Mathematica® computer program

Descriptions of symbolic mathematics programs and some of their applications can be found, for example, in W. H. Cropper, Mathematica Computer Programs for Physical Chemistry, Springer-Verlag, New York (1998) and in J. F. Ogilvie, Mathematics for Chemistry with Symbolic Computation, an electronic text containing Maple worksheets, available at http //www.maplesoft.com. 

As a second example, consider the integral (6.58) above, which we found using the Mathematica computer program. A first simplification would be to write X = ay so that... 

Many tables of indefinite and definite integrals have been published. They range from collections of certain common integrals presented in appendices to most elementary calculus books, the famous Peirce tables, to compendia such as that by Gradshteyn and Ryzhik. More recently, many integrals have become available in analytical form in computer programs. One of the most complete lists is included in Mathematica (see footnote in Section 3.2). 

Equation (27) expresses an error in the dynamic matrix element Lij obtained from full matrix analysis if the error in peak volumes is Aa [50]. It also assumes that volume errors are equal for all peaks and are uncorrelated Aa is volume error normalized to the volume of a single spin at Tm = 0. Modem computer programs (Matlab, Mathematica, Mapple) can calculate the dynamic matrix from eq. (11) directly. 

Principles of Thermodynamics should be accessible to scientifically literate persons who are either learning the subject on their own or reviewing the material. At Emory University, this volume forms the basis of the first semester of a one-year sequence in physical chemistry. Problems and questions are included at the end of each chapter. Essentially, the questions test whether the students understand the material, and the problems test whether they can use the derived results. More difficult problems are indicated by an asterisk. Some problems, marked with an M, involve numerical calculations that are most easily performed with the use of a computer program such as Mathcad or Mathematica. A brief survey of some of these numerical methods is included in Appendix B, for cases in which the programs are unavailable or cumbersome to use. 

In this exercise, the fractional conversion XA for the 4-min isothermal batch operation is 77.1%. Obviously, the required answer is obtained through trial-and-error. Equation 5-45 can be incorporated into a computer program or mathematical package such as Mathcad, Maple, Polymath, or Mathematica. 

Mathematical methods are discussed in references 21-27 (some of these are "golden oldies"). Integrals, series, and other tables can be found in [28-33]. Of course, much labor can now be saved by using the computer program package Mathematica. 

A number of standard computer programs easily handle problems of this type such as spreadsheet packages, Matlab, Mathcad, Polymath, and so on as well as symbolic manipulators such as Mathematica, Maple, Derive, etc. Most statistic packages and equation solvers will also solve linear equations and have a simple user interface. 

In Mathematica reactants need to named with words starting with lower case letters because words starting with capital letters refer to operations. Also the names of reactants need to be as short as convenient and cannot involve spaces, subscripts, superscripts, hyphens, dots or other symbols that are Mathematica operations. Therefore, ATP is referred to as atp both in the text and in computer programs. Most of these abbreviated names will be recognized immediately, but a glossary of names is provided in the Appendix. 

A number of computer programs are available for machine computation using product operators within programs such as Mathematica or Maple. These can be very labour saving. 

Most of the Mathematica symbols are the same as those used in Excel or various computer programming languages such as BASIC except for the use of a blank space for multiplication. Excel and BASIC use only the asterisk for multiplication. In ordinary formulas, placing two symbols together without a space between them can stand for multiplication. In Mathematica, if you write xy, the software will think you mean a variable called xy, and not the product of x and y. However, you can write either 2x or 2 x for 2 times x, but not x2. It is probably best to use the asterisk ( ) for multiplication rather than a space in input statements. Watch for the use of the blank space in output statements. Complex arithmetic is done automatically, using the capital letter I for. Several constants are available by using symbols Pi, E, I, Infinity, and Degree stand for n,e,i =, oo, and... 

As you can see, matrix multiplication with fairly large matrices can involve a lot of computation. Computer programs can be written to carry out the process, and such programs are built into Mathematica and also into computer languages such as BASIC so that a matrix multiplication can be carried out with a single statement. 

There exist several sources where special formulae for the n/-symbols are presented. One reference compilation is represented by the Landolt-Bornstein series [1]. Of great advantage is the computer system Mathematica which is capable of deriving 3/ - and 6y -symbols in both symbolic and numerical forms [2], Another source using the FORTRAN computer program is available [3]. 

For the formation of the differentials and the solution of the sets of equations, symbolic computational programs can be used in favor, such as DERIVE(r) or Mathematica . 

To summarize, we perform a singular value decomposition of the augmented formula matrix to obtain the matrices U, W, and V. With these, we use (11.2.10) to obtain a particular basis vector N for the range. From V, we form P and then use (11.2.7) to obtain all sets of stoichiometric coefficients Vy. Then we combine N and Vy into (11.2.5) to determine all sets of mole numbers that satisfy the elemental balances. Therefore, a singular value decomposition provides the number of independent reactions 91, all sets of 91 independent stoichiometric coefficients Vy, and all possible combinations of mole numbers N that satisfy the elemental balances. A computer program for performing the decomposition is contained in the book by Press et al. [9] routines for performing the decomposition are also available in MATLAB and in Mathematica . 

Equation 11.25 must be used if the acid is very dilute, very weak, or both, such that the H ion concentration is expected to be less than 10 M. (If can be assumed to be negligible, Equation 11.25 reduces to Equation 11.20, as expected.) Although there is no simple formula for the roots of a cubic equation, Equation 11.25 can be solved using a graphing calculator or a computer program such as Mathematica or Mathcad, as illustrated in Example 11.13. 

Expanding larger determinants by hand can be tedious. Computer programs such as Mathematica perform the manipulations automatically. 

The first entry is for the species with the fewest hydrogen atoms. Only species of interest in the range pH 5 to 9 are included. No complex ions are included in this table. The advantage of this table is that, with Mathematica in a personal computer, it is not necessary to copy these numbers to make a calculation. Programs for making these calculations are also included in the package. 

Although matrix multiplications, row reductions, and calculation of null spaces can be done by hand for small matrices, a computer with programs for linear algebra are needed for large matrices. Mathematica is very convenient for this purpose. More information about the operations of linear algebra can be obtained from textbooks (Strang, 1988), but this section provides a brief introduction to making calculations with Mathematica (Wolfram, 1999). 

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