{"id":1744,"date":"2021-09-17T15:37:31","date_gmt":"2021-09-17T15:37:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/?post_type=chapter&#038;p=1744"},"modified":"2024-09-11T14:40:03","modified_gmt":"2024-09-11T14:40:03","slug":"2-4-converting-units-prefix-conversions","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/chapter\/2-4-converting-units-prefix-conversions\/","title":{"raw":"2.3 Converting Units: Prefix Conversions","rendered":"2.3 Converting Units: Prefix Conversions"},"content":{"raw":"<div>\r\n<div id=\"ball-ch02_s04\" class=\"section\" lang=\"en\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Be able to write prefix conversion factors.<\/li>\r\n \t<li>Use dimensional analysis to carry out prefix unit conversions.<\/li>\r\n<\/ul>\r\n<\/div>\r\nSometimes we use units that are fractions or multiples of a base unit. Ice cream is sold in quarts (a familiar, non-SI base unit), pints (0.5 quart), or gallons (4 quarts). We also use fractions or multiples of units in the SI system, but these fractions or multiples are always powers of 10. Fractional or multiple SI units are named using a prefix and the name of the base unit. For example, a length of 1000 meters is also called a kilometer because the prefix <em>kilo<\/em> means \u201cone thousand,\u201d which in scientific notation is 10<sup>3<\/sup> (1 kilometer = 1000 m = 10<sup>3<\/sup> m). The prefixes used and the powers to which 10 are raised are listed in Table 1.\r\n<table style=\"width: 271px;\" summary=\"The prefix femto has the symbol lowercase f and a factor of 10 to the negative fifteenth power. Therefore, 1 femtosecond, F S, is equal to 1 times 10 to the negative 15 of a meter, or 0.000000000001 of a meter. The prefix pico has the symbol lowercase P and a factor of 10 to the negative twelfth power. Therefore, 1 picosecond, P S, is equal to 1 times 10 to the negative 12 of a meter, or 0.000000000001 of a meter. The prefix nano has the symbol lowercase N and a factor of 10 to the negative ninth power. Therefore, 4 nanograms, or NG, equals 4 times ten to the negative 9, or 0.000000004 g. The prefix micro has the greek letter mu as its symbol and a factor of 10 to the negative sixth power. Therefore, 1 microliter, or mu L, is equal to one times ten to the negative 6 or 0.000001 L. The prefix milli has a lowercase M as its symbol and a factor of 10 to the negative third power. Therefore, 2 millimoles, or M mol, are equal to two times ten to the negative 3 or 0.002 mol. The prefix centi has a lowercase C as its symbol and a factor of 10 to the negative second power. Therefore, 7 centimeters, or C M, are equal to seven times ten to the negative 2 meters or 0.07 M O L. The prefix deci has a lowercase D as its symbol and a factor of 10 to the negative first power. Therefore, 1 deciliter, or lowercase D uppercase L, are equal to one times ten to the negative 1 meters or 0.1 L. The prefix kilo has a lowercase K as its symbol and a factor of 10 to the third power. Therefore, 1 kilometer, or K M, is equal to one times ten to the third meters or 1000 M. The prefix mega has an uppercase M as its symbol and a factor of 10 to the sixth power. Therefore, 3 megahertz, or M H Z, are equal to three times 10 to the sixth hertz, or 3000000 H Z. The prefix giga has an uppercase G as its symbol and a factor of 10 to the ninth power. Therefore, 8 gigayears, or G Y R, are equal to eight times 10 to the ninth years, or 800000000 G Y R. The prefix tera has an uppercase T as its symbol and a factor of 10 to the twelfth power. Therefore, 5 terawatts, or T W, are equal to five times 10 to the twelfth watts, or 5000000000000 W.\">\r\n<thead>\r\n<tr>\r\n<th style=\"width: 271px;\" colspan=\"3\">Table 1. Common Unit Prefixes<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th style=\"width: 84px; text-align: center;\">Prefix<\/th>\r\n<th style=\"width: 100px; text-align: center;\">Symbol<\/th>\r\n<th style=\"width: 87px; text-align: center;\">Factor<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td style=\"width: 84px; text-align: center;\">femto<\/td>\r\n<td style=\"width: 100px; text-align: center;\">f<\/td>\r\n<td style=\"width: 87px; text-align: center;\">10<sup>\u221215<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 84px; text-align: center;\">pico<\/td>\r\n<td style=\"width: 100px; text-align: center;\">p<\/td>\r\n<td