OPERATIONS WITH SCIENTIFIC NOTATION. Writing Numbers in Scientific Notation Write g in scientific notation.
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1 Name Date Class HOLT ChemFile TEACHING RESOURCES 4. OPERATIONS WITH SCIENTIFIC NOTATION In chemistry you will work with a variety of numbers that range from very large to very small. Avogadro s constant is an example of a very large number that you will use. Avogadro s constant is In scientific notation this number is written as As you can see, scientific notation makes writing this number easier. The mass of an electron is an example of a very small number that is used in chemistry. The mass of an electron is kg. It is much easier to write kg. EXAMPLE 1 Writing Numbers in Scientific Notation Write g in scientific notation. Move the decimal to the left or right until there is only one digit to the left of the decimal : Determine the power of 10 by counting the number of places you moved the decimal. If you moved the decimal to the left, the exponent is positive. If you moved the decimal to the right, the exponent is negative : Determine the units, and place them after the number. Units do not change when a number is converted to scientific notation g PRACTICE 1. Work through the following problem. What is mm in scientific notation? a. Move the decimal to the left or right until there is only one digit to the left of the decimal : b. Determine the power of 10 by counting the number of places you moved the decimal. If you moved the decimal to the left, the exponent is positive. If you moved the decimal to the right, the exponent is negative : 10 c. Determine the units, and place them after the number. 2. Rewrite the following values in scientific notation. a L b L ChemFile TEACHING RESOURCES: Math Skills 1
2 c g d ml e mg 3. Rewrite the following expressions in decimal form. a km b ml c mg d g e kg SIGNIFICANT FIGURES AND SCIENTIFIC NOTATION When a number is written in scientific notation, all of the significant figures are represented in the first factor so that there is no confusion about which zeros are significant and which are not. Look at the following examples for expressing data in scientific notation with the correct number of significant figures (If both zeros are not significant, there are two significant figures.) (If one zero is significant, there are three significant figures.) (If all the zeros are significant, there are five significant figures.) When performing calculations in scientific notation, you must often round numbers to obtain the correct number of significant figures. Review the following rules for rounding to the correct number of significant figures. If the digit immediately to the right of the last significant figure you want to retain is greater than 5, then increase the last digit by 1. less than 5, then do not change the last digit. 5, followed by nonzero digit(s), then increase the last digit by 1. 5, not followed by nonzero digit(s), and preceded by an odd digit, then increase the last digit by 1. 5, not followed by nonzero digit(s), and the preceding significant digit is even, then do not change the last digit. Example (rounded to three significant figures) L L 2574 g g s s m m (because 3 is odd) cm cm (because 6 is even) 2 TEACHING RESOURCES: Math Skills ChemFile
3 PRACTICE 4. Write each of the following in scientific notation with the indicated number of significant figures. Round where appropriate. a. 525 ml (one significant figure) b kg (three significant figures) c m (two significant figures) d cm (three significant figures) e mg (four significant figures) CALCULATIONS USING SCIENTIFIC NOTATION When adding or subtracting numbers in scientific notation, all of the powers of 10 must be the same magnitude. When multiplying powers of 10, the exponents are added together When dividing powers of 10, the exponents are subtracted Following are some examples that demonstrate how to work through calculations with numbers in scientific notation. EXAMPLE 2 Adding and Subtracting Numbers in Scientific Notation m m All of the powers of 10 must be the same magnitude. The power of 10 in is 4. The power of 10 in is 3. To make the power of 10 equal to 4 in both numbers, move the decimal one place to the left in : Notice that these two numbers have the same value in decimal form. Add or subtract the first factors. In this case, we are adding Attach the second factor to the number Determine the number of significant figures, and round if necessary. Be sure all the numbers have the same power of 10 when you find the leftmost uncertain digit. The two numbers used to determine the number of significant figures in the answer in this problem are and (not ) ChemFile TEACHING RESOURCES: Math Skills 3
