Birch equation

Fig. 8. Reduced volume of the high-pressure Si phases. Experimental points are from Reference [67] (Si-I and Si-II) and Reference [63] (Si-III and Si-XII). Filled and open circles correspond to increasing and decreasing pressure, respectively. The solid lines are fits to the 3rd-order Birch equation of state [69, 70] with the values of bulk modulus Bq = 108 GPa and its first pressure derivative Bq = 4 for Si-I Bq = 117 GPa for Si-Ill and Bo = 108 GPa for Si-XII with Bq fixed at 5. Dashed line serves as a guide to the eye.

In order to elucidate the energetics of the cation site occupation, we have applied ab-initio techniques of density functional theory within the local density approximation. Two possible configurations corresponding to normal and (partially) inverse spinel were considered (Figs. 3a and b). The results are given in Tablet, with the bulk moduli obtained by fitting the Birch equation of state. 

For this purpose, we reanalyze all the available static EOS data for Th, as shown in table 1, with a set of three different EOS forms, and compare the effect of the different EOS forms with the effects resulting from different data sets. As EOS forms we use the Birch equation (Birch 1978) in second order, BE2, and two recently proposed forms (Holzapfel 1990,1991) in second-order form, H02 and HI2, which are related to the Thomas-Fermi theory and are distinguished by the fact that H12 is bound to approach the Fermi-gas limit at infinite compression. A close inspection of table 1 shows very clearly that most of the data are fitted almost equally well by any of these forms, without any significant difference in the fitted parameters Kq and K g or in the minimized standard deviation of the pressure, Tp. In contrast to many other publications, table 1 presents the unrestricted standard deviations of Kq and K, which correspond to the extreme values of the error ellipsoids presented in fig. 11, and not just to the width of the error ellipsoids along and K at the center points, which are usually given as (restricted) statistical errors. Thus, it becomes obvious that... 

In the following treatment we show how the usual Birch-Murnaghan finite strain equation of state is derived and is related to the Hugoniot parameters. Using the Eulerian definition of finite strain. 

If the pressure volume equations of state is given by the two parameter third-order, Birch-Murnaghan, Uj = 0-... 

Moreover, upon comparing (4.32) with (4.14), it can be seen that (Jeanloz and Grover, 1988) the Birch-Murnaghan equation (4.32) with a2 = 0 describes the isentropic equation of state provided the linear shock-particle velocity relation (4.5) describes the Hugoniot. In combination, these require that... 

Jeanloz, R., and Grover, R. (1988), Birch-Murnaghan and Us-Up Equations of State, in Proceedings of the American Physical Society Topical Conference on Shock Waves in Condensed Matter, Monterey, CA, 1987 (edited by Schmidt S.C. and N.C. Holmes), Plenum, New York, pp. 69-72. 

Dipyrrolo[l,2- 2, l - ]pyrazines, for example, 47, and related dipyrrolopyrimidines can be partially reduced in the central diazine ring under Birch reduction conditions to give the corresponding dihydro derivative 48 (Equation 5) <2005JOC2054>. 

Reduction of aromatic compounds to dihydro derivatives by dissolved metals in liquid ammonia (Birch reduction) is one of the fundamental reactions in organic chemistry308. When benzene derivatives are subjected to this reduction, cyclohexa-1,4-dienes are formed. The 1,4-dienes obtained from the reduction isomerize to more useful 1,3-dienes under protic conditions. A number of syntheses of natural products have been devised where the Birch reduction of a benzenoid compound to a cyclohex-1,3-diene and converting this intermediate in Diels-Alder fasion to polycyclic products is involved (equation 186)308f h. 

The two most usual equations of state for representation of experimental data at high pressure are the Murnaghan and Birch-Murnaghan equations of state. Both models are based on finite strain theory, the Birch-Murnaghan or Eulerian strain [26], The main assumption in finite strain theory is the formal relationship between compression and strain [27] ... 

Figure 2.15 Pressure-volume data for diamond, SiC>2-stishovite, MgSiC>3 and 8102-quartz based on third order Birch-Murnaghan equation of state descriptions. The isothermal bulk modulus at 1 bar and 298 K are given in the figure.

Volumetric data for four different substances represented by the third-order Birch-Murnaghan equation of state are shown in Figure 2.15. 

Figure 2.3 Total energy, E, of Cu in the fee crystal structure as a function of the lattice para meter, a. Data points are from DFT calculations and the dashed curve is the Birch Mumaghan equation of state.

Early examples of electron transfer processes are shown in equations (2), (12), and (13). Birch in 1944 followed up the findings of Wooster, and demonstrated that Na metal and ethanol in ammonia reduce benzene, anisole, and other aromatics to 1,4-cyclohexadienes. Birch speculated about the mechanism of this reaction, but did not explicitly describe a radical pathway involving 55 (equation 87) until later, as described in his autobiography. Electron transfer from arenes was found by Weiss in 1941, who obtained crystalline salts of Ci4H]o from oxidation of anthracene. ... 

Lactams.1 The ketene derived from (S)-phenyloxazolidylacetyl chloride (1), prepared from (S)-phenylglycine, undergoes cycloaddition with N-benzyl al-dimines to give two as-azetidinones with high stereochemical control (equation I). The chiral auxiliary and the benzyl group are cleaved by Birch reduction to provide enantiomerically pure azetidinones (3). 