style=\"width: 87px; text-align: center;\">10<sup>\u221212<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 84px; text-align: center;\">nano<\/td>\r\n<td style=\"width: 100px; text-align: center;\">n<\/td>\r\n<td style=\"width: 87px; text-align: center;\">10<sup>\u22129<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 84px; text-align: center;\">micro<\/td>\r\n<td style=\"width: 100px; text-align: center;\">\u00b5<\/td>\r\n<td style=\"width: 87px; text-align: center;\">10<sup>\u22126<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 84px; text-align: center;\">milli<\/td>\r\n<td style=\"width: 100px; text-align: center;\">m<\/td>\r\n<td style=\"width: 87px; text-align: center;\">10<sup>\u22123<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 84px; text-align: center;\">centi<\/td>\r\n<td style=\"width: 100px; text-align: center;\">c<\/td>\r\n<td style=\"width: 87px; text-align: center;\">10<sup>\u22122<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 84px; text-align: center;\">deci<\/td>\r\n<td style=\"width: 100px; text-align: center;\">d<\/td>\r\n<td style=\"width: 87px; text-align: center;\">10<sup>\u22121<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 84px; text-align: center;\">kilo<\/td>\r\n<td style=\"width: 100px; text-align: center;\">k<\/td>\r\n<td style=\"width: 87px; text-align: center;\">10<sup>3<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 84px; text-align: center;\">mega<\/td>\r\n<td style=\"width: 100px; text-align: center;\">M<\/td>\r\n<td style=\"width: 87px; text-align: center;\">10<sup>6<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 84px; text-align: center;\">giga<\/td>\r\n<td style=\"width: 100px; text-align: center;\">G<\/td>\r\n<td style=\"width: 87px; text-align: center;\">10<sup>9<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 84px; text-align: center;\">tera<\/td>\r\n<td style=\"width: 100px; text-align: center;\">T<\/td>\r\n<td style=\"width: 87px; text-align: center;\">10<sup>12<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Deriving Prefix Conversion Factors<\/h2>\r\n<\/div>\r\n<\/div>\r\nSuppose we are asked to convert 50.0 mg of ibuprofen to grams.\u00a0 How do we go about this conversion? The first thing to recognize is that we are dealing with a prefix conversion problem and in order for us to successfully convert milligrams to grams, we need to know what exactly does the prefix milli- mean.\u00a0 From Table 1, we see milli- is 10<sup>-3<\/sup>. How do we express this as a conversion factor? The relationship between mg and g can be expressed two ways (shown as equalities):\r\n<p style=\"text-align: center;\">[latex]\\large (1 \\text{ mg}=10^{-3}\\text{ g}) \\text{ or }(10^{3}\\text{ mg}=1\\text{ g})[\/latex]<\/p>\r\nWe use the equality to write the conversion factor needed to convert whatever unit we wish to convert, by showing the equality as a fraction, as shown below:\r\n<p style=\"text-align: center;\">[latex]\\large (\\frac{10^{-3}{\\text{ g }}}{1\\text{ mg}})\\text{ or }(\\frac{1{\\text{ mg }}}{10^{-3}{\\text{ g }}})\\text{ ; }(\\frac{10^{3}{\\text{ mg }}}{1\\text{ g}})\\text{ or }(\\frac{1{\\text{ g }}}{10^{3}{\\text{ mg }}})[\/latex]<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n\r\nNote: When writing the conversion factor, if you place the 10 to the power (10<sup>x<\/sup>) opposite the prefix unit, the sign of the power stays exactly the same as presented in Table 1. However, if you place the 10 to the power (10<sup>x<\/sup>) with the prefix unit, the sign of the power needs to be switched.\r\n\r\n<\/div>\r\n<h2>Prefix to Base Unit Conversions<\/h2>\r\nNow that we know the relationship between grams and milligrams we use dimensional analysis to determine how many grams are in 50.0 mg of ibuprofen. In dimensional analysis, we multiply what is given by one or more conversion factors. Conversion factors are applied in a manner to which cancel units until you arrive at the desired unit.\u00a0 In this example, we start with 50.0 mg and use the conversion factor derived earlier to cancel the mg:\r\n<p style=\"text-align: center;\">[latex]\\large 50.0 \\cancel{\\text{ mg}}\\times\\frac{10^{-3}{\\text{ g}}}{1\\cancel{\\text{ mg}}}= 5.00\\times 10^{-2}\\text{ g}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">or<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large 50.0 \\cancel{\\text{ mg}}\\times\\frac{1{\\text{ g}}}{10^{3}\\cancel{\\text{ mg}}}= 5.00\\times 10^{-2}\\text{ g}[\/latex]<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 1: <strong>Prefix to base Unit Conversions\r\n<\/strong><\/h3>\r\nHow many femtoliters are in [latex]\\large 2.00\\times 10^{2}[\/latex] liters?