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Dimensional Analysis

# Dimensional Analysis - Dimensional analysis A Introduction...

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Dimensional analysis. A. Introduction .................................................................................................... 1 B. Simple unit conversions; the idea of dimensional analysis ............................ 1 C. Multi-step conversions ................................................................................... 3 D. Velocity and density as conversion factors .................................................... 4 E. Answers .......................................................................................................... 7 A. Introduction This handout was originally written for another purpose. It was intended as a self-paced review for students to use on their own, with staff available for help. It was part of a set of several such handouts. Some of the other handouts are posted at the web site, and some may be referred to in class handouts as “supplementary materials” that are available. Dimensional analysis is a problem solving tool. It is particularly useful as a guide in solving problems that might be difficult to do otherwise. To emphasize learning the method -- the tool -- we introduce dimensional analysis with some easy examples. You could undoubtedly do many of these problems in your head. The point for now is to learn how to use the tool and to see that it works. The key to dimensional analysis is following the units. Many of the examples and problems in this handout use “common” English (USCS) units, for familiarity. (Most chem books should contain a list of conversion factors for these, including metric-English equivalents. See, for example Table 3.3. of Cracolice. If I use one you don’t know, please ask.) Metric problems are also included. If you need help with the metric units, see Sect 3.4 of Cracolice. Within the problem sets, some problems are marked with an *. This indicates that the problem introduces something new. If you are skipping around in the problems, you may well want to stop for a bit at a problem marked with an *. I did not intend significant figures (SF) to be an issue when I wrote the original version of this. However, some who use this now may care about SF. The answers do now show the correct number of SF. Exception: In some simple problems, with simple integer data, I have treated the data as “exact”. B. Simple unit conversions; the idea of dimensional analysis Example. Convert 2 feet to inches. There is a relationship between the two units in this problem, the “given” (feet) and the “wanted” (inches). 12 in = 1 ft. This simple equation can also be written 12 in 1 ft  = 1 or  = 1 1 ft 12 in

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Dimensional analysis. Page 2 You know that multiplying a number by 1 does not change its value. So multiplying by 12 in/1 ft, which equals 1, does not change the value. But it does change the form, the units; in fact, this is precisely the goal. 2 ft 12 in x  = 24 in 1 ft In writing out this dimensional analysis solution, we start by writing the “given” (2 ft). We then write an appropriate conversion factor. What is appropriate? First, it must be true; that is, the factor must equal 1. Second, it must be useful in the problem, relating the given to the wanted. (In this case, there is a simple, single step relationship. We will see more complex cases later.) Third, we write the conversion factor in a useful way . The conversion factor would be just as true if we had written it upside down. But we wrote it so that the ft cancel out, and the
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