Defining equation

The rotational spectrum of a molecule with internal rotation (torsion) is modified due to torsion-rotation interaction. The interpretation of this modification allows the determination of the internal rotation potential barrier [59Lin, 68Dre, 84Gorj. The molecule is generally taken to be rigid except for this torsional degree of freedom. However, special methods have been developed to include interactions with molecular vibrations. 

When internal rotation is present, a torsional term Hi and an interaction term //rt have to be added to the rotational Hamiltonian. In the case of a Csv group (methyl, silyl,. ..), they may be written [62Kir, 83Won] 

Generally, no torsion fine stracture appears in the rotational spectrum because the moments of inertia of a symmetric top do not depend directly on the angle of internal rotation, but the internal rotation affects the moments of inertia through interactions with the other vibrational modes, which in turn interact with the overall rotational motion. The effective rotational constant for the torsional state v is given by [54Kiv, 84Gor]  

When it is possible to identify rotational lines in higher torsional states, the magnitude of the potential barrier can also be determined by comparing intensities of rotational lines which have been assigned to different torsional states this is the Int. (= intensity comparison) method. 

An easier (and more accurate) method is to substitute the top asymmetrically (e.g. CH3 CH2D) and use the internal rotation theory for asymmetric tops see the Introduction of the asyrmnetric top sub volume. 

Watson has shown that of the six quaitic distortion constants (7) only five combinations are generally determinable from the spectra (one exception is nearly spherical tops such as SO2F2). Furthermore, only seven combinations of the ten sextic constants ( P) and only nine combinations of the fifteen octic constants (L) can be obtained from the spectra. Watson proposed two sets of constraints  

The relations between the different sets of parameters are given in [77Wat, 84Gor]. The notation of the centrifugal distortion constants permits to know which reduction is used, and therefore the rotational constants are simply called A, B, C (without the superscript A or S). There are six different ways (representations) to identify the (x, y, z) reference system with the (a, b, c) principal axis system. In practice two different representations are used F where x=b, y=c, z=a and which is best for prolate molecules (Ray s asymmetiy parameter k = 1B-A-C)I A-C) 0), and IIF where x=a, y=b, z=c which is thought to be better for oblate molecules (k 0). Representation IlF is also used where x = a, y = c, z = b ) xXiX is equivalent to representation mf Maity authors use codes written in F representation for oblate molecules, in particular for the analysis of inCnared spectra. 

The analysis of the spectra gives the constants for a given vibrational state o. These constants may be expanded as a function of (o, +1/2) where o, is the quantum number of the i-th normal mode. For instance, the rotational constants 5 in a given vibrational state w may be written 

Subscripts w like in eq. (2.4) do not appear in the tabulations below. Instead, the vibrational state where the measurement took place is indicated in the first column, and holds for all parameters listed on its right. Additional state information may be included, for example the symmetiy labels A and E if methyl internal rotation coupling is present. Normally, the parameters have been determined in the electronic ground state which is not notified explicitly. 

Coriolis interactions [63A11, 84Gor] are caused by the coupling of the total angular momentum Jg and the vibrational angular momentum pg. The interaction matrix element between two interacting states v = (u, 0 ) and o = (o + 1, o -1) may be written 

The analysis of the rotational spectrum of an asymmetric molecule in the vibrational state ui. vj. v u-6 normally allows the determination of the constants listed in this table. All rotating molecules show the influence of molecular deformation (centrifugal distortion, c.d.) in their spectra. The theory of centrifugal distortion was first developed by Kivelson and Wilson [52Kiv]. The rotational Hamiltonian in cylindrical tensor form has been given by Watson [77 Wat] in terms of the angular momentum operators J, J/and as follows  

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An alternative defining equation for surface diffusion coefficient Ds is that the surface flux Js is Js = - Ds dT/dx. Show what the dimensions of Js must be. 