\r\n\r\n[reveal-answer q=\"79228735\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"79228735\"]\r\n\r\nFirst, we look up what a femto means, 10<sup>-15<\/sup>.\u00a0 The conversion can be written as 1 fm = 10<sup>-15<\/sup> L.\r\n\r\n[latex] 2.00\\times 10^{2} \\cancel{\\text{ L}}\\times\\frac{1{\\text{ fL}}}{10^{-15}\\cancel{\\text{ L}}}= 2.00\\times 10^{17}\\text{ fL}[\/latex]\r\n\r\n[\/hidden-answer]\r\n<h4><strong>Check Your Learning<\/strong><\/h4>\r\nHow many seconds are in 5.0 gigaseconds (Gs)?\r\n\r\n[reveal-answer q=\"79871236\"]Show Answer[\/reveal-answer]\r\n\r\n[hidden-answer a=\"79871236\"]\r\n\r\n5.0 \u00d7 10<sup class=\"superscript\">9<\/sup> s\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Prefix to Prefix Conversions<\/h2>\r\n<p id=\"ball-ch02_s04_p33\" class=\"para editable block\">Sometimes there will be a need to convert from one unit with one numerical prefix to another unit with a different numerical prefix. How do we handle those conversions? Well, you could memorize the conversion factors that interrelate all numerical prefixes. Or you can go the easier route: first convert the quantity to the base unit, the unit with no numerical prefix, using the definition of the original prefix. Then convert the quantity in the base unit to the desired unit using the definition of the second prefix. You can do the conversion in two separate steps or as one long algebraic step. For example, to convert 2.77 kg to milligrams:<\/p>\r\nFirst convert to the base unit of grams:\r\n<p style=\"text-align: center;\">[latex]\\large 2.77 \\cancel{\\text{ kg}}\\times\\frac{10^{3}{\\text{ g}}}{1\\cancel{\\text{ kg}}}= 2.77\\times 10^{3} \\text{ g}[\/latex]<\/p>\r\nThen convert the base unit to milligrams:\r\n<p style=\"text-align: center;\">[latex]\\large 2.77\\times 10^{3}\\cancel{\\text{ g}}\\times\\frac{1{\\text{ mg}}}{10^{-3}\\cancel{\\text{ g}}}= 2.77\\times 10^{6} \\text{ mg}[\/latex]<\/p>\r\nAlternatively, it can be done in a single multistep process:\r\n<p style=\"text-align: center;\">[latex]\\large 2.77 \\cancel{\\text{ kg}}\\times\\frac{10^{3}\\cancel{\\text{ g}}}{1\\cancel{\\text{ kg}}}\\times\\frac{10^{3}{\\text{ mg}}}{1\\cancel{\\text{ g}}}= 2.77\\times 10^{6} \\text{ mg}[\/latex]<\/p>\r\n<p id=\"ball-ch02_s04_p35\" class=\"para editable block\">You get the same answer either way.<\/p>\r\nYou can also combine the two conversion factors into one, and convert directly from one prefix to another prefix:\r\n<p style=\"text-align: center;\">[latex]\\large 2.77 \\cancel{\\text{ kg}}\\times\\frac{10^{3}\\cancel{\\text{ mg}}}{10^{-3}\\cancel{\\text{ kg}}}= 2.77\\times 10^{6} \\text{ mg}[\/latex]<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n\r\nNote: When writing the prefix to prefix conversion factor, you place the 10 to the power (10<sup>x<\/sup>) opposite the prefix unit. In this case we have prefixes in the numerator and the denominator, so we end up with a 10<sup>x<\/sup> in both spots, with the 10<sup>3<\/sup> going opposite of the kilo prefix and the 10<sup>-3<\/sup> going opposite the milli prefix.\r\n\r\n<\/div>\r\n<p id=\"ball-ch02_s04_p40\" class=\"para editable block\">When considering the significant figures of a final numerical answer in a conversion, there is one important case where a number does not impact the number of significant figures in a final answer\u2014the so-called <span class=\"margin_term\"><a class=\"glossterm\">exact number<\/a><\/span>. An exact number is a number from a defined relationship, not a measured one. For example, the prefix <em class=\"emphasis\">kilo-<\/em> means 1,000\u2014<em class=\"emphasis\">exactly<\/em> 1,000, no more or no less. Thus, in constructing the conversion factor<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large \\frac{1000{\\text{ g}}}{\\text{ kg}}[\/latex]<\/p>\r\n<p id=\"ball-ch02_s04_p41\" class=\"para editable block\">neither the 1,000 nor the 1 enter into our consideration of significant figures. The numbers in the numerator and denominator are defined exactly by what the prefix <em class=\"emphasis\">kilo-<\/em> means. Another way of thinking about it is that these numbers can be thought of as having an infinite number of significant figures, such as<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large \\frac{1000.0000000000 . . .