An Alternate Derivation

According to the concept of free volume as the effective volume over which the centers of gravity of the molecules are distributed, the entropy may be taken as that of a perfect gas composed of the same number of molecules confined to a volume equal to the free volume. Since the entropy of a perfect gas consisting of n molecules depends on its volume as nk InF, the increase in entropy owing to the greater free volume available to the solvent molecules in the solution will be 

The sum of these expressions represents the entropy of mixing. The result obtained by substituting for the free volumes from Eqs. (13), (14), and (15), is identical with Eq. (10). According to this simple derivation, the terms appearing in Eq. (10) represent contributions to the entropy which originate in the greater spatial freedom of the molecules in the solution. By a simple extension of the derivation to a mixture of polymer species, Eq. (12) may be obtained. 

An obvious refinement suggested by this second derivation would consist in ascribing different free volume fractions Vf to the two pure 

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The analogue of the Clapeyron equation for multicomponent systems can be derived by a complex procedure of systematically eliminating the various chemical potentials, but an alternative derivation uses the Maxwell relation (A2.1.41)... 

An alternative derivation, which uses the Omstein-Zemicke equation ( equation (A2.4.12). was given by Lee [2], The Omstein-Zemicke equation can be written as... 

The Untruncated Hilbert Space An Alternative Derivation General Implications... 

Now, we recall the remarkable result of [72] that if the adiabatic electronic set in Eq. (90) is complete (N = oo), then the curl condition is satisfied and the YM field is zero, except at points of singularity of the vector potential. (An algebraic proof can be found in Appendix 1 in [72]. An alternative derivation, as well as an extension, is given below.) Suppose now that we have a (pure) gauge g(R), that satisfies the following two conditions ... 

This conclusion can be confirmed by an alternative derivation of Eq. (1.71). According to Mori [52], Eq. (1.71) may be obtained from a generalized Langevin equation ... 

Crison et al. [52] presented an alternative derivation of Eq. (29) that included individualized transport of the solute and micelle while still maintaining the basic assumption of equilibrium. This was accomplished by rewriting Eq. (24) to include the magnitude of the individual diffiisional boundary layers for free drug and drug-loaded micelle according to Eq. (10), as follows ... 

With applications to protein solution thermodynamics in mind, we now present an alternative derivation of the potential distribution theorem. Consider a macroscopic solution consisting of the solute of interest and the solvent. We describe a macroscopic subsystem of this solution based on the grand canonical ensemble of statistical thermodynamics, accordingly specified by a temperature, a volume, and chemical potentials for all solution species including the solute of interest, which is identified with a subscript index 1. The average number of solute molecules in this subsystem is... 

Finally, an alternative derivation of AEins that introduces the static... 

However, Olgin employed a somewhat unwieldy matrix partition method to locate the surplus value term in the output multiplier matrix. An alternative derivation is suggested by substituting (A2.4) into (A2.1) such that... 

An alternate derivation of the Washburn equation can be pursued as follows. For a pore of circular cross-section with radius r the surface tension acts to force a nonwetting liquid out of the pore. The force developed due to interfacial tensions is the product of the surface tension y of the liquid and the circumference (2nr) of the pore, that is. 

A promising method based on an integral equation formulation of the problem of scattering by an arbitrary particle has come into prominence in recent years. It was developed by Waterman, first for a perfect conductor (1965), later for a particle with less restricted optical properties (1971). More recently it has been applied to various scattering problems under the name Extended Boundary Condition Method, although we shall follow Waterman s preference for the designation T-matrix method. Barber and Yeh (1975) have given an alternative derivation of this method. 

Compounds of stoichiometry AX2 with six-coordinated A require (according to eqn (11.1)) that X be three coordinate. Since none of the close packed lattices have cage points with three coordination, these structures are less simple. The rutile (202240) and anatase (202242) forms of Ti02 are based on FICP and FCC lattices of Ti respectively, but fitting the ions into positions of three coordination results in distortions that lower the symmetry. An alternative derivation of these structures is described in Section 11.2.2.4 below. 