By differentiating the defining equations for H, A and G and combining the results with equation (A2.T25) and equation (A2.T27) for dU and U (which are repeated here) one obtains general expressions for the differentials dH, dA, dG and others. One differentiates the defined quantities on the left-hand side of equation (A2.1.34), equation (A2.1.35), equation (A2.1.36), equation (A2.1.37), equation (A2.1.38) and equation (A2.1.39) and then substitutes die right-hand side of equation (A2.1.33) to obtain the appropriate differential. These are examples of Legendre transformations. ... 

In the same section, we also see that the source of the appropriate analytic behavior of the wave function is outside its defining equation (the Schibdinger equation), and is in general the consequence of either some very basic consideration or of the way that experiments are conducted. The analytic behavior in question can be in the frequency or in the time domain and leads in either case to a Kramers-Kronig type of reciprocal relations. We propose that behind these relations there may be an equation of restriction, but while in the former case (where the variable is the frequency) the equation of resh iction expresses causality (no effect before cause), for the latter case (when the variable is the time), the restriction is in several instances the basic requirement of lower boundedness of energies in (no-relativistic) spectra [39,40]. In a previous work, it has been shown that analyticity plays further roles in these reciprocal relations, in that it ensures that time causality is not violated in the conjugate relations and that (ordinary) gauge invariance is observed [40]. 

They unfold a connection between parts of time-dependent wave functions that arises from the structure of the defining equation (2) and some simple properties of the Hamiltonian. 

The relationship of these quantum meehanieal operators to experimental measurement will be made elear later in this ehapter. For now, suffiee it to say that these operators define equations whose solutions determine the values of the eorresponding physieal property that ean be observed when a measurement is earried out only the values so determined ean be observed. This should suggest the origins of quantum meehanies predietion that some measurements will produee discrete or quantized values of eertain variables (e.g., energy, angular momentum, ete.). 

Each of the two laws of thermodynamics asserts the existence of a primitive thermodynamic property, and each provides an equation connecting the property with measurable quantities. These are not defining equations they merely provide a means to calculate changes in each property. 

Analogous to the defining equation for the residual Gibbs energy is the definition of a partial molar residual Gibbs energy (eq. 161) ... 

Foi an ideal solution, G, = 0, and tlieiefoie 7 = 1- Compatison shows that equation 203 relates to exactiy as equation 163 relates ( ) to GG Moreover, just as ( ) is a partial property with respect to G /E.T, so In y is a partial property with respect to G /RT. Equation 116, the defining equation for a partial molar property, in this case becomes equation 204 ... 

Capacitors. Ceramic materials suitable for capacitor (charge storage) use are also dependent on the dielectric properties of the material. Frequently the goal of ceramic capacitors is to achieve maximum capacitance in minimum volume. The defining equation for capacitance is given by ... 

All three quantities are for the same T, P, and physical state. Eq. (4-126) defines a partial molar property change of mixing, and Eq. (4-125) is the summability relation for these properties. Each of Eqs. (4-93) through (4-96) is an expression for an ideal solution property, and each may be combined with the defining equation for an excess property (Eq. [4-99]), yielding ... 

A generally apphcable alternative to the gamma/phi approach results when both the hquid and vapor phases are described by the same equation of state. The defining equation for the fugacity coefficient, Eq. (4-79), may be applied to each phase ... 

For a given degree of reaction and value of or flow rate, a choice can be made of the stagger angle or setting of the blades. From the consideration of two defining equations ... 

Many equilibrium calculations are accomplished using tlie plrase equilibrium constant K,. Tliis constant has been referred to in industry as a coiiiponential split factor, since it provides the ratio of the mole fractions of a component in two equilibrium pluises. The defining equation is... 

You probably noted that the original papers were couched in terms of HF-LCAO theory. From Chapter 6, the defining equation for a Hamiltonian matrix element (in the usual doubly occupied molecular orbital, closed-shell case) is... 

From the defining equations of H (equation 20.131) and G (equation 20.146) differentiation gives... 

We can obtain a relation between AH and AE for a chemical reaction at constant temperature by starting with the defining equation relating enthalpy, H, to energy, E ... 