{\\text{ g}}}{1.0000000000 . . .\\text{ kg}}[\/latex]<\/p>\r\n<p id=\"ball-ch02_s04_p42\" class=\"para editable block\">The other numbers in the calculation will determine the number of significant figures in the final answer.<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 2: <strong>Prefix to Prefix Conversion\r\n<\/strong><\/h3>\r\nHow many nanoseconds are in 368.09 \u03bcs?\r\n\r\n[reveal-answer q=\"798735\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"798735\"]\r\n\r\nYou can either do this as a one-step conversion from microseconds to nanoseconds or convert to the base unit first and then to the final desired unit. We will use the second method here, showing the two steps in a single line. Using the definitions of the prefixes <em class=\"emphasis\">micro-<\/em> and <em class=\"emphasis\">nano-<\/em>,\r\n\r\n[latex]368.09 \\cancel{{\\text{ \u03bcs}}}\\times\\frac{10^{-6}\\cancel{\\text{ s}}}{1\\cancel{\\text{ \u03bcs}}}\\times\\frac{1\\text{ ns}}{10^{-9}\\cancel{\\text{ s}}}= 3.6809\\times 10^{5} \\text{ ns}[\/latex]\r\n\r\nor solve using one conversion directly between the prefix units:\r\n\r\n[latex]368.09 \\cancel{{\\text{ \u03bcs}}}\\times\\frac{10^{-6}\\text{ ns}}{10^{-9}\\cancel{\\text{ \u03bcs}}}= 3.6809\\times 10^{5} \\text{ ns}[\/latex]\r\n\r\n[\/hidden-answer]\r\n<h4><strong>Check Your Learning<\/strong><\/h4>\r\nHow many milliliters are in 607.8 kL?\r\n\r\n[reveal-answer q=\"798736\"]Show Answer[\/reveal-answer]\r\n\r\n[hidden-answer a=\"798736\"]\r\n\r\n6.078 \u00d7 10<sup class=\"superscript\">8<\/sup> mL\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"ball-ch02_s04_n07\" class=\"callout block\"><\/div>\r\n<div>\r\n<div id=\"ball-ch02_s04\" class=\"section\" lang=\"en\">\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\n1. Perform each of the following conversions:\r\n<p style=\"padding-left: 30px;\">a.\u00a0 3.4 m to nm<\/p>\r\n<p style=\"padding-left: 30px;\">b.\u00a0 72.6 mL to L<\/p>\r\n<p style=\"padding-left: 30px;\">c.\u00a0 0.50 g to mg<\/p>\r\n<p style=\"padding-left: 30px;\">d.\u00a0 60.0 s to Ms<\/p>\r\n2. Perform each of the following conversions:\r\n<p style=\"padding-left: 30px;\">a.\u00a0 5.83 km to m<\/p>\r\n<p style=\"padding-left: 30px;\">b.\u00a0 5 \u00d7 10<sup>\u22124<\/sup> L to cL<\/p>\r\n<p style=\"padding-left: 30px;\">c.\u00a0 10.0 kg to g<\/p>\r\n<p style=\"padding-left: 30px;\">d.\u00a0 3.6 \u00d7 10<sup>3<\/sup> s to ms<\/p>\r\n3. Perform each of the following conversions:\r\n<p style=\"padding-left: 30px;\">a.\u00a0 1.2 \u00d7 10<sup>\u22123<\/sup>\u00a0mL to pL.<\/p>\r\n<p style=\"padding-left: 30px;\">b.\u00a0 200.0 cm to Tm.<\/p>\r\n<p style=\"padding-left: 30px;\">c.\u00a0 1.2 \u00d7 10<sup class=\"superscript\">6<\/sup> Ms to ks<\/p>\r\n<p style=\"padding-left: 30px;\">d.\u00a0 0.674 kL to mL<\/p>\r\n4. Perform each of the following conversions:\r\n<p style=\"padding-left: 30px;\">a.\u00a0 588.2\u00a0dg to Mg.<\/p>\r\n<p style=\"padding-left: 30px;\">b.\u00a0 200.0 km to fm.<\/p>\r\n<p style=\"padding-left: 30px;\">c.\u00a0 0.150\u00a0ms to Gs<\/p>\r\n<p style=\"padding-left: 30px;\">d.\u00a0\u00a03.0 \u00d7 10<sup class=\"superscript\">9<\/sup> nm to Tm<\/p>\r\n[reveal-answer q=\"983991\"]Show Selected Answers[\/reveal-answer]\r\n[hidden-answer a=\"983991\"]\r\n\r\n1. a.\u00a03.4 \u00d7 10<sup class=\"superscript\">9<\/sup> nm; b. 7.26\u00a0\u00d7 10<sup>-2<\/sup>\u00a0L ; c. 5.0 \u00d7 10<sup>2<\/sup>\u00a0mg; d.\u00a06.00 \u00d7 10<sup>-5<\/sup> Ms\r\n\r\n3.\u00a0a.\u00a01.2 \u00d7 10<sup class=\"superscript\">6<\/sup> pL; b. 2.000 \u00d7 10<sup>-12<\/sup>\u00a0Tm ; c. 1.2 \u00d7 10<sup>9<\/sup>\u00a0ks; d.\u00a06.74 \u00d7 10<sup>5<\/sup>\u00a0mL\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div>\n<div id=\"ball-ch02_s04\" class=\"section\" lang=\"en\">\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Be able to write prefix conversion factors.<\/li>\n<li>Use dimensional analysis to carry out prefix unit conversions.