Chemical potential equation

For a polydisperse polymer, analysis of sedimentation equilibrium data becomes complex, because the molecular weight distribution significantly affects the solute distribution. In 1970, Scholte [62] made a thermodynamic analysis of sedimentation equilibrium for polydisperse flexible polymer solutions on the basis of Flory and Huggins chemical potential equations. From a similar thermodynamic analysis for stiff polymer solutions with Eqs. (27) for IT and (28) for the polymer chemical potential, we can show that the right-hand side of Eq. (29) for the isotropic solution of a polydisperse polymer is given, in a good approximation, by Eq. (30) if M is replaced by Mw [41],... 

Without the standard chemical potential, Equation 5.19 becomes in terms of absolute chemical potential (fx = kTinX),... 

The repulsive frequency shift, Av0, is expressed explicitly in terms of the first and second derivatives of the excess chemical potential (equation 2) along with the vapor phase vibrational transition frequency, vvib, equilibrium bond length, re, and harmonic and anharmonic vibrational force constants, f and g (232528). 

If we had expressed the chemical potential of the third component in terms of its molality and activity coefficient rather than the mole fraction and excess chemical potential, Equation (10.258) would be expressed as... 

With the expression of the chemical potentials, equation (1) gives the Darcy s law [9] ... 

When transformed Gibbs energies of formation are used rather than chemical potentials, equation 4.3-4 can be written... 

The Nature of Logarithmic Terms in Chemical Potential Equations... 

If the relativistic kinetic energy functional T[p] is completely local, then one can rewrite the general chemical potential equation of DFT as... 

As noted above, the chemical potential equation can be generalized from the non-relativistic form in Eq. (42) to include non-zero fine structure constant ot. The result in Eq. (58) can be rewritten as [6,40]... 

These equations now allow us to express the affinity as a function of the chemical potentials. Equation (4.30) gives us immediately (c/. 2.35)... 

By extending the definition of chemical potentials (equation 3.47) to cations In ( )), and converting equation 3.51 from logarithmic to exponential form, the equilibrium expression becomes ... 

The LFHB model or the former LFAS (lattice fluid associated solution) modeP° can provide the needed equations for the chemical potential as a function of composition. The picture that emerges from application of the LFHB and LFAS models in this case is, essentially, identical. For the chemical potential. Equation 2.30 can be combined with Equation 2.A23 of Appendix 2.A to provide the required expression. On the other hand, the experimental data can be correlated to provide the appropriate expressions a(X2> for the surface tension. - ... 

Unfortunately, these simple relationships between pAAB and PA/PA do not hold for large values of pAAB. The general relation between the two quantities follows from the general expression for the chemical potential (equation 6.2, chapter 6)... 

Triad 1 is also designed to explore one way to couple a photoinduced electron transfer process to a change in proton chemical potential. Equations 2 and 3 illustrate two processes involved in the decay of the final charge-separated state to the ground state. 

Solving chemical potential equations for concentrations in the solution between clay particles we receive... 

In a real solution the energy of electrostatic field included in the equation of chemical potential (equation (1.57)) as additional third addend ... 

The Smoluchowski equation can be obtained more formally from the concept of chemical potential. Equation (3.10) can be rewritten as... 

From the dependence of the gain on the dipole chemical potential (Equation D3.7), the capacitive relationship for a physical chemical pole is deduced (see Section 7.4.2) ... 

The Langmuir isotherm (and the Langmuir chemical potential. Equation 4.42) is the ideal adsorption case, similar in some aspects to the ideal gas law. Despite this, it is found to describe satisfactorily the experimental data in many cases, either in gas phase or in solution, including many cases in soil science. Presumably, in a nnmber of cases, a compensation of effects takes place. 

Table 5.1 collects the different conventions discussed in the preceding paragraphs, including the equations linking the selectivity coefficients with the true equilibrium constant, whenever possible. It should be remarked now that the numerical value of and, of course, of the selectivity coefficients depends on the convention chosen. This should be clear considering Equation 5.33 and the general equation for the chemical potential (Equation 2.5) ... 

The osmotic pressure of an ideal solution can be calculated from the phenomenological expression for the chemical potential (Equation 5.12). The pressure dependence of the activity is ignored ... 

Phase separation occurs at temperatures satisfying x>Xc Equation [27] reveals that then the second derivative will be negative for a range of values. Because / is often inversely proportional to temperature (cf. solubility approach discussion), this implies that phase separation will occur at reduced temperatures. The calculation of the critical point and the spinodal within the Flory-Huggins theory is rather simple, but the determination of the coexistence curve is slightly more involved. For this, the chemical potential equations ) and A/U2 (p ) = Afi2 (p ) have to be... 

We close this section with a reminder of a fnndamental issue in electrochemistry Not all the quantities in Equations 13.8 throngh 13.13 are accessible to measurement by electrochemical or thermodynamic methods. Only the electrochemical potential ( i ), the work function (W ) or equivalently the real potential (a ) and the Volta potential ( / ) are. Equations 13.9, 13.11, and 13.13 are therefore formal resolutions. It is not possible to assign actual values to the separate terms, the chemical potential ( t ), the Galvani potential (cp ), nor the surface potential (x ), without making extrathermodynamic assumptions. These quantities must therefore be considered unphysical, at least from the point of view of thermodynamics. This statement, which is called the Gibbs-Guggenheim Principle in [42], is often met with disbelief from theoretical and computational chemists, particularly in the case of the chemical potential (Equation 13.10). The standard chemical potential is essentially the (absolute) solvation free energy AjG of species i. One would hope that a molecular simulation contains all information needed to compute AjG . Indeed, there seems to be a way around this thermodynamic verdict for computation and also mass spectroscopic. This continues to be, however, hazardous territory, particularly for DFT calculations in periodic systems. ... 

The chemical potential equation is arguably one of the most important thermodynamic equations vis k vis phase science and solubility. It states that the actual chemical potential of each component in a mixture is the sum of two very different terms. One of these, x , is called the standard-state chemical potential of component i. Like all energy parameters, the absolute values of standard-state chemical potentials cannot be numerically determined, but they are, nevertheless, very important. They have two characteristic attributes ... 

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