Problem Solving and Dimensional Analysis

I AIM To learn how dimensional analysis can be used to solve various types of problems. 

Suppose that the boss at the store where you work on weekends asks you to pick up 2 dozen doughnuts on the way to work. However, you find that the doughnut shop sells by the doughnut. How many doughnuts do you need  

This problem is an example of something you encounter all the time converting from one unit of measurement to another. Examples of this occur in cooking (The recipe calls for 3 cups of cream, which is sold in pints. How many pints do 1 buy ) traveling (The purse costs 250 pesos. How much is that in dollars ) sports (A recent Tour de France bicycle race was 3215 kilometers long. How many miles is that ) and many other areas. 

How do we convert from one unit of measurement to another Let s explore this process by using the doughnut problem. 

You can use this information to make the needed conversion as follows  

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Measurements of Length, Volume, and Mass Uncertainty in Measurement Significant Figures Problem Solving and Dimensional Analysis Temperature Conversions ... 

Throughout the text we use an approach called dimensional analysis as an aid in problem solving. In dimensional analysis we carry units through all calculations. Units are multiplied together, divided into each other, or "canceled." Dimensional analysis will help ensure lhat the solutions to problems yield the proper units. Moreover, dimensional analysis provides a systematic way of solving many numerical problems and of checking our solutions for possible errors. 

See Schmidt, R., and Housen, K. 1995. Problem Solving with Dimensional Analysis, The Industrial Physicist, Vol. 1, pp. 21-24.)... 

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. 

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 units are an important part of every measurement. The units and dimensional analysis will even help solve mathematical problems. 

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 ... 

This section presents a method for arranging numbers that will work for most of the numerical problems you will encounter in this course. This method has a number of names, including the factor-unit method, the factor-label method, and dimensional analysis. We will call it the factor-unit method. It is a systematic approach to solving numerical problems and consists of the following steps ... 

This is a problem with an easily calculated answer whose purpose is to stress the mechanics of dimensional analysis. It is helpful to plan how to solve a dimensional analysis problem by identifying the Given quantity and the Wanted units, identifying... 

You can solve this problem by using dimensional analysis. Identify the Given quantity and units, the Wanted units, and the Per/Path that you will use. 

The problem can be solved by dimensional analysis if the Given and Wanted can be linked by one or more Per expressions and you know the conversion factor for each expression. 

In this case, Equation 4.4 gives you a Per relationship between the Given and Wanted quantities, 1 atm = 14.69 psi. Therefore, the problem is solved by dimensional analysis. 

Several introductory comments may help you learn how to solve stoichiometry problems. The problem-solving strategy from Section 3.9 is used repeatedly. You will soon see that a series of Per relationships link the Given and Wanted quantities in all problems in this chapter. Therefore, the problems are solved by dimensional analysis. 

In Section 1.6, we learned the SI unit system, the prefix multipliers, and a few other units. Knowing how to work with and manipulate these units in calculations is central to solving chemical problems. In calculations, units help to determine correctness. Using units as a guide to solving problems is called dimensional analysis. Units should always be included in calculations they are multiplied, divided, and canceled like any other algebraic quantity. 

Detection in 2DLC is the same as encountered in one-dimensional HPLC. A variety of detectors are presented in Table 5.2. The choice of detector is dependent on the molecule being detected, the problem being solved, and the separation mode used for the second dimension. If MS detection is utilized, then volatile buffers are typically used in the second-dimension separation. Ultraviolet detection is used for peptides, proteins, and any molecules that contain an appropriate chromophore. Evaporative light scattering detection has become popular for the analysis of polymers and surfactants that do not contain UV chromophores. Refractive index (RI) detection is generally used with size exclusion chromatography for the analysis of polymers. 

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. 

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. 

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