A Physical Interpretation

Usually we c culate the fugacity through the i gacity coefficient. Before, however, we discuss how the fugacity coefficient is calculated, let us consider a physical interpretation of fugacity. 

Fugacity has units of pressure as Eq.9.10.4 indicates, and a comparison of Eqs 9.10.1 and 9.10.2 suggests. Beyond this, however, its physical meaning becomes elusive. 

Consider, now, the case of liquid water at 100°C and 1 atm, in equilibrium with its vapor, i.e. saturated water. According to Webster s definition of fugacity we would expect that, due to the equilibrium between the two phases, the molecules of the liquid are as apt to flee their 

We will demonstrate in Chapter 12, through a combination of the first and second laws, that this also represents a thermodynamic requirement  

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Bain A D and Duns G J 1996 A unified approach to dynamic NMR based on a physical interpretation of the transition probability Can. J. Chem. 74 819-24... 

The calculated energy differences give a good correlation with The p parameter (p = —17) is larger than that observed experimentally for proton exchange (p — 8). A physical interpretation of this is that the theoretical results pertain to the gas phase, where... 

Kennedy and Benedick [67K02, 68K03] were successful in carrying out difficult Hall effect measurements in germanium samples explosively loaded at the upper end of the elastic range. Nevertheless, the measurements did not provide sufficient information to develop a physical interpretation. 

If there is more than one constraint, one additional multiplier term is added for each constraint. The optimization is then performed on the Lagrange function by requiring that the gradient with respect to the x- and A-variable(s) is equal to zero. In many cases the multipliers A can be given a physical interpretation at the end. In the variational treatment of an HF wave function (Section 3.3), the MO orthogonality constraints turn out to be MO energies, and the multiplier associated with normalization of the total Cl wave function (Section 4.2) becomes the total energy. 

In order to give the Reynolds number a physical interpretation, we can look at the typical magnitudes of individual terms of the Navier-Stokes equation. Since I V Vv V /L and vVjl , we see that... 

Zsako [519] accepts certain points in Gam s reappraisal of compensation behaviour but stresses that the existence of a linear relation between log A and E is a more general characteristic. Thus, while a physical interpretation for obedience to eqn. (21) cannot at present be provided, the relationship provides parameters which are useful in describing the reactivities of groups of related rate processes. 

Khanna et al. [136] proposed a mechanism of the reactions of aluminum based clusters with O, which lends a physical interpretation as to why the HOMO-LUMO gap of the clusters successfully predicts the oxygen etching behaviors. The importance of the HOMO-LUMO gap strongly suggests that the reactions of the metal clusters belong to the pseudoexcitation band. 

Formal verification that this result actually satisfies Equation (14.13) is an exercise in partial differentiation, but a physical interpretation will confirm its validity. Consider a small group of molecules that are in the reactor at position z at time t. They entered the reactor at time i = t — (zju) and had initial composition a t, z) = ai (t ) = ai (t — z/u). Their composition has subsequently evolved according to batch reaction kinetics as indicated by the right-hand side of Equation (14.14). Molecules leaving the reactor at time t entered it at time t — t. Thus,... 

Give examples of desorption systems following first-, second- and zero-order kinetics. Can you give a physical interpretation for the latter ... 

As an example of the interest to scrutinise the UHF solution, one may quote the Bea problem [19]. The bond is weak but it takes plaee at short interatomic distance and is definitely not the dispersion well which one might expect from two closed shell atoms (and which occurs in Mga and heavier eompounds). Quantum chemical calculations only reproduce this bond when using large basis sets and extensive Cl calculations [20]. It is amazing to notice that the UHF solution gives a qualitatively correct behaviour, and suggests a physical interpretation of this bond since in... 

From the results described above it is clear that a different QSPR model can be obtained depending on what data is used to train the model and on the method used to derive the model. This state of affairs is not so much a problem if, when using the model to predict the solubility of a compound, it is clear which model is appropriate to use. The large disparity between models also highlights the difficulty in extrapolating any physical significance from the models. Common to all models described above is the influence of H-bonding, a feature that does at least have a physical interpretation in the process of aqueous solvation. 

Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle. Several experiments, which lead to a physical interpretation of the wave function, are also described. In Chapter 2, the Schrodinger differential wave equation is introduced and the wave function concept is extended to include particles in an external potential field. The formal mathematical postulates of quantum theory are presented in Chapter 3. 

The essential features of the particle-wave duality are clearly illustrated by Young s double-slit experiment. In order to explain all of the observations of this experiment, light must be regarded as having both wave-like and particlelike properties. Similar experiments on electrons indicate that they too possess both particle-like and wave-like characteristics. The consideration of the experimental results leads directly to a physical interpretation of Schrodinger s wave function, which is presented in Section 1.8. 

Young s double-slit experiment and the Stem-Gerlaeh experiment, as described in the two previous sections, lead to a physical interpretation of the wave function associated with the motion of a particle. Basic to the concept of the wave function is the postulate that the wave function contains all the... 

