Scientific Notation and Dimensional Analysis

Extremely small and extremely large numbers can be compared more easily when they are converted into a form called scientific notation. Scientific notation expresses numbers as a multiple of two factors a number between 1 and 10 and ten raised to a power, or exponent. The exponent tells you how many times the first factor must be multiplied by ten. When numbers larger than 1 are expressed in scientific notation, the power of ten is positive. When numbers smaller than 1 are expressed in scientific notation, the power of ten is negative. For example, 2000 is written as 2 X 10 in scientific notation, and 0.002 is written as 2 x 10 .  

The surface area of the Pacific Ocean is 166 000 000 000 000 m. Write this quantity in scientific notation. 

To write the quantity in scientific notation, move the decimal point to after the first digit to produce a factor that is between 1 and 10. Then count the number of places you moved the decimal point this number is the exponent ( ). Delete the extra zeros at the end of the first factor, and multiply the result by 10 . When the decimal point moves to the left, n is positive. When the decimal point moves to the right, n is negative. In this problem, the decimal point moves 14 places to the left thus, the quantity is written as 1.66 X 10 in scientific notation. 

Adding Quantities Written in Scientific Notation Solve the following problem. 

First express both quantities to the same power of ten. Either quantity can be changed. For example, you might change 2.45 X Qi to 245 X 1012, Then add the quantities 245 X IO12 kg + 4.00 X IO12 kg = 249 X 1012 kg. Write the final answer in scientific notation 2.49 X 1014 kg. 

Scientists often express numbers in scientific notation and solve problems using dimensional analysis. 

Real-World Reading Link If you have ever had a job, one of the first things you probably did was figure out how much you would earn per week. If you make 10 dollars per hour and work 20 hours per week, how much money will you make Performing this calculation is an example of dimensional analysis. 

Scientific notation can be used to express any number as a number between 1 and 10 (known as the coefficient) multiplied by 10 raised to a power (known as the exponent). When written in scientific notation, the two numbers above appear as follows. 

Note that a carat is a unit of measure used for gemstones (1 carat = 200 mg). 

Determining the exponent to use when writing a number in scientific notation is easy simply count the number of places the decimal point must be moved to give a coefficient between 1 and 10. The number of places moved equals the value of the exponent. The exponent is positive when the decimal moves to the left and the exponent is negative when the decimal moves to the right. 

A proton s mass is 0.000 000 000 000 000 000 000 000 001 672 62 kg. An electron s mass is 0.000 000 000 000 000 000 000 000 000 000 910 939 kg. If you try to compare the mass of a proton with the mass of an electron, the zeros get in the way. Numbers that are extremely small or large are hard to handle. You can convert such numbers into a form called scientific notation. 

Change the following data into scientific notation. 

You are given two measurements. One measurement is much larger than 10. The other is much smaller than 10. In both cases, the answers will be factors between 1 and 10 that are multiplied by a power of ten. 

Move the decimal point to produce a factor between 1 and 10. Count the number of places the decimal point moved and the direction. 

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A good portion of the AP Chemistry Test deals with calculations, either with or without the aid of a calculator. For all of these problems, there are two different components—the chemistry component and the math component. Most of this book is devoted to a review of the chemistry component of the problems, but this chapter is designed to review a few important mathematical skills that you will need to know as you work through the problems. Three skills that are critical to success on the AP Chemistry Test use significant figures, scientific notation, and 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. 

Organize Create a flowchart that outlines when to use dimensional analysis and when to use scientific notation. 

Chapter 4 provides an opportunity for you to apply the chemical calculation skills you learned as a result of studying Chapter 3. Scientific notation, dimensional analysis, metric units, significant figures, temperature, proportionality, and density are needed to understand the concepts and work the problems in this introduction to gases. You may find that you occasionally need to review Chapter 3 as you study this chapter. If so, don t be concerned. All successful science students review and refine their understanding of prior material-even content from prior coursework—as they learn new ideas. In fact, we selected the topics of Chapter 4, in part to give you a chance to apply your calculating skills immediately after you learned them. 

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