Thermodynamic Corrections

Many semiempirical methods compute energies as heats of formation. The researcher should not add zero-point corrections to these energies because the thermodynamic corrections are implicit in the parameterization. 

A rapid increase in diffusivity in the saturation region is therefore to be expected, as illustrated in Figure 7 (17). Although the corrected diffusivity (Dq) is, in principle, concentration dependent, the concentration dependence of this quantity is generally much weaker than that of the thermodynamic correction factor d ap d a q). The assumption of a constant corrected diffusivity is therefore an acceptable approximation for many systems. More detailed analysis shows that the corrected diffusivity is closely related to the self-diffusivity or tracer diffusivity, and at low sorbate concentrations these quantities become identical. 

For concentrated binary liquid nonhydrocarbon systems, the method of Caldwell and Babb, " Eq. (2-156) has been modified by introduction of a thermodynamic correction term as shown in Eq. (2-158). 

Since the infinite dilution values D°g and Dba. re generally unequal, even a thermodynamically ideal solution hke Ya = Ys = 1 will exhibit concentration dependence of the diffusivity. In addition, nonideal solutions require a thermodynamic correction factor to retain the true driving force for molecular diffusion, or the gradient of the chemical potential rather than the composition gradient. That correction factor is ... 

Zeohte ciyst lite diffusivities for sorbed gases range from 10" to lO" " cmVs. These diffusivities generally show a strong increase with the adsorbate concentration that is accounted for by the Darken thermodynamic correction factor... 

Calculating the electronic barrier with an accuracy of 0.1 kcal/mol is only possible for very simple systems. An accuracy of 1 kcal/mol is usually considered a good, but hard to get, level of accuracy. The situation is slightly better for relative energies of stable species, but a 1 kcal/mol accuracy still requires a significant computational effort. Thermodynamic corrections beyond the rigid rotor/harmonic vibrations approximation are therefore rarely performed. 

It should be stressed that the pH value of an actual buffer solution prepared by mixing quantities of the weak acid or base and its conjugate base or acid based on the calculated ratio will likely be different from what was calculated. The reason for this is the use of approximations in the calculations. For example, the molar concentration expressions found in Equations (5.23) to (5.30), e.g., [H+], are approximations. To be thermodynamically correct, the activity of the chemical should be used rather than the concentration. Activity is directly proportional to concentration, the activity coefficient being the proportionality constant ... 

To be thermodynamically correct, the tendencies for half-reactions to occur depend on the activities of the chemical species involved, not the concentrations. See Chapter 5 (Section 5.2.12) for a brief discussion of activity. 

Once again, to be thermodynamically correct, activity should be used rather than concentration. Use of concentration is an approximation. 

For single-component gas permeation through a microporous membrane, the flux (J) can be described by Eq. (10.1), where p is the density of the membrane, ris the thermodynamic correction factor which describes the equilibrium relationship between the concentration in the membrane and partial pressure of the permeating gas (adsorption isotherm), q is the concentration of the permeating species in zeolite and x is the position in the permeating direction in the membrane. Dc is the diffusivity corrected for the interaction between the transporting species and the membrane and is described by Eq. (10.2), where Ed is the diffusion activation energy, R is the ideal gas constant and T is the absolute temperature. 

We can answer these questions using Eq. (30.1), which defines relative efficiency. The calculated numerical value of relative efficiency means nothing The equation may be used only to compare two sets of operating data. The equation is not even thermodynamically correct. But it is sufficiently correct, provided the services represented by the two sets of data are reasonably similar. 

In hydrate equilibrium, it may seem slightly unusual to apply it to binary systems (water and one guest component) of three-phase (Lw-H-V or I-H-V) equilibrium to obtain the heats of dissociation. As van der Waals and Platteeuw (1959b) point out, however, the application of the Clapeyron equation is thermodynamically correct, as long as the system is univariant, as is the case for simple hydrates. 

Finally, it appears that the kinetic models of complex reactions contain two types of components independent of and dependent on the complex mechanism structure [4—7]. Hence the thermodynamic correctness of these models is ensured. The analysis of simple classes indicates that an unusual analog arises for the equation of state relating the observed characteristics of the open chemical system, i.e. a kinetic polynomial [7]. This polynomial distinctly shows how a complex kinetic relationship is assembled from simple reaction equations. 

Thermodynamic correction factor Y is defined using the Gibbs-Duhem relation... 

The thermodynamic correction factor T can be expressed in terms of the activity using Eq. (6.39)... 

Rathbun and Babb [20] suggested that Darkens equation could be improved by raising the thermodynamic correction factor PA to a power, n, less than unity. They looked at systems exhibiting negative deviations from Raoult s law and found n = 0.3. Furthermore, for polar-nonpolar mixtures, they found n = 0.6. In a separate study, Siddiqi and Lucas [22] followed those suggestions and found an average absolute error of 3.3 percent for nonpolar-nonpolar mixtures, 11.0 percent for polar-nonpolar mixtures, and 14.6 percent for polar-polar mixtures. Siddiqi, Krahn, and Lucas (ibid.) examined a few other mixtures and... 

Figure 5.4 Finite-size correction of the probability densities of the electrostatic energies of positively charged imidazolium ion in water. The uncorrected distributions are shown with symbols, together with corresponding Gaussian distributions. In addition to the electrostatic correction, a thermodynamic correction is also applied, but this correction is small in magnitude, see Hummer etal. (1998ft). With the corrections, the distributions collapse and agree closely for all system sizes of 16 < iV < 512 water molecules.

Figure 11 shows that our rate equation, with appropriate thermodynamic corrections, satisfactorily predicts observed rate in sea water over a wide range of pH and PCO2 for values of Q between 0.0 and 0.6. Nearer equilibrium, uncertainties in the calculations and the experimental data ( 3) are too large for a reliable test of our model. 

Geometry level al which the structure is optimized higher-order correlation method(s) for estimating higher-order correlation effects thermo level at which the thermodynamical corrections are calculated [vibrational scale factor] MAD Mean Absolute Deviation for reference data set in kcal/mol. 

The surface diffiisivity (Dp) is known to be a function of adsorbed phase concentration and is equal to the corrected diffiisivity (D p) multiplied by a thermodynamic correction factor (dlnP/dlnCp). Assuming that the driving force for surface diffusion is the gradient of the chemical potential and that the mobility constant is a function of adsorbed phase concentration. Do and Do [6] have introduced the following form for the corrected sur ce diffiisivity ... 

Conversely, the correct approach to formulate the diffusion of a single component in a zeolite membrane is to use the MaxweU-Stefan (M-S) framework for diffusion in a nonideal binary fluid mixture made up of species 1 and 2 where 1 and 2 stands for the gas and the zeohtic material, respectively. In the M-S theory it is recognized that to effect relative motions between the species 1 and 2 in a fluid mixture, a force must be exerted on each species. This driving force is the chemical potential gradient, determined at constant temperature and pressure conditions [68]. The M-S diffiisivity depends on coverage and fugacity, and, therefore, is referred to as the corrected diffiisivity because the coefficient is corrected by a thermodynamic correction factor, which can be determined from the sorption isotherm. 

In order to determine the activation energy of the difiuaon, the uptake experiments were conducted at temperatures in the range 398 to 473 K. In Table 2, results are compiled. The errors of the transport diffusion coefficient are estimated to be 0.75 10 cmVs. A thermodynamic correction [13] of the transport diffiiavity has not been applied. However, since the... 

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