{"id":7303,"date":"2021-06-08T21:55:36","date_gmt":"2021-06-08T21:55:36","guid":{"rendered":"https:\/\/opentextbc.ca\/introductorychemistry\/chapter\/expressing-units\/"},"modified":"2021-09-21T22:40:50","modified_gmt":"2021-09-21T22:40:50","slug":"expressing-units","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/introductorychemistry\/chapter\/expressing-units\/","title":{"raw":"Expressing Units","rendered":"Expressing Units"},"content":{"raw":"[latexpage]\r\n
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Learning Objectives<\/p>\r\n\r\n<\/header>\r\n

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  1. Learn the units that go with various quantities.<\/li>\r\n \t
  2. Express units using their abbreviations.<\/li>\r\n \t
  3. Make new units by combining numerical prefixes with units.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nA number indicates \u201chow much,\u201d but the unit indicates \u201cof what.\u201d The \u201cof what\u201d is important when communicating a quantity. For example, if you were to ask a friend how close you are to Lake Erie and your friend says \u201csix,\u201d then your friend isn\u2019t giving you complete information. Six what<\/em>? Six miles? Six inches? Six city blocks? The actual distance to the lake depends on what units you use.\r\n\r\nChemistry, like most sciences, uses the International System of Units, or SI for short. (The letters SI<\/em> stand for the French \u201cle Syst\u00e8me International d\u2019unit\u00e9s.\u201d) SI specifies certain units for various types of quantities, based on seven [pb_glossary id=\"8136\"]fundamental units[\/pb_glossary]\u00a0for various quantities. We will use most of the fundamental units in chemistry. Initially, we will deal with three fundamental units. The metre (m) is the SI unit of length. It is a little longer than a yard (see Figure 2.4 \"The Metre\"). The SI unit of mass is the kilogram (kg), which is about 2.2 pounds (lb). The SI unit of time is the second (s).\r\n\r\n[caption id=\"attachment_7300\" align=\"aligncenter\" width=\"300\"]\"1 Figure 2.4 \"The Metre.\" The SI standard unit of length, the metre, is a little longer than a yard.[\/caption]\r\n\r\nTo express a quantity, you need to combine a number with a unit. If you have a length that is 2.4 m, then you express that length as simply 2.4 m. A time of 15,000 s can be expressed as 1.5 \u00d7 104<\/sup> s in scientific notation.\r\n\r\nSometimes, a given unit is not an appropriate size to easily express a quantity. For example, the width of a human hair is very small, and it doesn\u2019t make much sense to express it in metres. SI also defines a series of [pb_glossary id=\"8137\"]numerical prefixes[\/pb_glossary]\u00a0that refer to multiples or fractions of a fundamental unit to make a unit more conveniently sized for a specific quantity. Table 2.4 \"Multiplicative Prefixes for SI Units\" lists the prefixes, their abbreviations, and their multiplicative factors. Some of the prefixes, such as kilo-, mega-, and giga-, represent more than one of the fundamental unit, while other prefixes, such as centi-, milli-, and micro-, represent fractions of the original unit. Note, too, that once again we are using powers of 10. Each prefix is a multiple of or fraction of a power of 10.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Table 2.4 Multiplicative Prefixes for SI Units<\/caption>\r\n
    Prefix<\/th>\r\nAbbreviation<\/th>\r\nMultiplicative Amount<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    giga-<\/td>\r\nG<\/td>\r\n1,000,000,000 \u00d7<\/td>\r\n<\/tr>\r\n
    mega-<\/td>\r\nM<\/td>\r\n1,000,000 \u00d7<\/td>\r\n<\/tr>\r\n
    kilo-<\/td>\r\nk<\/td>\r\n1,000 \u00d7<\/td>\r\n<\/tr>\r\n
    deci-<\/td>\r\nd<\/td>\r\n[latex]\\dfrac{1}{10}\\times[\/latex]<\/td>\r\n<\/tr>\r\n
    centi-<\/td>\r\nc<\/td>\r\n[latex]\\dfrac{1}{100}\\times[\/latex]<\/td>\r\n<\/tr>\r\n
    milli-<\/td>\r\nm<\/td>\r\n[latex]\\dfrac{1}{1000}\\times[\/latex]<\/td>\r\n<\/tr>\r\n
    micro-<\/td>\r\n\u03bc[footnote]The letter \u03bc is the Greek letter lowercase equivalent to an m and is called \u201cmu\u201d (pronounced \u201cmyoo\u201d).[\/footnote]<\/td>\r\n[latex]\\dfrac{1}{1,000,000}\\times[\/latex]<\/td>\r\n<\/tr>\r\n
    nano-<\/td>\r\nn<\/td>\r\n[latex]\\dfrac{1}{1,000,000,000}\\times[\/latex]<\/td>\r\n<\/tr>\r\n
    pico-<\/td>\r\np<\/td>\r\n[latex]\\dfrac{1}{1,000,000,000,000}\\times[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo use the fractions to generate new units, simply combine the prefix with the unit itself; the abbreviation for the new unit is the combination of the abbreviation for the prefix and the abbreviation of the unit. For example, the kilometre (km) is 1,000 \u00d7 metre, or 1,000 m. Thus, 5 kilometres (5 km) is equal to 5,000 m. Similarly, a millisecond (ms) is [latex]\\dfrac{1}{1,000}[\/latex] \u00d7 second, or one one-thousandth of a second. Thus, 25 ms is 25 thousandths of a second. You will need to become proficient in combining prefixes and units. (You may recognize that one of our fundamental units, the kilogram, automatically has a prefix-unit combination, the kilogram. The word kilogram<\/em> means 1,000 g.)\r\n\r\nIn addition to the fundamental units, SI also allows for [pb_glossary id=\"8138\"]derived units[\/pb_glossary]\u00a0based on a fundamental unit or units. There are many derived units used in science. For example, the derived unit for area comes from the idea that area is defined as width times height. Because both width and height are lengths, they both have the fundamental unit of metre, so the unit of area is metre \u00d7 metre, or metre2<\/sup> (m2<\/sup>). This is sometimes spoken as \u201csquare metres.\u201d A unit with a prefix can also be used to derive a unit for area, so we can also have cm2<\/sup>, mm2<\/sup>, or km2<\/sup> as acceptable units for area.\r\n\r\nVolume is defined as length times width times height, so it has units of metre \u00d7 metre \u00d7 metre or metre3<\/sup> (m3<\/sup>), sometimes spoken as \u201ccubic metres.\u201d The cubic metre is a rather large unit, however, so another unit is defined that is somewhat more manageable: the litre (L). A litre is one one-thousandth of a cubic metre and is a little more than 1 quart in volume (see Figure 2.5 \"The Litre\"). Prefixes can also be used with the litre unit, so we can speak of millilitres (one one-thousandth of a litre; mL) and kilolitres (1,000 L; kL).\r\n\r\n[caption id=\"attachment_7301\" align=\"aligncenter\" width=\"300\"]\"1 Figure 2.5 \"The Litre.\" The SI unit of volume, the litre, is slightly larger than 1 quart.[\/caption]\r\n\r\nAnother definition of a litre is one-tenth of a metre cubed. Because one-tenth of a metre is 10 cm, then a litre is equal to 1,000 cm3<\/sup> (Figure 2.6 \"The Size of 1 Litre\"). Because 1 L equals 1,000 mL, we conclude that 1 mL equals 1 cm3<\/sup>; thus, these units are interchangeable.\r\n\r\n[caption id=\"attachment_7302\" align=\"aligncenter\" width=\"300\"]\"A Figure 2.6 \"The Size of 1 Litre.\" One litre equals 1,000 cm3<\/sup>, so 1 cm3<\/sup> is the same as 1 mL.[\/caption]\r\n\r\nUnits not only are multiplied together but also can be divided. For example, if you are travelling at one metre for every second of time elapsed, your velocity is 1 metre per second, or 1 m\/s. The word per<\/em> implies division, so velocity is determined by dividing a distance quantity by a time quantity. Other units for velocity include kilometres per hour (km\/h) or even micrometres per nanosecond (\u03bcm\/ns). Later, we will see other derived units that can be expressed as fractions.\r\n
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    Example 2.15<\/p>\r\n\r\n<\/header>\r\n

