Problem Solving and Symbolic Mathematics

EXAMPLE 2.18 Under ordinary conditions, ordinary gases nearly obey the 

SOLUTION Since there are four variables, we can calculate the value of one of them if the values of the other three are given. We solve the ideal gas equation for V by symbolically dividing both sides of the equation by P, obtaining 

We substitute the numerical values into Eq. (2.113), convert the pressure from atmospheres to pascals by use of the factor label method, and carry out the numerical operations  

We give the answer to four significant digits, because both the pressure and the amount of gas are specified to four significant digits. We can now make an additional conversion to express the volume in liters  

We know that this solution is in the correct range of values, since a mole of gas at room temperature and one atmosphere pressure occupies about 25 liters.  

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CHAPTER I 3 Problem Solving and Symbolic Mathematics Algebra... 

Formulate a complete mathematical statement of the problem, and label each individual part, identifying the objective function and constraints with the correct units (, days, etc.). Make a list of the variables by names and symbol plus units. Do not solve. 

In this chapter we have introduced symbolic mathematics, which involves the manipulation of symbols instead of performing numerical operations. We have presented the algebraic tools needed to manipulate expressions containing real scalar variables, real vector variables, and complex scalar variables. We have also introduced ordinary and hyperbolic trigonometric functions, exponentials, and logarithms. A brief introduction to the techniques of problem solving was included. 

In the previous sections, we discussed how to use MATLAB to solve eng eering problems with numerical values. In this section, we briefly explain the symbolic capabilities of MATLAB. In symbolic mathematics, as the name implies, the problem and the solution are presented using symbols, such as x instead of numerical values. We will demonstrate MATLAB s symbolic capabilities using Examples 15.7 and 15.8. 

Algebra is used to solve many simple problems encountered by process technicians. Basic mathematics is useful but inadequate for all process problems. Algebra uses letters and symbols to represent variables that are known and unknown. This form of mathematics allows unknown variables to be identified by following well-defined principles. 

A mathematical device, the NSS, which can be related to Artificial Intelligence techniques, has been defined and applied in order to solve or reformulate some quantum chemical problems. This symbol is related to computer formulae generation. It has been shown that by means of the use of NSS s many applications of such symbols can be found in mathematics as well as in Mathematical Chemistry in particular. 

Alternative technologies should also be considered, and the reasons for not using them should be justifiable. For example, database technology is not the right choice if a task requires reasoning that goes beyond retrieval of stored data based on well-defined criteria. At the same time, many problems that are stated in a symbolic way can be formulated mathematically, and in fact can be better solved numerically. For such problems, knowledge-based systems are not the appropriate answer. 

This book is designed to help you leam the fundamentals of chemistry. To be successful, you must master the concepts of chemistry and acquire the mathematical skills necessary to solve problems in this quantitative science. If your algebra is rusty, you should polish it up. Appendix 1 reviews the algebra used in basic chemistry and also shows how to avoid mistakes while solving chemistry problems with your scientific calculator. The factor label method is introduced in Chapter 2 to show you how to use units to help with problem solutions. You can help yourself by using the standard symbols and abbreviations for various quantities (such as m for mass, m for meter, mol for moles, and M for molarity). Always use the proper units with your numerical answers it makes a big difference whether your roommate s pet is 6 inches long or 6 feet long ... 

We now turn to more detailed consideration of material balance problems. Your basic task is to turn the problem, expressed in words, into a quantitative form, expressed-in-mathematical-symbols-and-numbers, and-then solve the.resulting math i ematical equations. 

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