressed in terms of the self-diffusion coefficients of polymers 1 and 2, as well as to the... 

The implementation of PCM methods to compute magnetic response properties has started later but has rapidly reached computational effectiveness. Cammi [54] has elaborated the formalism for the magnetic susceptibility x and the magnetic shielding Both quantities can be expressed as second derivatives of the free energy G with respect to Cartesian components of the magnetic field (B) and of the magnetic moment of nucleus N (/I ) ... 

The glass transition is a second-order transition, that is, the second derivative of the free energy function. In comparison, a first-order transition, such as the melting of a material, is the first derivative of Gibbs free energy. The equation used to determine the glass transition is ... 

Now, the modulus B is related to the second derivative of the free energy with respect to the average layer spacing i.e., imagine a uniform expansion or compression of the system. TTie restoring force is just the effective value of B which is proportional to the macroscopic compressibility of the system. Thus, following Helfrich we can obtain a self-consistent equation to determine B from... 

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