MATHEMATICAL TABLES 1 Constants

Zwillinger, D., ed. 1996. CRC Standard Mathematical Tables and Formulae, 30th ed. CRC Press, Boca Raton, FL Greek alphabet, physical constants. 

The 85th Edition includes updates and expansions of several tables, such as Aqueous Solubility of Organic Compounds, Thermal Conductivity of Liquids, and Table of the Isotopes. A new table on Azeotropic Data for Binary Mixtures has been added, as well as tables on Index of Refraction of Inorganic Crystals and Critical Solution Temperatures of Polymer Solutions. In response to user requests, several topics such as Coefficient of Friction and Miscibility of Organic Solvents have been restored to the Handbook. The latest recommended values of the Fundamental Physical Constants, released in December 2003, are included in this edition. Finally, the Appendix on Mathematical Tables has been revised by Dr. Daniel Zwillinger, editor of the CRC Standard Mathematical Tables and Formulae it includes new information on factorials, Clebsch-Gordan coefficients, orthogonal polynomials, statistical formulas, and other topics. 

The system of atomic units was developed to simplify mathematical equations by setting many fundamental constants equal to 1. This is a means for theorists to save on pencil lead and thus possible errors. It also reduces the amount of computer time necessary to perform chemical computations, which can be considerable. The third advantage is that any changes in the measured values of physical constants do not affect the theoretical results. Some theorists work entirely in atomic units, but many researchers convert the theoretical results into more familiar unit systems. Table 2.1 gives some conversion factors for atomic units. 

Table 6.1 gives the mathematical forms of energy terms often used in popular force fields. The constants may vary from one force field to another according to the designer s choice of unit system, zero of energy, and fitting procedure. 

Section 2 combines the former separate section on Mathematics with the material involving General Information and Conversion Tables. The fundamental physical constants reflect values recommended in 1986. Physical and chemical symbols and definitions have undergone extensive revision and expansion. Presented in 14 categories, the entries follow recommendations published in 1988 by the lUPAC. The table of abbreviations and standard letter symbols provides, in a sense, an alphabetical index to the foregoing tables. The table of conversion factors has been modified in view of recent data and inclusion of SI units cross-entries for archaic or unusual entries have been curtailed. 

Kinds oi Inputs Since a tracer material balance is represented by a linear differential equation, the response to anv one kind of input is derivable from some other known input, either analytically or numerically. Although in practice some arbitrary variation of input concentration with time may be employed, five mathematically simple input signals supply most needs. Impulse and step are defined in the Glossaiy (Table 23-3). Square pulse is changed at time a, kept constant for an interval, then reduced to the original value. Ramp is changed at a constant rate for a period of interest. A sinusoid is a signal that varies sinusoidally with time. Sinusoidal concentrations are not easy to achieve, but such variations of flow rate and temperature are treated in the vast literature of automatic control and may have potential in tracer studies. 

A simple, time-honoured illustration of the operation of the Monte Carlo approach is one curious way of estimating the constant n. Imagine a circle inscribed inside a square of side a, and use a table of random numbers to determine the cartesian coordinates of many points constrained to lie anywhere at random within the square. The ratio of the number of points that lies inside the circle to the total number of points within the square na l4a = nl4. The more random points have been put in place, the more accurate will be the value thus obtained. Of course, such a procedure would make no sense, since n can be obtained to any desired accuracy by the summation of a mathematical series... i.e., analytically. But once the simulator is faced with a eomplex series of particle movements, analytical methods quickly become impracticable and simulation, with time steps included, is literally the only possible approach. That is how computer simulation began. 

Data given in Tables 1-6 clearly show a significant dependence of P2 and p4 on amine concentration, that is, at least one of the apparent rate constants kj contains a concentration factor. Thus, according to the mathematical considerations outlined in the Analysis of Data Paragraph, both p2, P4 exponents and the derived variables -(P2 + p)4> P2 P4 ind Z (see Eqns. 8-12) are the combinations of the apparent rate constants (kj). To characterize these dependences, derived variables -(p2+p)4, P2 P4 and Z (Eqns. 8,11 and 12) were correlated with the amine concentration using a non-linear regression program to find the best fit. Computation resulted in a linear dependence for -(p2 + p)4 and Z, that is... 

