Convenience properties

All of thermodynamics could be worked out in terms of P, T, m, V, u, and 5 some purists do it that way. However, some combinations of these variables occur together so often that we can save writing and calculations by defining convenience properties, which are combinations of these basic variables. The most widely seen of these are 

The typical problems addressed in an elementary ME thermodynamics course ask the following questions  

How much heat and/or work must be transferred into or out of some batch or flow system as the material in it goes from an initial to a final state  

If the amount of heat and/or work is specified, what is the final or outlet state of the matter  

If you have a table of properties of the material in the system, you can solve such problems using the equations in the previous section. Such tables are available (and included as appendices in all elementary thermodynamics books) for commonly used materials, such as steam, common refrigerants, and low molecular weight hydrocarbons. A typical table is made up of several parts, with the description shown in Table 2.1. 

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Apart from the distance variable x that Dunham used in his function V(x) for potential energy, other variables are amenable to production of term coefficients in symbolic form as functions of the corresponding coefficients in a power series of exactly the same form as in formula 16. Through any method to derive algebraic expressions for Dunham coefficients l j, the hamiltonian might have x as its distance variable, but after those expressions are produced they are convertible to contain coefficients of other variables possessing more convenient properties. To replace x, two defined variables are y [38],... 

The physical properties of many macrocyclic polyethers and their salt complexes have been already described. - Dibenzo-18-crown-6 polyether is useful for the preparation of sharpmelting salt complexes. Dicyclohexyl-18-crown-6 polyether has the convenient property of solubilizing sodium and potassium salts in aprotic solvents, as exemplified by the formation of a toluene solution of the potassium hydroxide complex (Note 13). Crystals of potassium permanganate, potassium Lbutoxide, and potassium palladium(II) tetrachloride (PdClj + KCl) can be made to dissolve in liquid aromatic hydrocarbons merely by adding dicyclohexyl-18-crown-6 polyether. The solubilizing power of the saturated macrocyclic polyethers permits ionic reactions to occur in aprotic media. It is expected that this [)ropcrty will find practical use in catalysis, enhancement of... 

In crystallography, heavy atom derivatives are required to solve the phase problem before electron density maps can be obtained from the diffraction patterns. In nmr, paramagnetic probes are required to provide structural parameters from the nmr spectrum. In other forms of spectroscopy a metal atom itself is often studied. Now many proteins contain metal atoms, but even these metal atoms may not be suitable for crystallographic or spectroscopic purposes. Thus isomorphous substitution has become of major importance in the study of proteins. Isomorphous substitution refers to the replacement of a given metal atom by another metal that has more convenient properties for physical study, or to the insertion of a series of metal atoms into a protein that in its natural state does not contain a metal. In each case it is hoped that the substitution is such that the structural and/or chemical properties are not significantly perturbed. 

All of the matrices we have just worked out, as well as all others which describe the transformations of a set of orthogonal coordinates by proper and improper rotations, are called orthogonal matrices. They have the convenient property that their inverses are obtained merely by transposing rows and columns. Thus, for example, the inverse of the matrix... 

Monocarboxylic acid complexes have occupied a central position in coordination chemistry from the 19th century through to the present day.1,2 Ubiquitous is an adjective frequently and appropriately used in conjunction with this ligand in any overview of coordination chemistry. Reasons for this omnipresence of, particularly, acetic acid (ethanoic acid), an early member of this series, include its convenient properties (b.p. 117.7°C, f.p. 16.6 °C), ready availability (millions of tonnes produced annually) and its versatile coordination behaviour. Despite this long and intensive exposure as a ligand, carboxylic acids still continue to provide coordination chemists with exciting new species as any survey of the current literature will confirm. 

In order to demonstrate the convenient properties of lenses combined with an electrostatic energy analyser, three examples will be discussed first the important case in which an acceleration/retardation lens is placed in front of the analyser which allows an operation mode with a constant pass energy in the analyser second the case in which increased acceptance angles are used, which is important for coincidence experiments and third the case in which 0 eV electrons are handled and analysed with high resolution and high acceptance. 

The preceding sections have used standard molar concentration units for RNA and ions, indicated by brackets or the abbreviation M. Thermodynamic definitions of interaction coefficients are made in terms of molal units, abbreviated m, the moles of solute per kilogram of solvent water. Molal units have the convenient properties that the concentration of water is a constant 55.5 m regardless of the amount ofsolute(s) present, and the molality of one solute is unaffected by addition of a second solute. For dilute solutions, M and m units are interchangeable. We use molal units for the thermodynamic derivations in this section, and indicate later (Section 3.1) the salt concentrations where a correction for molar-molal conversion is required. 

The work of Feldberg [231] indirectly provides a useful transformation function, that has some convenient properties. The function is... 

Derivatives other than esters and silyl compounds were mostly developed for particular groups of acids, if not for individual substances, containing in addition to carboxylic also other functional groups, in an attempt to obtain derivatives possessing convenient properties, e.g., derivatives of lower acids less volatile than methyl esters, or more stable, or derivatives suitable for selective and sensitive detection. 

