Order of a derivative

C What is the difference between the degree and the order of a derivative ... 

As we demonstrate later, in Chapter 10, linear systems of equations can only be solved exactly when there exists one BC or IC for each order of a derivative. The above system is now properly posed, and will be solved as an example in Chapter 10 using Laplace transform. 

In the derivation of the Boltzmann equation, it was noted that the distribution function must not change significantly in times of the order of a collision time, nor in distances of the order of the maximum range of the interparticle force. For the usual interatomic force laws (but not the Coulomb force, which is of importance in ionized gases), this distance is less than about 10 T cm the corresponding collision times, which are of the order of the force range divided by a characteristic particle velocity (of the order of 10 cm/sec for hydrogen at 300° C), is about 10 12 seconds. 

Second-order kinetics, (a) Derive expressions for the half-time and lifetime of A if the rate law for its disappearance is v = fc[A]2 (b) calculate t]/i and t for the data presented in Section 2.2 (c) calculate the second half-life, t /i(2), i.e., the time elapsed between 50 percent and 75 percent completion, for the same reaction (d) compare fj/2(l) and fi/>(2), and contrast this result with that from first-order kinetics. 

The order of the above reaction is, therefore, 1.5 + 0.5 = 2. This is typical of situations where the order of reaction and the molecularity of the reaction are the same. It may, however, be noted that the form of rate law, which determines the order of a reaction, can only be derived by actual experiment, and that may or may not be equal to the molecularity of the reaction as provided by the equation representing that reaction. Thus, a general reaction... 

The EA/CA ratio was proposed as a measure of hardness of the Lewis acid, and EB/CB as hardness of the Lewis base in aqueous solution (17). It now seems that the E/C ratio is not a measure of hardness in the sense in which Pearson (5,5a) defined hardness. Rather, the E/C ratio for a Lewis acid or base is a measure of the tendency to ionicity in the M-L bonds formed. The EAICA ratio should rather be called IA, and the EbICb ratio IB, the tendency to ionic bonding in forming the M-L bond. Acids and bases in Tables I and II are placed in order of increasing tendency towards ionicity in the M-L bond, according to the E/C ratios IA and 7b. A justification for this interpretation is that the order of IA values for metal ions in aqueous solution strongly resembles the order of hardness derived by Pearson (19) from enthalpies of complex forma-... 

With these results for the angular-momentum operators it is possible to obtain die Hamiltonian for the rotation of a symmetric top by direct substitution in Eq. (13). The leader is warned that care must be taken in this substitution, as die order of the derivatives is to be rigorously respected. However, given sufficient patience one can show that the classical energy becomes the Hamiltonian operator in the form (problem 12)... 

The interpretation of the elements of the matrix 0 is slightly more subtle, as they represent the derivatives of unknown functions fi(x) with respect to the variables x at the point x° = 1. Nevertheless, an interpretation of these parameters is possible and does not rely on the explicit knowledge of the detailed functional form of the rate equations. Note that the definition corresponds to the scaled elasticity coefficients of Metabolic Control Analysis, and the interpretation is reminiscent to the interpretation of the power-law coefficients of Section VII.C Each element 6% of the matrix measures the normalized degree of saturation, or likewise, the effective kinetic order, of a reaction v, with respect to a substrate Si at the metabolic state S°. Importantly, the interpretation of the elements of does again not hinge upon any specific mathematical representation of specific... 

Four/V-[(acyloxy)methyl] derivatives of 5-FU (8.152, R = AcOCH2, Et-COOCH2, PrCOOCH2, and (7-Bu)COOCH2) were very stable in buffer at pH 7.4 and 37° (t1/2 70, 90, 140, and 700 h, respectively) [219], In 80% human plasma under the same conditions of pH and temperature, the f1/2 values were 14, 9.6, 2.3, and 40 h, respectively, i.e., a marked substrate-dependent acceleration. In rat liver homogenates, the tm values for decomposition were in the order of a few minutes or less. Clearly enzymatic hydrolysis is possible for such compounds and may even be quite significant. Additional data on A-[(acyloxy)methyl] prodrugs have confirmed their potential for topical delivery of 5-FU [220],... 

OPA in combination with chiral thiols is one method used to determine amino acid enantiomers. A highly fluorescent diastereomeric isoindole is formed and can be separated on a reverse-phase column. Some of these chiral thiols include N-acetyl-L-cysteine (NAC), N-tert-butyloxy-carbonyl- L-cysteine (Boc-L-Cys), N-isobutyryl- L-cysteine (IBLC), and N-isobutyryl- D -cysteine (IBDC). Replacing OPA-IBLC with OPA-IBDC causes a reversal in the elution order of the derivatives of D- and L-amino acids on an ODS column (Hamase et al., 2002). Nimura and colleagues (2003) developed a novel, optically active thiol compound, N-(tert-butylthiocarbamoyl)- L-cysteine ethyl ester (BTCC). This reagent was applied to the measurement of D-Asp with a detection limit of approximately 1 pmol, even in the presence of large quantities of L-ASP. 

Again, the molecularity of a reaction is always an integer and only applies to elementary reactions. Such is not always the case for the order of a reaction. The distinction between molecularity and order can also be stated as follows molecularity is the theoretical description of an elementary process reaction order refers to the entire empirically derived rate expression (which is a set of elementary reactions) for the complete reaction. Usually a bimolecular reaction is second order however, the converse need not always be true. Thus, unimolecular, bimolecular, and termolecular reactions refer to elementary reactions involving one, two, or three entities that combine to form an activated complex. 

Two additional corrections should be mentioned here. First one is the relativistic correction which is of order of a where a is the fine stmcture constant and the other is the radiative correction which can be derived from quantum electrodynamics. (Bethe and Salpeter, 1957 Kolos, 1993)... 

The BP utilises formation of a derivative in order to quantify penicillins in formulations. Some penicillins do not have distinctive chromophores a further problem with these molecules is that when they are in suspensions they are not readily extracted away from excipients since they are quite insoluble in organic solvents which are immiscible with water. Using the formation of a complex with the mercuric ion in the presence of imidazole as a catalyst, a derivative of the penicillin structure is produced, which has an absorption maximum between 325 and 345 nm. In the assay, comparison with pure standard for the particular penicillin is carried out rather than relying on a standard A(l%, 1 cm) value. This assay is used by the BP for... 

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