NR with standard recipe with 10 phr CB (NR 10) was prepared as the sample. The compound recipe is shown in Table 21.2. The sectioned surface by cryo-microtome was observed by AFM. The cantilever used in this smdy was made of Si3N4. The adhesion between probe tip and sample makes the situation complicated and it becomes impossible to apply mathematical analysis with the assumption of Hertzian contact in order to estimate Young s modulus from force-distance curve. Thus, aU the experiments were performed in distilled water. The selection of cantilever is another important factor to discuss the quantitative value of Young s modulus. The spring constant of 0.12 N m (nominal) was used, which was appropriate to deform at rubbery regions. The FV technique was employed as explained in Section 21.3.3. The maximum load was defined as the load corresponding to the set-point deflection. 

Transform) the content of a given column ( vector) can be mathematically modified in various ways, the result being deposited in the (N + 1) column. The available operators are addition of and multiplication with a constant, square and square root, reciprocal, log(w), Infn), 10 , exp(M), clipping of digits, adding Gaussian noise, normalization of the column, and transposition of the table. More complicated data work-up is best done in a spreadsheet and then imported. 

Runaway criteria developed for plug-flow tubular reactors, which are mathematically isomorphic with batch reactors with a constant coolant temperature, are also included in the tables. They can be considered conservative criteria for batch reactors, which can be operated safer due to manipulation of the coolant temperature. Balakotaiah et al. (1995) showed that in practice safe and runaway regions overlap for the three types of reactors for homogeneous reactions (1) batch reactor (BR), and, equivalently, plug-flow reactor (PFR), (2) CSTR, and (3) continuously operated bubble column reactor (BCR). 

The mathematical model describing the two-phase dynamic system consists of modeling of the flow and description of its boundary conditions. The description of the flow is based on the conservation equations as well as constitutive laws. The latter define the properties of the system with a certain degree of idealization, simplification, or empiricism, such as equation of state, steam table, friction, and heat transfer correlations (see Sec. 3.4). A typical set of six conservation equations is discussed by Boure (1975), together with the number and nature of the necessary constitutive laws. With only a few general assumptions, these equations can be written, for a one-dimensional (z) flow of constant cross section, without injection or suction at the wall, as follows. 

To build a machine based on the models suggested by Pour-El and Rubel [157-159] and on the biochemical units developed in this work, the biochemical units should be able to perform the mathematical operations dehned above. In this work it was shown that networks A, B, and C can act as constant multipliers (see Table 5.1). Other units will have to be dehned in the future. 

The model is seen to be a series sequence of N equal sized CSTRs which have a total volume V and through which there is a constant flowrate Q. From the physical standpoint, it is natural to restrict N, the number of tanks, to integer values but, mathematically, this need not be the case. When N is considered as a continuous variable which lies between one and infinity, a model results which can be used to interpolate continuously between the bounds of mixing associated with the CSTR and PFR. For N less than unity, the model represents systems with partial bypassing [41]. For integral values of N eqn. (43) may be inverted directly (see Table 9, Appendix 1) to give... 

The mathematical notion of an operator may be unfamiliar it is a rule for modifying a function. A comparison of the ideas of operator and function may be useful Whereas a function acts to take an argument, called the independent variable, as input, and produces a value, called the dependent variable an operator takes 2l function as input and produces a function as output. Multiphcation of a function by a constant, taking a square or square root, differentiation or integration, are examples of operators. Table 8.1 contains examples of functions and operators. 

Figure 2 is a good representation of almost all the 112 runs made with 1-octanol. There was curvature on only a few runs, undoubtedly caused by experimental error since they could not be reproduced. This feature was checked carefully after mathematical analysis indicated reasons to expect curvature. The conditions of reactions and values of k0 have been tabulated (Tables I and II). Since k0 depends upon 1-octanol and TMAE, it is called the pseudo-first-order rate constant. 

Owing to the inadequacy of the mathematical model available for analysis of the amine extraction system (7), accurate values of the stability constants could not be evaluated for the Hs Pbn1-" system in the presence of NaCl. However, using the values of stability constants obtained by Bertazzi for the system (C2H5)3PbCln1 n in LiCl at 8.0 m (10), viz. 0 = 3.5, 02 = 1.0, 0s = 0.1, the neutral species Hs PbCl0 (n = 1) is seen to be dominant. Therefore a simple solvent extraction would be expected to remove a certain amount of triethyl lead from solution. As shown in Table II, this is seen to be so. However,... 

Fortunately, the Madelung constant may be obtained mathematically from a converging series, and there are computer programs that converge rapidly. However, we need not delve into these procedures, but may simply employ the values obtained by other workers (Table 4.1). The value of the Madelung constant is determined... 