<\/li>\n<\/ul>\n<\/div>\n<p>Sometimes we use units that are fractions or multiples of a base unit. Ice cream is sold in quarts (a familiar, non-SI base unit), pints (0.5 quart), or gallons (4 quarts). We also use fractions or multiples of units in the SI system, but these fractions or multiples are always powers of 10. Fractional or multiple SI units are named using a prefix and the name of the base unit. For example, a length of 1000 meters is also called a kilometer because the prefix <em>kilo<\/em> means \u201cone thousand,\u201d which in scientific notation is 10<sup>3<\/sup> (1 kilometer = 1000 m = 10<sup>3<\/sup> m). The prefixes used and the powers to which 10 are raised are listed in Table 1.<\/p>\n<table style=\"width: 271px;\" summary=\"The prefix femto has the symbol lowercase f and a factor of 10 to the negative fifteenth power. Therefore, 1 femtosecond, F S, is equal to 1 times 10 to the negative 15 of a meter, or 0.000000000001 of a meter. The prefix pico has the symbol lowercase P and a factor of 10 to the negative twelfth power. Therefore, 1 picosecond, P S, is equal to 1 times 10 to the negative 12 of a meter, or 0.000000000001 of a meter. The prefix nano has the symbol lowercase N and a factor of 10 to the negative ninth power. Therefore, 4 nanograms, or NG, equals 4 times ten to the negative 9, or 0.000000004 g. The prefix micro has the greek letter mu as its symbol and a factor of 10 to the negative sixth power. Therefore, 1 microliter, or mu L, is equal to one times ten to the negative 6 or 0.000001 L. The prefix milli has a lowercase M as its symbol and a factor of 10 to the negative third power. Therefore, 2 millimoles, or M mol, are equal to two times ten to the negative 3 or 0.002 mol. The prefix centi has a lowercase C as its symbol and a factor of 10 to the negative second power. Therefore, 7 centimeters, or C M, are equal to seven times ten to the negative 2 meters or 0.07 M O L. The prefix deci has a lowercase D as its symbol and a factor of 10 to the negative first power. Therefore, 1 deciliter, or lowercase D uppercase L, are equal to one times ten to the negative 1 meters or 0.1 L. The prefix kilo has a lowercase K as its symbol and a factor of 10 to the third power. Therefore, 1 kilometer, or K M, is equal to one times ten to the third meters or 1000 M. The prefix mega has an uppercase M as its symbol and a factor of 10 to the sixth power. Therefore, 3 megahertz, or M H Z, are equal to three times 10 to the sixth hertz, or 3000000 H Z. The prefix giga has an uppercase G as its symbol and a factor of 10 to the ninth power. Therefore, 8 gigayears, or G Y R, are equal to eight times 10 to the ninth years, or 800000000 G Y R. The prefix tera has an uppercase T as its symbol and a factor of 10 to the twelfth power. Therefore, 5 terawatts, or T W, are equal to five times 10 to the twelfth watts, or 5000000000000 W.\">\n<thead>\n<tr>\n<th style=\"width: 271px;\" colspan=\"3\">Table 1. Common Unit Prefixes<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 84px; text-align: center;\">Prefix<\/th>\n<th style=\"width: 100px; text-align: center;\">Symbol<\/th>\n<th style=\"width: 87px; text-align: center;\">Factor<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td style=\"width: 84px; text-align: center;\">femto<\/td>\n<td style=\"width: 100px; text-align: center;\">f<\/td>\n<td style=\"width: 87px; text-align: center;\">10<sup>\u221215<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 84px; text-align: center;\">pico<\/td>\n<td style=\"width: 100px; text-align: center;\">p<\/td>\n<td style=\"width: 87px; text-align: center;\">10<sup>\u221212<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 84px; text-align: center;\">nano<\/td>\n<td style=\"width: 100px; text-align: center;\">n<\/td>\n<td style=\"width: 87px; text-align: center;\">10<sup>\u22129<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 84px; text-align: center;\">micro<\/td>\n<td style=\"width: 100px; text-align: center;\">\u00b5<\/td>\n<td style=\"width: 87px; text-align: center;\">10<sup>\u22126<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 84px; text-align: center;\">milli<\/td>\n<td style=\"width: 100px; text-align: center;\">m<\/td>\n<td style=\"width: 87px; text-align: center;\">10<sup>\u22123<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 84px; text-align: center;\">centi<\/td>\n<td style=\"width: 100px; text-align: center;\">c<\/td>\n<td style=\"width: 