A second steady-state method involves the analysis of the broadening of the nuclear magnetic resonance spectra of phospholipids in bilayers containing low concentrations of spin-labeled phospholipids. A theoretical analysis of the relation between this line broadening and diffusion rates has been given by Brulet and McConnell.3 [In this paper (6) is not correct the subsequent equations are nonetheless correct. For an alternative derivation, see Brulet.2] In this paper it is shown that a number of measurements of nuclear relaxation rates T71 of nuclei in phospholipids are consistent with lateral diffusion constants in the range 10 7 to 10 R cm2/s. 

The Laplace equation applied specifically to spherical surfaces can be derived in a variety of ways. Example 6.1 considers an alternative derivation that points out the thermodynamic character of the result quite clearly. 

Often the thermochemical properties, for example, p,°, are known for the gas-phase species. An alternate derivation is to equate Eqs. 9.58 and 11.76 at equilibrium, leading to the following expression for Kp ... 

Classroom exercise to derive Beer s law R. W. Ricci, M. A. Ditzler, and L. P. Nestor, Discovering the Beer-Lambert Law, J. Chem. Ed. 1994, 71, 983. An alternate derivation W. D. Bare, A More Pedagogically Sound Treatment of Beer s Law A Derivation Based on a Corpuscular-Probability Model, J. Chem. Ed. 2000, 77, 929. 

Suppose one is faced with a one-step problem in which the coefficients rn and g are nonlinear but can be represented by smooth functions r(n), g(n). Smooth means not only that r(n) and g(n) should be continuous and a sufficient number of times differentiable, but also that they vary little between n and n+ 1. Suppose furthermore that one is interested in solutions pn(t) that can similarly be represented by a smooth function P(n, t). It is then reasonable to approximate the problem by means of a description in which n is treated as a continuous variable. Moreover, since the individual steps of n are small compared to the other lengths that occur, one expects that the master equation can be approximated by a Fokker-Planck equation. The general scheme of section 2 provides the two coefficients, but we shall here use an alternative derivation, particularly suited to one-step processes. 

This section represents the conclusive part of our work on the quantum-classical equations of motion derived in section 5, following the prescriptions of Ref. [15]. We will show an alternative derivation of the quantum-classical equation of motion (60), obtained by taking the limit hi —> h, h2 — 0 in eq.(44), which is an ansatz on the mixed dynamical generator, after making some remarks on the equation of motion itself and on the operators used as generators in the representation of the group D", i.e., the position Xj>ol and momentum haD a operators. 

Even the layout of the periodic table of the elements cannot be derived from quantum theory without assuming an empirical concept, known as the Pauli exclusion principle. An alternative derivation (4.6.1) through number theory predicts the correct periodicity, without assuming the exclusion principle. In fact, the operation of an exclusion principle can be inferred from this periodic structure and reduced to a property of space, but it remains impossible to reconstruct or predict from more basic principles. 

The Gibbs-Duhem equation (50.6), derived below, proves to be a useful starting point for an alternative derivation of the Clausius Claperyron equation to that offered in Frame 26) and offers an alternative proof of the Phase Rule to that given in Frame 30. 

A. The Thermal Fluctuations of the Interfaces for Arbitrary Interactions. After the Helfrich initial theory,18 Helfrich and Servuss17 suggested an alternate derivation of the entropic repulsion due to the confinement of a membrane between rigid walls, by considering the lipid bilayer composed of many independent pieces , whose area is related to the root mean square fluctuations of the positions of the undulatingbilayer. As shown below, this representation can be extended to interfaces interacting via arbitrary potentials. 

An alternative derivation proceeds as follows Consider a function F(x1,x2,. .. xr) for which we write dF - FjdXi + F2dx2 +. .. + Frdxr, with F (dF/dXi). Now change all independent... 

We provide an alternative derivation leading to Eqs. (1.6.13a) and (1.6.15a) by starting with the Maxwell field equations... 

The reader is asked in Exercise 1.17.1 to demonstrate that the specification of T and S is independent of the choice of working material - a most important result. An alternative derivation is provided in Section 1.18, after the development of further thermodynamic interrelations. 

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