The wave function P itself is not observable. A physical interpretation can only be associated with the square of the wave function in that... 

Finally, we should note Koopmans theorem (Koopmans, 1934) which provides a physical interpretation of the orbital energies e from equation (1-24) it states that the orbital energy e obtained from Hartree-Fock theory is an approximation of minus the ionization energy associated with the removal of an electron from that particular orbital i. e., 8 = EN - Ey.j = —IE(i). The simple proof of this theorem can be found in any quantum chemistry textbook. 

The vanishing of the second term in the optimum state arises from a cancelation that lends itself to a physical interpretation. This expression for the second entropy may be rearranged as... 

O Kedem, A Katchalsky. A physical interpretation of the phenomenological coefficients of membrane permeability. J Gen Physiol 45 143-179, 1961. 

In order to give a physical interpretation of special relativity it is necessary to understand the implications of the Lorentz rotation. Within Galilean relativity the three-dimensional line element of euclidean space (r2 = r r) is an invariant and the transformation corresponds to a rotation in three-dimensional space. The fact that this line element is not Lorentz invariant shows that world space has more dimensions than three. When rotated in four-dimensional space the physical invariance of the line element is either masked by the appearance of a fourth coordinate in its definition, or else destroyed if the four-space is not euclidean. An illustration of the second possibility is the geographical surface of the earth, which appears to be euclidean at short range, although on a larger scale it is known to curve out of the euclidean plane. 

We obtain a physical interpretation of this approach by a suitable definition of the Lagrange multipliers /3U and /3jy- Thus we assume that... 

Equation 19.4-30 actually allows any value N > 0. See Stokes and Nauman (1970) for discussion of this and a physical interpretation of nonintegral values of N for the US model the case of N < 1 corresponds to bypassing of some entering fluid (see also Nauman and Buffham, 1983, pp. 61-2). 

Thus the XC energy is the energy of interaction between the electrons and a charge distribution represented by p cM(f, r ). The question we wish to answer now is whether the expression in Equation 7.18 is just the rewriting of the XC energy in a different way or does it have a physical interpretation. We now show that it indeed has a physical interpretation the term represents the deficit in the density of electrons at r when an electron is at r. 

The quantity h of the second procedure has a physical interpretation One must consider interactions between at least h ligands in order to obtain a nonvanishing chirality function belonging to the given representation. 

In late fall 1925, the Dutch physicists G. Uhlenbeck and Samuel Goudsmit gave a physical interpretation to Pauli s postulate of a fourth quantum number. The electron, they proposed, may spin in one of two directions. In a given atom, a pair of electrons having three identical quantum-number values must have their spin axes oriented in opposite directions, and if paired oppositely in a single orbital, they neutralize each other magnetically. 22... 

One example when the harmonic limit provides a physical interpretation is that of the dipole operator (2.79). The limit of the operator F+ + F is... 

Equation (5.42a) clearly depicts what determines P2, and indeed it appears that the average molecular weight of the unbumed gas mixtures is a major factor [16]. A physical interpretation as to the molecular-weight effect can be... 

The consequence of this small B assumption may not be immediately apparent. One may obtain a physical interpretation by again writing the mass burning rate expression for the two assumptions made (i.e., B 1 and B = [im<, H]/Lw)... 

The determination of which features the underlying factors are composed of provides a basis for attaching a physical Interpretation to the factors. Varlmax rotation of the PGA may be utilized to aid In the Interpretation of the factors. Hierarchical dendrograms Indicate feature clusters whose composition are analogous to PC factors. The physical Interpretation of the clusters and principal components Indicates the Influence of pollution emission sources or meteorological processes on the rainwater composition at an Individual monitoring site. 

These are semi-empirical equations of state that are formulated to describe experimental data accurately, instead of conforming to theoretical descriptions of molecular behavior, and each parameter does not necessarily have a physical interpretation. 

In order to define the transitions in fig. 4, we need to go back and examine the transition probability, which is usually given by time-dependent perturbation theory [36]. If (j>i is the initial state, 0f the final state and Ix the perturbation, then the transition probability is given in eq. (4). Two factors make up the transition probability, and they are complex conjugates, so the intensity is real. In the generalization presented here, these two factors have a physical interpretation they are the projections of the individual transitions along the total magnetization. In a dynamic system, the two factors are not complex conjugates, so the lineshapes in fig. 4 are more complicated. In spite of this, we may still treat them as transitions, as in the static case. 

This relation gives a physical interpretation for the parameter cr a 1/2 equals the average length of a helical sequence in a sufficiently long chain at the midpoint of a helix-coil transition. Thus, as a becomes smaller, the helical portion of such a chain consists, on the statistical average, of a smaller number of sequences. 

Equation (65) also permits us to assign a physical interpretation to the diffusion coefficient in addition to the macroscopic meaning it has from Fick s laws. Rearranging and factoring in a way that admittedly ignores the averaging procedure, we write Equation (65) as... 

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