Dimensional Analysis in Solving Problems

Careful measurements and the proper use of significant figures, along with correct calculations, will yield accurate numerical results. But to be meaningfiil, 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, we know that the monetary imit dollar is different from the unit penity. However, 1 dollar is equivalent to 100 pennies because they both represent the same amount of money that is, 

This equivalence enables us to write a conversion factor 

Because this is a dollar-to-penity conversion, we choose the conversion factor that has the unit dollar in the denominator (to cancel the dollars in 2.46 dollars) and write 

Note that the conversion factor 100 pennies/1 dollar contains exact numbers, so it does not affect the number of significant figures in the final answer. 

let us consider the conversion of 57.8 meters to centimeters. This problem can be expressed as 

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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... 

Use dimensional analysis to solve these problems. Remember that numbers in the numerator should be preceded by the multiplication key, whereas numbers in the denominator should be preceded by the division key. 

There are a variety of problem-solving strategies that you will use as you prepare for and take the AP test. Dimensional analysis, sometimes known as the factor label method, is one of the most important of the techniques for you to master. Dimensional analysis is a problem-solving technique that relies on the use of conversion factors to change measurements from one unit to another. It is a very powerful technique but requires careful attention during setup. The conversion factors that are used are equalities between one unit and an equivalent amount of some other unit. In financial terms, we can say that 100 pennies is equal to 1 dollar. While the units of measure are different (pennies and dollars) and the numbers are different (100 and 1), each represents the same amount of money. Therefore, the two are equal. Let s use an example that is more aligned with science. We also know that 100 centimeters are equal to 1 meter. If we express this as an equation, we would write ... 

Lack of geometrical similarity often is the main obstacle in applying the dimensional analysis to solving the scale-up problems. It was shown, for example, that Collette Oral 10, 75, and 300 are not geometrically similar.f In such cases, a proper correction to the resulting equations is required. 

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. 

This chemistry course may have been the first time you have encoimtered the method of dimensional analysis in problem solving. Explain what are meant by a conversion factor and an equivalence statement. Give an everyday example of how you might use dimensional analysis to solve a simple problem. 

Dimensional analysis (1.5) Problem-solving strategy in which one inspects the units on all quantities in a calculation to check for correctness. 

Section 1.6 In the dimensional analysis approach to problem solving, we keep track of units as we carry meas-... 

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

Chemistry is full of calculations. Our basic goal is to help you develop the knowledge and strategies you need to solve these problems. In this chapter, you will review the Metric system and basic problem solving techniques, such as the Unit Conversion Method. Your textbook or instructor may call this problem solving method by a different name, such as the Factor-Label Method and Dimensional Analysis. Check with your instructor or textbook as to for which SI (Metric) prefixes and SI-English relationships will you be responsible. Finally, be familiar with the operation of your calculator. (A scientific calculator will be the best for chemistry purposes.) Be sure that you can correctly enter a number in scientific notation. It would also help if you set your calculator to display in scientific notation. Refer to your calculator s manual for information about your specific brand and model. Chemistry is not a spectator sport, so you will need to Practice, Practice, Practice. 

In this section, we will introduce one of the two common methods for solving problems. (You will see the other method in Chapter 5.) This is the Unit Conversion Method. It will be very important for you to take time to make sure you fully understand this method. You may need to review this section from time to time. The Unit Conversion Method, sometimes called the Factor-Label Method or Dimensional Analysis, is a method for simplifying chemistry problems. This method uses units to help you solve the problem. While slow initially, with practice it will become much faster and second nature to you. If you use this method correctly, it is nearly impossible to get the wrong answer. For practice, you should apply this method as often as possible, even though there may be alternatives. 

Now, solve the problem using the dimensional analysis method. We want the answer to be in inches per second. Set up the fractions with inches on the top and seconds on the bottom, so that the centimeter and minute units cancel. 

The basic approach that GEORGE uses to solve problems is dimensional analysis, the same technique that many of us use in our own classrooms to teach students how to solve problems. Instead of having numerous formulas for different kinds of problems, GEORGE simply contains a set of heuristic rules which he follows to search for a solution. One result of using these heuristic rules is that he can solve problems never worked by the authors of the program. 

Dimensional analysis is a technique for solving problems that involve units or conversions that is taught in many engineering schools. It is a very useful technique in some areas of the emergency services, especially in EMS, where drug and fluid administration rates need to be calculated. 

Two approaches have been used to describe the effect of concentration polarization. One has its origins in the dimensional analysis used to solve heat transfer problems. In this approach the resistance to permeation across the membrane and the resistance in the fluid layers adjacent to the membrane are treated as resistances in series. Nothing is assumed about the thickness of the various layers or the transport mechanisms taking place. 

General buckling in a slender column with a slenderness ratio, L/D, greater than 100, occurs when it is subjected to a critical compressive load. This load is much lower than the maximum load allowable for compressive yield. Although this problem can be easily solved using Euler s equation1, which predicts the critical load applied to the slender column, it lends itself very well to illustrate dimensional analysis. 

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