Analysis in Solving Problems

The difference between accuracy and precision is a subtle but important one. Suppose, for example, that three students are asked to determine the mass of a piece of copper wire. The results of two successive weighings by each smdent are 

The true mass of the wire is 2.000 g. Therefore, Student B s results are more precise than those of Student A (1.972 g and 1.968 g deviate less from 1.970 g than 1.964 g and 1.978 g from 1.971 g), but neither set of results is very accurate. Student C s results are not only the most precise, but also the most accurate, because the average value is closest to the true value. Highly accurate measurements are usually precise too. On the other hand, highly precise measurements do not necessarily guarantee accurate results. For example, an improperly calibrated meterstick or a faulty balance may give precise readings that are in error. 

Careful measurements and the proper use of significant figures, along with correct calculations, will yield accurate numerical results. But to be meaningful, the answers also must be expressed in the desired units. The procedure we use to convert between units in solving chemistry problems is called dimensional analysis (also called the factor-label method). A simple technique requiring little memorization, dimensional analysis is based on the relationship between different units that express the same physical quantity. For example, by definition 1 in = 2.54 cm (exactly). This equivalence enables us to write a conversion factor as follows  

Because both the numerator and the denominator express the same length, this fraction is equal to 1. Similarly, we can write the conversion factor as 

Dimensional analysis might also have led Einstein to his famous mass-energy equation E = mc.  

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Throughout the text we use dimensional analysis in solving problems. In this approach, units are multiplied together, divided into each other, or canceled. Using dimensional analysis helps ensure that solutions to problems yield the proper units. Moreover, it provides a systematic way of solving many numerical problems and of checking solutions for possible errors. 

A strong suggestion As you use dimensional analysis in solving problems, label each entry completely. Specifically, include the chemical formula of each substance in the calculation... 

Whereas many scientists shared Mulliken s initial skepticism regarding the practical role of theory in solving problems in chemistry and physics, the work of London (6) on dispersion forces in 1930 and Hbckel s 7t-electron theory in 1931 (7) continued to attract the interest of many, including a young scientist named Frank Westheimer who, drawing on the physics of internal motions as detailed by Pitzer (8), first applied the basic concepts of what is now called molecular mechanics to compute the rates of the racemization of ortho-dibromobiphenyls. The 1946 publication (9) of these results would lay the foundation for Westheimer s own systematic conformational analysis studies (10) as well as for many others, eg, Hendrickson s (11) and Allinger s (12). These scientists would utilize basic Newtonian mechanics coupled with concepts from spectroscopy (13,14) to develop nonquantum mechanical models of structures, energies, and reactivity. 

Antropov, P.Y., 1981. Laser gas analysis in solving geological and production problems. Inti. Gcol. Rev., 23 314-318. 

Do not let recycle streams confuse you. The steps in the analysis and solution of material balance problems involving recycle are the same as described in Table 2.4. With a little practice in solving problems involving recycle, you should experience little difficulty in solving recycle problems in general. The essential point you should grasp with respect to recycle calculations in this chapter is that the processes such as shown in Fig. 2.4 or 2.16 are in the steady state. 

In solving problems, one can be guided consciously by the units to the proper way of combining the given values. Such techniques are referred to in textbooks as the factor-label method, the unit-factor method, or dimensional analysis. In essence one goes from a given unit to the desired unit by multiplying by a fraction called a unit-factor in which the numerator and the denominator must represent the same quantity. 

Batch reactors are used primarily to determine rate law parameters for hotr geneous reactions. This determination is usually achieved by measuring cc centraiion as a function of time and then using either the differential, integr or nonlinear regression method of data analysis to determine the reacti order, a, and specific reaction rate constant, k. If some reaction parame other than concentration is monitored, such as pressure, the mole balance mi be rewritten in terms of the measured variable (e.g.. pressure as shown in t example in Solved Problems on the CD). 

Although these utilitarian reasons are often cited for a synthesis, the challenging problems usually encountered in a total synthesis provide many opportunities to find new reactions, processes, or strategies which may be of value to other organic chemists. Corey and Wipke stated a few of the u.seful applications that may be found in a retrosynthetic analysis-synthesis problem.Many of the.se may be useful in solving problems for another target or in applications to a completely different area of organic chemistry. 

Plan In solving problems of this type, we can use dimensional analysis. 

If in the analysis of a problem there is a set of simultaneous equations then the use of matrices can be a very convenient shorthand way of expressing and solving the equations. For example, consider the following set of equations ... 

Case study 5 provides an example from the offshore oil and gas production industry, and illustrates the fact that in solving a specific practical problem, a practitioner will utilize a wide variety of formal and informal methods. Table 7.1, which describes some of the methods used in the study, includes several techniques discussed in Chapter 4, including interviews, critical incident techniques, walk-throughs and task analysis. 

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