The Kronecker delta functions, 5 and 6,p, resulting from Eq. [21], cannot be simplified to 1 or 0 because the indices p and q may refer to either occupied or virtual orbitals. The important point here, however, is that the commutator has reduced the number of general-index second-quantized operators by one. Therefore, each nested commutator from the Hausdorff expansion of H and T serves to eliminate one of the electronic Hamiltonian s general-index annihilation or creation operators in favor of a simple delta function. Since f contains at most four such operators (in its two-electron component), all creation or annihilation operators arising from f will be eliminated beginning with the quadruply nested commutator in the Hausdorff expansion. All higher order terms will contain commutators of only the cluster operators, T, and are therefore zero. Hence, Eq. [52] truncates itself naturally after the first five terms on the right-hand side. ° This convenient property results entirely from the two-electron property of the Hamiltonian and from the fact that the cluster opera-... 

A convenient property of the glassy matrix for determining extensive physical ageing is the excess enthalpy. Results from Rueda et al. (1995) indicate that the... 

A convenient property of the partition function emerges if one makes the usual approximation that energies... 

Enthalpy is a mathematical property defined for convenience in solving flow problems ([42], p. 48). Enthalpy is termed a convenience property becanse we have specifically defined it to be nseful in problems where irreversible heat flow and pressnre are manipnlated ([42], p. 175). 

We now turn to the effective operator definitions produced by (2.14) with model eigenfunctions that incorporate the normalization factors of (2.16) so their true counterparts are unity normed. Equations (2.27) and (2.38) show these model eigenfunctions to be the a)o and ( that are defined in (2.33) and (2.34). Substituting Eqs. (2.27) and (2.38) into (2.14) and proceeding as in the derivation of the forms / = I-I1I, yields the state-independent definitions A, A" and A" of Table I. Notice that the effective Hamiltonian H is identically produced upon taking A = // in the effective operator A". Table I indicates that this convenient property is not shared by all the effective operator definitions. 

Dyes - Dyes in general have convenient properties for the investigation of primary processes by the whole range of fast reaction techniques. An example of a very much state of the art... 

If the substituent R is in the P position, it is possible to obtain copolymers of THF-P substituted THF, which, fortunately, are liquid at room temperature, a very convenient property for PU applications. An interesting example of the liquid copolymer of THF-P substituted THF is the random copolymer THF-2-methyl-THF (reaction 7.16). 

Various titration techniques are based on the same principle. The agent used in titration has to react irreversibly and quantitatively with living polymers, but not with dead ones. The end-point is determined by a sharp change in some convenient property of the system, e.g. by the disappearance of the color of living polymers, provided that the dead polymers are colorless. 

For near-degenerate vibrational levels of any two crossing potential curves, the i -centroid has the convenient property of being nearly equal to Rc, the F-value where the two curves cross (Schamps, 1977). Thus,... 

The presence of l,m > 1 powers of v and J is irrelevant to whether the model is linear. Least-squares fits to a linear model have the convenient property of converging, in a single step, to a unique best set of parameter values. 

The correction constants do not have this convenient property, and they must be calculated for the actual choice of the ratio S /S. Of course this applies only to the cases with two or more inequivalent atoms in the cell. 

Germanium-73 has several low-lying excited levels, of which the 13-5-keV first excited state suffers from excessive internal conversion (ax > 1000) and a long lifetime (4 us), as does the 66-8-keV second excited state (0-S3 s). The 67 03-keV third excited state has convenient properties for Mbssbauer spectroscopy but is only weakly populated by the decay of Ga. All Mossbauer experiments have therefore used a direct population of this level by Coulomb excitation in an ion beam. The ground state has I — f+, while the 67-03-keV level is probably / = J-H and decays directly to the ground state. 

The resultant transport equation and the variables used in its development are summarized in Table 1. Transport equations unknowns are the total concentrations (/, that includes precipitated minerals in local equilibrium. They have the convenient property of not depending on the equilibrium reactions. Dependent variables in transport equation (as the total aqueous... 

This explicit removal of the self-interaction amongst the electrons is a great strength of the (algebraic approximation to the) Hartree-Fock equations. We shall see later that separate approximations to parts of the total interaction energy of a system of electrons do not have this convenient property and may often include spurious energies of interaction between different parts of a given electron . The so-called self-interaction correction (SIC) must be invoked in such... 

The fact remains that, in the solution of the matrix Hartree-Fock equations, virtual orbitals are generated and, whatever we might think or even prove about the physical interpretation of these orbitals, they can be used to form determinants and a sum of determinants is always a better approximation to the true wavefunction than a single determinant if the linear parameters it contains are optimised. Further, these virtual orbitals are orthogonal to the occupied orbitals the whole set, occupied plus virtual, is orthonormal and so they have many convenient properties. In particular, we can write an approximation to the wave-function of any system for which we have a solution of the LCAO Hartree-Fock equations as ... 

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