Recall from Chapter 5 the defining equation for mole fraction (X) of a component A ... 

Strategy First (1) calculate the number of moles of C2H602 (MM = 62.07 g/mol). Then (2) apply the defining equation to calculate the molality. Finally (3), use the equation ATf = (1.86°C/m) X molality to find the freezing point lowering. 

Figure 13.2 shows the relationship between pH and [H+]. Notice that, as the defining equation implies, pH increases by one unit when the concentration of H+ decreases by a power of 10. Moreover, the higher the pH, the less acidic the solution. Most aqueous solutions have hydrogen ion concentrations between 1 and 10-14 M and hence have a pH between 0 and 14. 

From the defining equation for free energy, it follows that at constant temperature... 

The most spectacular feature of a conductivity-concentration function is its maximum, attained for every electrolyte if the solubility of the salt is sufficiently high. For electrolytes which do not show strong ion association, the maxima can be understood on the basis of the defining equation of specific conductivity at the maximum [205], yielding... 

SW shall often indicate factors ft and o in the defining equations, although we shall consistently adopt a system of units wherein ft sx c = 1. 

The defining equations (9-133) and (9-134) for H and P reflect the fact that the particles are free and do not interact with one another. The total energy and total momentum of the system is, therefore, the sum of energies and momenta of the individual particles as indicated by Eqs. (9-135) to (9-137). We shall see that we may consider the operators H, P as the time and space components of the four-vector P ... 

In fact, by virtue of the circumstance that in the defining equation for (+)(x) only positive frequencies occur (k0 = +Vk2 + m2>0), (+ >(x) obeys the following first order equation... 

In obtaining Eq. (9-375) we have made use of the defining equation for det A, namely... 

Equations (9-510) and (9-511) together with the defining equation for the fields in terms of the potential, Eq. (9-506), are equivalent to the original Maxwell equations. 

Pl.l Use the properties of the exact differential and the defining equations for the derived thermodynamic variables as needed to prove the following relationships ... 

In Chapter 1 we gave the defining equations for enthalpy, Helmholtz free... 

The defining equation for fugacity fc in a condensed phase (solid or liquid) is the same as in the gas phase... 

Fugacity of a Component in a Gaseous Mixture One could guess that the determination of fugacities, /, for the individual components in a gaseous mixture can become complicated as one takes into account the different types of interactions that are present. The mathematical relationship that applies is obtained by starting with the defining equations... 

Here, the value of k is defined by the choice shown. Of course, one just as well could have written a different defining equation,... 

The temperature control was modeled by using these defining equations for a PID (Proportional-Integral-Derivative controller) algorithm ... 

Although the protonic theory is not confined to aqueous solutions, it does not cover aprotic solvents. The solvent system theory predates that of Bronsted-Lowry and represents an extension of the Arrhenius theory to solvents other than water. It may be represented by the defining equation ... 

This theory is associated in its early protonic form with Franklin (1905, 1924). Later it was extended by Germaim (1925a,b) and then by Cady Elsey (1922,1928) to a more general form to include aprotic solvents. Cady Elsey describe an acid as a solute that, either by direct dissociation or by reaction with an ionizing solvent, increases the concentration of the solvent cation. In a similar fashion, a base increases the concentration of the solvent anion. Cady Elsey, in order to emphasize the importance of the solvent, modified the above defining equation to ... 

The pharmacokinetic parameters of the model are then readily derived from the defining equations and the results of the regression ... 

The kinetically controlled current 7k at 0.02 V was determined from a well-defined equation [Levich, 1962 Gerischer et al., 1965], i.e., plotting the inverse of the current... 

The system mass balance equations are often the most important elements of any modelling exercise, but are themselves rarely sufficient to completely formulate the model. Thus other relationships are needed to complete the model in terms of other important aspects of behaviour in order to satisfy the mathematical rigour of the modelling, such that the number of unknown variables must be equal to the number of defining equations. 

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