Another derivation

A second approach to the NVT ensemble found in Feynman s lecture notes on statistical mechanics [55] is also based on the central idea from NVE ensemble theory that the probability of a microstate is proportional to the number of microstates available to the system. Thus 

Next Feynman makes use of the fact that energy is defined only up to an additive constant. In other words, there is no absolute energy value, and we can always add a constant, say e, so long as we add the same e to all relevant values of energy. 

Without changing its physical meaning Equation (12.21) can be modified  

Next we define the function g(x) = Q(E0 - E2 + x). Equating the right-hand sides of Equations (12.21) and (12.22) results in 

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The alkylidene dimethone (dimedone) (I) upon boiling with glacial acetic acid, acetic anhydride, hydrochloric acid and other reagents frequently loses water and passes into a substituted octahydroxanthene or the anhydride (II), which often serves as another derivative. The derivatives (I) are soluble in dilute alkali and the resulting solutions give colourations with ferric chloride solution on the other hand, the anhydrides (II) are insoluble in dilute alkali and hence can easily be distinguished from the alkylidene dimedones (I). 

Two points must be stressed that a 3-R-pyrazole (199a) gives a 1,5-disubstituted derivative (200) and that often the less abundant tautomeric species (Section 4.04.1.5.1) is the more reactive. It is theoretically possible to use the tautomeric composition to direct the synthesis towards one or another derivative. For instance, if (199a) is the more abundant tautomer, the condition to obtain a large proportion of (200) is that The factors... 

It has been seen thus far that the first law, when applied to thermodynamic processes, identifies the existence of a property called the internal energy. It may in other words be stated that analysis of the first law leads to the definition of a derived property known as internal energy. Similarly, the second law, when applied to such processes, leads to the definition of a new property, known as the entropy. Here again it may in other words be said that analysis of the second law leads to the definition of another derived property, the entropy. If the first law is said to be the law of internal energy, then the second law may be called the law of entropy. The three Es, namely energy, equilibrium and entropy, are centrally important in the study of thermodynamics. It is sometimes stated that classical thermodynamics is dominated by the second law. 

Another derivative that is a useful intermediate for synthesizing other derivatives is the mercury compound that can be prepared from ferrocene by reaction of mercuric acetate in a solution also containing chloride ion. 

Similar cross-coupling reaction of another derivative of the same ring system 141 with dihydrofuran has also been described. In this case, palladium diacetate, tributylamine, triphenylphosphine, and tetrabutylammonium chloride were used to afford the product 142 in high yield (70%) <1995NN105>. 

In a similar way fulminic acid, another derivative of bivalent carbon, which has been accurately investigated and for which the carboxime >C NOH constitution has been established, is decomposed into formic acid and hydroxylamine (see experiment on p. 159). 

Amino acid derivatives can be examined for enantiomeric purity by the same procedures after removal of the protecting groups. Another approach is to couple them directly with another derivative to give protected dipeptides whose diastereomeric forms are usually easy to separate by HPLC (see Section 4.11). An A-protected amino acid is coupled with an amino acid ester, and vice versa. Use of soluble carbodiimide as reagent (see Section 1.16), followed by aqueous washes, gives clean HPLC profiles. It is understood that the derivative that serves as reagent must have been demonstrated to be enantiomerically pure.43 84-89... 

The DSO term involves a produet of ordering parameters and formally a second-order term. As seen from equation (81), the DSO term does not arise in our derivation and requires further investigation. In this regard, first, we will present another derivation whieh will eventually lead to the DSO term in addition to all the other terms given in equation (81). 

Let s move on to another derivative of retinol, retinoic acid ... 

In Eq. (14), /max is the maximum of the orbital angular momentum quantum numbers of the active electron in either the initial or final states, I nl, n l ) is the radial transition integral, that contains only the radial part of both initial and final wavefunctions of the jumping electron and a transition operator. Two different forms for this have been employed, the standard dipole-length operator, P(r) = r, and another derived from the former in such a way that it accounts explicitly for the polarization induced in the atomic core by the active electron [9],... 

Another derivative of butadiene, hexamethylenediamine (HMDA), is used in the synthesis of nylon. We have already met this compound earlier in this chapter since it is made from acrylonitrile through adiponitrile. 

Another derivative evaluated at the Hopf bifurcation point of interest, which we will need later on, is that of the imaginary part of frequency (d[Pg.115]

CHa - CHa NHa with (COOH)a, followed by nitration. As an example of a sulfonyl (sulfuryl) derivative may be cited [OaNO-CHa. CHa -N(NOa)]SOa, prepd by condensing HO-CHa. CHa -NHa with SOaCla followed by nitration(Ref 3). The same inventor proposed another deriv of amino ethanol,... 

Another derivation has been given by Resibois and De Leener. In principle, eqn. (287) can be applied to describe chemical reactions in solution and it should provide a better description than the diffusion (or Smoluchowski) equation [3]. Reaction would be described by a spatial- and velocity-dependent term on the right-hand side, — i(r, u) W Sitarski has followed such an analysis, but a major difficulty appears [446]. Not only is the spatial dependence of the reactive sink term unknown (see Chap. 8, Sect. 2,4), but the velocity dependence is also unknown. Nevertheless, small but significant effects are observed. Harris [523a] has developed a solution of the Fokker—Planck equation to describe reaction between Brownian particles. He found that the rate coefficient was substantially less than that predicted from the diffusion equation for aerosol particles, but substantially the same as predicted by the diffusion equation for molecular-scale reactive Brownian particles. 

The second comment concerns yet another derivation, based on the idea that the bath variables are much faster than those of the system. One writes for the total Hamiltonian... 

Simpler names for compounds of type 4 (mentioned in the guiding index of Chemical Abstracts) will be used in this survey the fundamental heterocycle 6f/-benzofuro[3,2-c][l]benzopyran-6-one is called cou-mestan.8 Coumestrol (4) becomes 3,9-dihydroxycoumestan. Another derivative of the same family, waedelolactone,9 is 3-methoxy-l,8,9-trihydroxycoumestan. In this connection, it would be advisable, for natural compounds of botanical origin, to put the corresponding chemical name in brackets. 

Another derivative of this kind is l,6-dimethyl-8/ -[(benzyloxy-carbonyl)aminomethyl]ergoline (71) [MCE] (109, 111). It shows a... 

Another derivative, the Me2Si(NLiSiMe3)2, was prepared by Zimmer and Veith49. In the two lithium derivatives 56 and 57 the lithium has no contact to nitrogen. Whereas... 

Another derivative was synthesized starting from 5-aminopyridazine-4-carboxylic acid (156). Its reaction with benzoyl chloride in pyridine gave 2-phenylpyridazino[4,5-[Pg.762]

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