4 If the first factor is greater than 10 or less than 1, move the decimal and change the power of 10. This number does not need to be changed. Determine the units, and place them after the number. Notice that the units are the same for both numbers. In order for two numbers to be added or subtracted, they must have the same units. The units do not change during addition or subtraction m PRACTICE 5. Work through the following problem kg kg a. All of the powers of 10 must be the same magnitude. b. Add or subtract the first factors. c. Attach the second factor to the number. d. Determine the number of significant figures, and round if necessary. e. If the first factor is greater than 10 or less than 1, move the decimal and change the power of 10. f. Determine the units, and place them after the number. 4 TEACHING RESOURCES: Math Skills ChemFile
5 6. Solve the following problems. Write the answers with the correct number of significant figures, and round if necessary. a. ( g) ( g) b. ( cm) ( cm) c. ( mol) ( mol) d. ( kg) ( kg) e. ( mg) ( mg) f. ( L) ( L) g. ( km) ( km) h. ( m) ( m) EXAMPLE 3 Multiplying and Dividing in Scientific Notation ( m)( m) When multiplying, add the powers of 10. When dividing, subtract the powers of 10. In this case, we are multiplying Multiply or divide the first factors. In this case, we are multiplying Attach the second factor to the number ChemFile TEACHING RESOURCES: Math Skills 5
6 Determine the number of significant figures, and round if necessary If the first factor is greater than 10 or less than 1, move the decimal and change the power of : Determine the units, and place them after the number. Because m m m 2, the unit is m 2. The answer is m 2. PRACTICE 7. Work through the following problem g L a. When multiplying, add the powers of 10. When dividing, subtract the powers of 10. b. Multiply or divide the first factors. c. Attach the second factor to the number. d. Determine the number of significant figures, and round if necessary. e. If the first factor is greater than 10 or less than 1, move the decimal and change the power of 10. f. Determine the units, and place them after the number. 6 TEACHING RESOURCES: Math Skills ChemFile
7 8. Solve the following problems. Write the answers with the correct number of significant figures, and round if necessary. a. ( km) ( km) b. ( m) ( m) c. ( M) ( M) d. ( cm) ( cm) e. ( mol/l) ( L) kg f. _ kg mg g mg kg h cm L i min g j g/mol ChemFile TEACHING RESOURCES: Math Skills 7
8 SCIENTIFIC NOTATION ON A CALCULATOR Many scientific calculators can perform calculations and give answers in scientific notation. When entering scientific notation on a calculator, enter only the first factor and the exponent of the power of 10. The exponent entry key is usually labeled or. This automatically includes the 10. When performing calculations, most scientific calculators will display the result in scientific notation when there are too many digits for the display. However, you must be careful because the calculator will not determine the number of significant figures. It will also cut off any digits that will not fit in the display. EXAMPLE 4 PRACTICE Solving Problems in Scientific Notation with a Calculator Calculate ( )( ) on your calculator. Press. Press or, then press. (This is the exponent of 10 4.) Press. Press. Press the or, then press. (This is the exponent of 10 5.) Press, and the calculator will display This is equivalent to Round the answer to three significant figures. The answer is The mass of an electron is kg. The mass of a proton is kg. The mass of a neutron is kg. a. Which has more mass, an electron or a proton? b. How much more mass does a neutron have than a proton? c. A hydrogen atom is made of one electron and one proton. What is the mass of one hydrogen atom? d. Tritium is a type of hydrogen that has one electron, one proton, and two neutrons. What is the mass of one tritium atom? e. Helium has two protons, two neutrons, and two electrons. What is the mass of one helium atom? f. Which has more mass, a helium atom or a tritium atom? g. What is the combined mass of one helium atom and one tritium atom? 8 TEACHING RESOURCES: Math Skills ChemFile
9 10. Some solid ionic compounds dissolve easily in water, while others dissolve only very slightly. When a compound only slightly dissolves, the amount that will dissolve in water is referred to as the solubility-product constant. To find the solubility product of a compound, multiply the concentration of each ion. The equation for the solubility product for silver chloride, AgCl, is as follows. K sp [Ag ][Cl ] If [Ag + ] = and [Cl ] = , what is the K sp for silver chloride? Answer 11. Assume Avogadro s constant is When you have that number of molecules of glucose, C 6 H 12 O 6, you have a quantity called a mole. If 1 mole of glucose has a mass of 180. g, find the mass in grams of molecules of glucose by working through the following problem. mass (g) molecules 180. g molecules mass (g) ( molecules)(180. g) molecules Answer ChemFile TEACHING RESOURCES: Math Skills 9
10 ANSWER KEY HOLT ChemFile TEACHING RESOURCES 4. Operations with Scientific Notation 1. a b f L g km c mm h m 2. a L b L 7. a b. 1.8 c g c d ml d (no change) e mg e (no change) 3. a km b ml c mg 8. f g/l a km 2 b m 2 d g c M 2 e kg d cm a ml b kg c m d cm e mg a. They are already the same. b c d (no change) e f kg a g b cm c mol e mol f g h kg/cm 3 i L/min j mol a. proton b kg c kg d kg e kg f. a helium atom g kg d kg e mg g ChemFile TEACHING RESOURCES: Math Skills Answer Key 1
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