87px; text-align: center;\">10<sup>\u22122<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 84px; text-align: center;\">deci<\/td>\n<td style=\"width: 100px; text-align: center;\">d<\/td>\n<td style=\"width: 87px; text-align: center;\">10<sup>\u22121<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 84px; text-align: center;\">kilo<\/td>\n<td style=\"width: 100px; text-align: center;\">k<\/td>\n<td style=\"width: 87px; text-align: center;\">10<sup>3<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 84px; text-align: center;\">mega<\/td>\n<td style=\"width: 100px; text-align: center;\">M<\/td>\n<td style=\"width: 87px; text-align: center;\">10<sup>6<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 84px; text-align: center;\">giga<\/td>\n<td style=\"width: 100px; text-align: center;\">G<\/td>\n<td style=\"width: 87px; text-align: center;\">10<sup>9<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 84px; text-align: center;\">tera<\/td>\n<td style=\"width: 100px; text-align: center;\">T<\/td>\n<td style=\"width: 87px; text-align: center;\">10<sup>12<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Deriving Prefix Conversion Factors<\/h2>\n<\/div>\n<\/div>\n<p>Suppose we are asked to convert 50.0 mg of ibuprofen to grams.\u00a0 How do we go about this conversion? The first thing to recognize is that we are dealing with a prefix conversion problem and in order for us to successfully convert milligrams to grams, we need to know what exactly does the prefix milli- mean.\u00a0 From Table 1, we see milli- is 10<sup>-3<\/sup>. How do we express this as a conversion factor? The relationship between mg and g can be expressed two ways (shown as equalities):<\/p>\n<p style=\"text-align: center;\">[latex]\\large (1 \\text{ mg}=10^{-3}\\text{ g}) \\text{ or }(10^{3}\\text{ mg}=1\\text{ g})[\/latex]<\/p>\n<p>We use the equality to write the conversion factor needed to convert whatever unit we wish to convert, by showing the equality as a fraction, as shown below:<\/p>\n<p style=\"text-align: center;\">[latex]\\large (\\frac{10^{-3}{\\text{ g }}}{1\\text{ mg}})\\text{ or }(\\frac{1{\\text{ mg }}}{10^{-3}{\\text{ g }}})\\text{ ; }(\\frac{10^{3}{\\text{ mg }}}{1\\text{ g}})\\text{ or }(\\frac{1{\\text{ g }}}{10^{3}{\\text{ mg }}})[\/latex]<\/p>\n<div class=\"textbox examples\">\n<p>Note: When writing the conversion factor, if you place the 10 to the power (10<sup>x<\/sup>) opposite the prefix unit, the sign of the power stays exactly the same as presented in Table 1. However, if you place the 10 to the power (10<sup>x<\/sup>) with the prefix unit, the sign of the power needs to be switched.<\/p>\n<\/div>\n<h2>Prefix to Base Unit Conversions<\/h2>\n<p>Now that we know the relationship between grams and milligrams we use dimensional analysis to determine how many grams are in 50.0 mg of ibuprofen. In dimensional analysis, we multiply what is given by one or more conversion factors. Conversion factors are applied in a manner to which cancel units until you arrive at the desired unit.\u00a0 In this example, we start with 50.0 mg and use the conversion factor derived earlier to cancel the mg:<\/p>\n<p style=\"text-align: center;\">[latex]\\large 50.0 \\cancel{\\text{ mg}}\\times\\frac{10^{-3}{\\text{ g}}}{1\\cancel{\\text{ mg}}}= 5.00\\times 10^{-2}\\text{ g}[\/latex]<\/p>\n<p style=\"text-align: center;\">or<\/p>\n<p style=\"text-align: center;\">[latex]\\large 50.0 \\cancel{\\text{ mg}}\\times\\frac{1{\\text{ g}}}{10^{3}\\cancel{\\text{ mg}}}= 5.00\\times 10^{-2}\\text{ g}[\/latex]<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1: <strong>Prefix to base Unit Conversions<br \/>\n<\/strong><\/h3>\n<p>How many femtoliters are in [latex]\\large 2.00\\times 10^{2}[\/latex] liters?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q79228735\">Show Answer<\/span><\/p>\n<div id=\"q79228735\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we look up what a femto means, 10<sup>-15<\/sup>.\u00a0 The conversion can be written as 1 fm = 10<sup>-15<\/sup> L.<\/p>\n<p>[latex]2.00\\times 10^{2} \\cancel{\\text{ L}}\\times\\frac{1{\\text{ fL}}}{10^{-15}\\cancel{\\text{ L}}}= 2.00\\times 10^{17}\\text{ fL}[\/latex]<\/p>\n<\/div>\n<\/div>\n<h4><strong>Check Your Learning<\/strong><\/h4>\n<p>How many seconds are in 5.0 gigaseconds (Gs)?