Table II shows the mathematical forms of the normal, log-normal, Rosin-Rammler, and Nukiyama-Tanasawa distributions for an arbitrary ptb-weighted size distribution. As in Table I, the formulas yield the surface increment in the size interval, t to t + dt, for p = 2 number or volume increments are obtained by setting p equal to 0 or 3, respectively. In these expressions 6 and n are constants, and a denotes an appropriate shape factor.

Classes of Estimation Methods Table 1.1.1 summarizes the property estimation methods considered in this book. Quantitative property-property relationships (QPPRs) are defined as mathematical relationships that relate the query property to one or several properties. QPPRs are derived theoretically using physicochemical principles or empirically using experimental data and statistical techniques. By contrast, quantitative structure-property relationships (QSPRs) relate the molecular structure to numerical values indicating physicochemical properties. Since the molecular structure is an inherently qualitative attribute, structural information has first to be expressed as a numerical values, termed molecular descriptors or indicators before correlations can be evaluated. Molecular descriptors are derived from the compound structure (i.e., the molecular graph), using structural information, fundamental or empirical physicochemical constants and relationships, and stereochemcial principles. The molecular mass is an example of a molecular descriptor. It is derived from the molecular structure and the atomic masses of the atoms contained in the molecule. An important chemical principle involved in property estimation is structural similarity. The fundamental notion is that the property of a compound depends on its structure and that similar chemical stuctures (similarity appropriately defined) behave similarly in similar environments. 

The validity of Boyle s law can be demonstrated by making a simple series of pressure-volume measurements on a gas sample (Table 9.2) and plotting them as in Figure 9.6. When V is plotted versus P, as in Figure 9.6a, the result is a curve in the form of a hyperbola. When V is plotted versus 1/P, as in Figure 9.6b, the result is a straight line. Such graphical behavior is characteristic of mathematical equations of the form y = mx + b. In this case, y = V,m = the slope of the line (the constant k in the present instance), x = 1 /P, and b = the y-intercept (a constant 0 in the present instance). (See Appendix A.3 for a review of linear equations.)... 

As indicated by the plots in Figure 10.13a, the vapor pressure of a liquid rises with temperature in a nonlinear way. A linear relationship is found, however, when the logarithm of the vapor pressure, In Pvap, is plotted against the inverse of the Kelvin temperature, 1 /T. Table 10.8 gives the appropriate data for water, and Figure 10.13b shows the plot. As noted in Section 9.2, a linear graph is characteristic of mathematical equations of the form y = mx + b. In the present instance, y = lnPvap, x = 1/T, m is the slope of the line (- AHvap/R), and b is the y-intercept (a constant, C). Thus, the data fit an expression known as the Clausius-Clapeyron equation. ... 

The temperature dependence of the dielectric properties of foods has been extensively measured and reviewed by Bengtsson and Risman (1971) and Buffler (1993). Mudgett et al. (1977) has pioneered the prediction of dielectric properties of foods as a function of constituency and temperature. Prediction of the temperature behavior of dielectric properties is crucial for accurate mathematical modeling of foods. Many workers today still use constant room temperature values or a look-up table at best. In the author s opinion, dielectric prediction of food properties is still a very fertile and useful research field. 

It is important to note that both H202 and Fe2+ have to be overdosed to maintain a steady-state concentration of hydroxyl radical and to obtain a satisfactory approximation of the mathematical model with the experimental data. When H202 and Fe2+ concentrations are 5 x 10-3 M and 2 x 1th4 M, respectively, the relative rate constants of 2-chlorophenol (2-CP) and 2,4,6-TCP with respect to 2,4-DCP can be calculated. The oxidation and dechlorination constants of 2,4-DCP were found to be 0.995 1/min (fc4) and 0.092 1/min (k2), as reported in a previous study (Tang and Fluang, 1996). For comparison, Table 6.1 summarizes all the kinetic constants as determined in this study and in the related literature. 

It was previously normal practice to use linear forms of rate equations to simplify determination of rate constants by graphical methods. For example, the logarithmic version of the first-order rate law (Table 3.1), Equation 3.17a, allows k to be determined easily from the gradient of a graph of In Ct against time, by fitting the data to the mathematical model, y = a + bx ... 

To test whether the reaction is first order, we simply fit the data to the exponential integrated first-order rate equation (Table 3.1) using a non-linear optimisation procedure and the result is shown in Fig. 3.3. The excellent fit shows that the reaction follows the mathematical model and, therefore, that the process is first order with respect to [N2O5], i.e. the rate law is r = A bsI Os]. The rate constant is also obtained in the fitting procedure, k0bs = (6.10 0.06) x 10 4 s 1. We see that, even with such a low number of experimental points, the statistical error is lower than 1%, which shows that many data points are not needed if... 

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