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    Problems<\/h1>\r\n
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    1. A human hair has a diameter of about 6.0 \u00d7 10\u22125<\/sup> m. Suggest an appropriate unit for this measurement and write the diameter of a human hair in terms of that unit.<\/li>\r\n \t
    2. What is the velocity of a car if it goes 25 m in 5.0 s?<\/li>\r\n<\/ol>\r\n

      Solutions<\/h2>\r\n
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      1. The scientific notation 10\u22125<\/sup> is close to 10\u22126<\/sup>, which defines the micro- prefix. Let us use micrometres as the unit for hair diameter. The number 6.0 \u00d7 10\u22125<\/sup> can be written as 60 \u00d7 10\u22126<\/sup>, and a micrometre is 10\u22126<\/sup> m, so the diameter of a human hair is about 60 \u03bcm.<\/li>\r\n \t
      2. If velocity is defined as a distance quantity divided by a time quantity, then velocity is 25 metres\/5.0 seconds. Dividing the numbers gives us 25 \u00f7 5.0 = 5.0, and dividing the units gives us metres\/second, or m\/s. The velocity is 5.0 m\/s.<\/li>\r\n<\/ol>\r\n

        Test Yourself<\/h1>\r\n
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        1. Express the volume of an Olympic-sized swimming pool, 2,500,000 L, in more appropriate units.<\/li>\r\n \t
        2. A common garden snail moves about 6.1 m in 30 min. What is its velocity in metres per minute (m\/min)?<\/li>\r\n<\/ol>\r\n

          Answers<\/h2>\r\n
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          1. 2.5 ML<\/li>\r\n \t
          2. 0.203 m\/min<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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            Key Takeaways<\/p>\r\n\r\n<\/header>\r\n

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