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q79871236\">Show Answer<\/span><\/p>\n<div id=\"q79871236\" class=\"hidden-answer\" style=\"display: none\">\n<p>5.0 \u00d7 10<sup class=\"superscript\">9<\/sup> s<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Prefix to Prefix Conversions<\/h2>\n<p id=\"ball-ch02_s04_p33\" class=\"para editable block\">Sometimes there will be a need to convert from one unit with one numerical prefix to another unit with a different numerical prefix. How do we handle those conversions? Well, you could memorize the conversion factors that interrelate all numerical prefixes. Or you can go the easier route: first convert the quantity to the base unit, the unit with no numerical prefix, using the definition of the original prefix. Then convert the quantity in the base unit to the desired unit using the definition of the second prefix. You can do the conversion in two separate steps or as one long algebraic step. For example, to convert 2.77 kg to milligrams:<\/p>\n<p>First convert to the base unit of grams:<\/p>\n<p style=\"text-align: center;\">[latex]\\large 2.77 \\cancel{\\text{ kg}}\\times\\frac{10^{3}{\\text{ g}}}{1\\cancel{\\text{ kg}}}= 2.77\\times 10^{3} \\text{ g}[\/latex]<\/p>\n<p>Then convert the base unit to milligrams:<\/p>\n<p style=\"text-align: center;\">[latex]\\large 2.77\\times 10^{3}\\cancel{\\text{ g}}\\times\\frac{1{\\text{ mg}}}{10^{-3}\\cancel{\\text{ g}}}= 2.77\\times 10^{6} \\text{ mg}[\/latex]<\/p>\n<p>Alternatively, it can be done in a single multistep process:<\/p>\n<p style=\"text-align: center;\">[latex]\\large 2.77 \\cancel{\\text{ kg}}\\times\\frac{10^{3}\\cancel{\\text{ g}}}{1\\cancel{\\text{ kg}}}\\times\\frac{10^{3}{\\text{ mg}}}{1\\cancel{\\text{ g}}}= 2.77\\times 10^{6} \\text{ mg}[\/latex]<\/p>\n<p id=\"ball-ch02_s04_p35\" class=\"para editable block\">You get the same answer either way.<\/p>\n<p>You can also combine the two conversion factors into one, and convert directly from one prefix to another prefix:<\/p>\n<p style=\"text-align: center;\">[latex]\\large 2.77 \\cancel{\\text{ kg}}\\times\\frac{10^{3}\\cancel{\\text{ mg}}}{10^{-3}\\cancel{\\text{ kg}}}= 2.77\\times 10^{6} \\text{ mg}[\/latex]<\/p>\n<div class=\"textbox examples\">\n<p>Note: When writing the prefix to prefix conversion factor, you place the 10 to the power (10<sup>x<\/sup>) opposite the prefix unit. In this case we have prefixes in the numerator and the denominator, so we end up with a 10<sup>x<\/sup> in both spots, with the 10<sup>3<\/sup> going opposite of the kilo prefix and the 10<sup>-3<\/sup> going opposite the milli prefix.<\/p>\n<\/div>\n<p id=\"ball-ch02_s04_p40\" class=\"para editable block\">When considering the significant figures of a final numerical answer in a conversion, there is one important case where a number does not impact the number of significant figures in a final answer\u2014the so-called <span class=\"margin_term\"><a class=\"glossterm\">exact number<\/a><\/span>. An exact number is a number from a defined relationship, not a measured one. For example, the prefix <em class=\"emphasis\">kilo-<\/em> means 1,000\u2014<em class=\"emphasis\">exactly<\/em> 1,000, no more or no less. Thus, in constructing the conversion factor<\/p>\n<p style=\"text-align: center;\">[latex]\\large \\frac{1000{\\text{ g}}}{\\text{ kg}}[\/latex]<\/p>\n<p id=\"ball-ch02_s04_p41\" class=\"para editable block\">neither the 1,000 nor the 1 enter into our consideration of significant figures. The numbers in the numerator and denominator are defined exactly by what the prefix <em class=\"emphasis\">kilo-<\/em> means. Another way of thinking about it is that these numbers can be thought of as having an infinite number of significant figures, such as<\/p>\n<p style=\"text-align: center;\">[latex]\\large \\frac{1000.0000000000 . . .{\\text{ g}}}{1.0000000000 . . .\\text{ kg}}[\/latex]<\/p>\n<p id=\"ball-ch02_s04_p42\" class=\"para editable block\">The other numbers in the calculation will determine the number of significant figures in the final answer.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 2: <strong>Prefix to Prefix Conversion<br \/>\n<\/strong><\/h3>\n<p>How many nanoseconds are in 368.09 \u03bcs?