The deep blue solutions formed by dissolving alkali metals in ammonia do not rapidly generate the amide unless a catalyst is added.9 However, a hydrogen acceptor will also initiate the reaction and this forms the basis of the important Birch reduction of aromatic compounds (equation 2).10... 

A method for preparing specific alkyl-substituted dienyl complexes takes advantage of the propensity of (diene)Fe(CO)3 complexes to rearrange under acidic conditions, coupled with the acid-promoted dehydration illustrated earlier for the conversion of (9) to (8). Birch and Haas14 discovered that complexes derived from methylanisoles could be converted to methyl-substituted cyclohexadienyliron complexes, whose substitution pattern is defined by the relative positions of the methoxy and methyl substituents in the precursor. Several examples are given in Schemes 5 and 6 and equation (17). 

Birch and Pearson22 have studied electrophilic substitution of a triphenylphosphine-substituted system, [Fe(cyclohexadiene)(CO)2PPh3], (8 equation 7). Several features of their results are instructive. First, substitution of CO by the better o-donor, poorer ir-acceptor PPh3 ligand renders the complex more reac-... 

The silver ion assisted carbon-halogen bond cleavage and the unraveling of the cyclopropane ring by the cyclopropyl-allyl rearrangement was first noted in the formation of 2-bromocyclohexen-l-ol from dibromobicyclo[3.l.0]hexane under solvolytic conditions (equation 86).220 The silver ion assisted solvolysis of the dihalocyclopropane adduct (43), derived from a Birch reduction product, smoothly rearranges to the tropone (equation 87).221 A number of other synthetic applications222-226 have beien reported... 

Cathodic reduction of arylsilanes in methylamine using LiCl as a supporting electrolyte in an undivided cell gives 1,4-cyclohexadiene derivatives. The reaction seems to proceed in a manner similar to the Birch-type reduction. The cathodic reduction in a divided cell provides desilylation products (equation 53)68. 

Recently, two completely different methods which are compatible with an alkene have been used successfully. The first of these involves121 prior Birch reduction of the phenyl group to a cyclohexadiene. Subsequent treatment with fluoride ion and basic hydrogen peroxide then completes the overall cleavage (equation 31). 

The Stern-Volmer equation (see sect. 4) may be used to determine small amounts of a species which would behave as an inhibitor of a given luminescent probe. The detection limit depends, among other parameters, on and on the detection limits of the setup. The potentials of this method for analytical purposes are discussed, on a general aspect, in Borissevitch (1999), Rakicioglu et al. (1998) and the specific cases of Eu(III) or U(VI) are presented in Georges (1993), Lopez and Birch (1996), Kessler (1998). For example, a detection limit of 7 ngl-1 for Cu2+ is obtained (Lopez and Birch, 1996). Numerous factors may render the method difficult to apply besides the variations of ksv as a function of ionic strength, if more than one quencher is present in solution, it becomes difficult to determine their individual concentrations. This problem has been studied in the case of solutions that more or less mimic the nuclear fuel solutions in Katsumura et al. (1989). 

Bulk moduli and pressure derivatives. Results for the bulk modulus and its pressure derivative for all three HMX polymorphs obtained from fitting simulation-predicted isotherms to the equations of state discussed above are summarized in Table 7. For all data sets, we include fits to the Us-Up form (Eq. 18) and both weighting schemes for the third-order Birch-Mumaghan equation of state (Eqs. 20 and 21). In the case of the experimental data for /THMX, values for the moduli based on Eqs. 18 and 20 were taken from the re-analysis of Menikoff and Sewell. Two sets of results are included in the case of Yoo and Cynn, since they reported on the basis of shifts in the Raman spectra a phase transition with zero volume change at 12 GPa. Simulation data of the /T HMX isotherm due to Sorescu et al. were extracted by hand from Fig. 3b of their work. 

Several empirical equations of state (EOS), representing correlations between pressure and molar volume data have been defined, one of which is the Birch-Mumaghan EOS,... 

Birch reduction of phenanthrene or 9-alkylphenanthrenes results in tetrahydro derivatives, but 9,10-dihydro derivatives are obtained in high yield by Li/NH3 reduction of phenanthrenes substituted by a 9-trimethylsilyl group (equation I). The trimethylsilyl group of 1 can be removed in quantitative yield with Bu4NF in... 

Several examples of the Birch reduction of substituted benzene derivatives are shown in the following equations. Note that substituents such as alkyl and alkoxy groups prefer to be attached to one of the carbons of the double bonds of the product, while a carboxyl group prefers to be attached to one of the singly bonded carbons. Benzene derivatives with other types of substituents are usually not employed as reactants in the Birch reduction because the substituents are not stable to the reaction conditions. 

In a new asymmetric synthesis of chiral 1,4-diols, the dioxocane 179 was transformed into diols 180 and 181 by either the Birch reduction or catalytic hydrogenolysis, respectively (Equation 40) <1996TL2245> (cf. Equation (46), Section 14.06.6.6). 

Nonalkylated 3,4-dehydroprolines 914 were obtained in 76-81% yields by diastereoselective protonation of an enolate resulting from Birch reduction of the A -BOC-pyrrole-2-carboxamide 913 (Equation 223) <1999T12309>. The reaction was quenched by addition of solid ammonium chloride after a reaction time of 1 h. The results using lithium and sodium are similar but the reaction with potassium failed. Remarkably, asymmetric protonation is more selective (de 88-90%) than methylation (de 50%). The selectivity decreases with increasing temperature (de 82% at —30°C). The diastereoselectivity of the reaction was detected by HPLC. 

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