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q798735\">Show Answer<\/span><\/p>\n<div id=\"q798735\" class=\"hidden-answer\" style=\"display: none\">\n<p>You can either do this as a one-step conversion from microseconds to nanoseconds or convert to the base unit first and then to the final desired unit. We will use the second method here, showing the two steps in a single line. Using the definitions of the prefixes <em class=\"emphasis\">micro-<\/em> and <em class=\"emphasis\">nano-<\/em>,<\/p>\n<p>[latex]368.09 \\cancel{{\\text{ \u03bcs}}}\\times\\frac{10^{-6}\\cancel{\\text{ s}}}{1\\cancel{\\text{ \u03bcs}}}\\times\\frac{1\\text{ ns}}{10^{-9}\\cancel{\\text{ s}}}= 3.6809\\times 10^{5} \\text{ ns}[\/latex]<\/p>\n<p>or solve using one conversion directly between the prefix units:<\/p>\n<p>[latex]368.09 \\cancel{{\\text{ \u03bcs}}}\\times\\frac{10^{-6}\\text{ ns}}{10^{-9}\\cancel{\\text{ \u03bcs}}}= 3.6809\\times 10^{5} \\text{ ns}[\/latex]<\/p>\n<\/div>\n<\/div>\n<h4><strong>Check Your Learning<\/strong><\/h4>\n<p>How many milliliters are in 607.8 kL?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q798736\">Show Answer<\/span><\/p>\n<div id=\"q798736\" class=\"hidden-answer\" style=\"display: none\">\n<p>6.078 \u00d7 10<sup class=\"superscript\">8<\/sup> mL<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"ball-ch02_s04_n07\" class=\"callout block\"><\/div>\n<div>\n<div id=\"ball-ch02_s04\" class=\"section\" lang=\"en\">\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p>1. Perform each of the following conversions:<\/p>\n<p style=\"padding-left: 30px;\">a.\u00a0 3.4 m to nm<\/p>\n<p style=\"padding-left: 30px;\">b.\u00a0 72.6 mL to L<\/p>\n<p style=\"padding-left: 30px;\">c.\u00a0 0.50 g to mg<\/p>\n<p style=\"padding-left: 30px;\">d.\u00a0 60.0 s to Ms<\/p>\n<p>2. Perform each of the following conversions:<\/p>\n<p style=\"padding-left: 30px;\">a.\u00a0 5.83 km to m<\/p>\n<p style=\"padding-left: 30px;\">b.\u00a0 5 \u00d7 10<sup>\u22124<\/sup> L to cL<\/p>\n<p style=\"padding-left: 30px;\">c.\u00a0 10.0 kg to g<\/p>\n<p style=\"padding-left: 30px;\">d.\u00a0 3.6 \u00d7 10<sup>3<\/sup> s to ms<\/p>\n<p>3. Perform each of the following conversions:<\/p>\n<p style=\"padding-left: 30px;\">a.\u00a0 1.2 \u00d7 10<sup>\u22123<\/sup>\u00a0mL to pL.<\/p>\n<p style=\"padding-left: 30px;\">b.\u00a0 200.0 cm to Tm.<\/p>\n<p style=\"padding-left: 30px;\">c.\u00a0 1.2 \u00d7 10<sup class=\"superscript\">6<\/sup> Ms to ks<\/p>\n<p style=\"padding-left: 30px;\">d.\u00a0 0.674 kL to mL<\/p>\n<p>4. Perform each of the following conversions:<\/p>\n<p style=\"padding-left: 30px;\">a.\u00a0 588.2\u00a0dg to Mg.<\/p>\n<p style=\"padding-left: 30px;\">b.\u00a0 200.0 km to fm.<\/p>\n<p style=\"padding-left: 30px;\">c.\u00a0 0.150\u00a0ms to Gs<\/p>\n<p style=\"padding-left: 30px;\">d.\u00a0\u00a03.0 \u00d7 10<sup class=\"superscript\">9<\/sup> nm to Tm<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q983991\">Show Selected Answers<\/span><\/p>\n<div id=\"q983991\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. a.\u00a03.4 \u00d7 10<sup class=\"superscript\">9<\/sup> nm; b. 7.26\u00a0\u00d7 10<sup>-2<\/sup>\u00a0L ; c. 5.0 \u00d7 10<sup>2<\/sup>\u00a0mg; d.\u00a06.00 \u00d7 10<sup>-5<\/sup> Ms<\/p>\n<p>3.\u00a0a.\u00a01.2 \u00d7 10<sup class=\"superscript\">6<\/sup> pL; b. 2.000 \u00d7 10<sup>-12<\/sup>\u00a0Tm ; c. 1.2 \u00d7 10<sup>9<\/sup>\u00a0ks; d.\u00a06.74 \u00d7 10<sup>5<\/sup>\u00a0mL<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":6181,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1744","chapter","type-chapter","status-web-only","hentry"],"part":36,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/1744","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/wp\/v2\/users\/6181"}],"version-history":[{"count":34,"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/1744\/revisions"}],"predecessor-version":[{"id":1972,"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/1744\/revisions\/1972"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/pressbooks\/v2\/parts\/36"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/1744\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/wp\/v2\/media?parent=1744"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/pressbooks\/v2\/chapter-type?post=1744"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/wp\/v2\/contributor?post=1744"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/wp\/v2\/license?post=1744"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}