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  • Chapter 2Measurement

    2.1Measurements*

    By the end of this section, you will be able to:

    • Explain the process of measurement

    • Identify the three basic parts of a quantity

    • Describe the properties and units of length, mass, volume, density, temperature, and time

    • Perform basic unit calculations and conversions in the metric and other unit systems

    Measurements provide the macroscopic information that is the basis of most of the hypotheses, theories, and laws that describe the behavior of matter and energy in both the macroscopic and microscopic domains of chemistry. Every measurement provides three kinds of information: the size or magnitude of the measurement (a number); a standard of comparison for the measurement (a unit); and an indication of the uncertainty of the measurement. While the number and unit are explicitly represented when a quantity is written, the uncertainty is an aspect of the measurement result that is more implicitly represented and will be discussed later.

    The number in the measurement can be represented in different ways, including decimal form and scientific notation. (Scientific notation is also known as exponential notation; a review of this topic can be found in Appendix B.) For example, the maximum takeoff weight of a Boeing 777-200ER airliner is 298,000 kilograms, which can also be written as 2.98 × 105 kg. The mass of the average mosquito is about 0.0000025 kilograms, which can be written as 2.5 × 10−6 kg.

    Units, such as liters, pounds, and centimeters, are standards of comparison for measurements. When we buy a 2-liter bottle of a soft drink, we expect that the volume of the drink was measured, so it is two times larger than the volume that everyone agrees to be 1 liter. The meat used to prepare a 0.25-pound hamburger is measured so it weighs one-fourth as much as 1 pound. Without units, a number can be meaningless, confusing, or possibly life threatening. Suppose a doctor prescribes phenobarbital to control a patient’s seizures and states a dosage of “100” without specifying units. Not only will this be confusing to the medical professional giving the dose, but the consequences can be dire: 100 mg given three times per day can be effective as an anticonvulsant, but a single dose of 100 g is more than 10 times the lethal amount.

    We usually report the results of scientific measurements in SI units, an updated version of the metric system, using the units listed in Table 2.1. Other units can be derived from these base units. The standards for these units are fixed by international agreement, and they are called the International System of Units or SI Units (from the French, Le Système International d’Unités). SI units have been used by the United States National Institute of Standards and Technology (NIST) since 1964.

    Table 2.1.
    Base Units of the SI System
    Property MeasuredName of UnitSymbol of Unit
    lengthmeterm
    masskilogramkg
    timeseconds
    temperaturekelvinK
    electric currentampereA
    amount of substancemolemol
    luminous intensitycandelacd

    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 kilo means “one thousand,” which in scientific notation is 103 (1 kilometer = 1000 m = 103 m). The prefixes used and the powers to which 10 are raised are listed in Table 2.2.

    Table 2.2.
    Common Unit Prefixes
    PrefixSymbolFactorExample
    femtof10−15 1 femtosecond (fs) = 1 × 10−15 s (0.000000000000001 s)
    picop10−12 1 picometer (pm) = 1 × 10−12 m (0.000000000001 m)
    nanon10−9 4 nanograms (ng) = 4 × 10−9 g (0.000000004 g)
    microµ10−6 1 microliter (μL) = 1 × 10−6 L (0.000001 L)
    millim10−3 2 millimoles (mmol) = 2 × 10−3 mol (0.002 mol)
    centic10−2 7 centimeters (cm) = 7 × 10−2 m (0.07 m)
    decid10−11 deciliter (dL) = 1 × 10−1 L (0.1 L )
    kilok1031 kilometer (km) = 1 × 103 m (1000 m)
    megaM1063 megahertz (MHz) = 3 × 106 Hz (3,000,000 Hz)
    gigaG1098 gigayears (Gyr) = 8 × 109 yr (8,000,000,000 Gyr)
    teraT10125 terawatts (TW) = 5 × 1012 W (5,000,000,000,000 W)

    SI Base Units

    The initial units of the metric system, which eventually evolved into the SI system, were established in France during the French Revolution. The original standards for the meter and the kilogram were adopted there in 1799 and eventually by other countries. This section introduces four of the SI base units commonly used in chemistry. Other SI units will be introduced in subsequent chapters.

    Length

    The standard unit of length in both the SI and original metric systems is the meter (m). A meter was originally specified as 1/10,000,000 of the distance from the North Pole to the equator. It is now defined as the distance light in a vacuum travels in 1/299,792,458 of a second. A meter is about 3 inches longer than a yard (Figure 2.1); one meter is about 39.37 inches or 1.094 yards. Longer distances are often reported in kilometers (1 km = 1000 m = 103 m), whereas shorter distances can be reported in centimeters (1 cm = 0.01 m = 10−2 m) or millimeters (1 mm = 0.001 m = 10−3 m).

    CNX_Chem_01_04_MYdCmIn.jpg
    Figure 2.1
    The relative lengths of 1 m, 1 yd, 1 cm, and 1 in. are shown (not actual size), as well as comparisons of 2.54 cm and 1 in., and of 1 m and 1.094 yd.

    Mass

    The standard unit of mass in the SI system is the kilogram (kg). A kilogram was originally defined as the mass of a liter of water (a cube of water with an edge length of exactly 0.1 meter). It is now defined by a certain cylinder of platinum-iridium alloy, which is kept in France (Figure 2.2). Any object with the same mass as this cylinder is said to have a mass of 1 kilogram. One kilogram is about 2.2 pounds. The gram (g) is exactly equal to 1/1000 of the mass of the kilogram (10−3 kg).

    CNX_Chem_01_04_Kilogram.jpg
    Figure 2.2
    This replica prototype kilogram is housed at the National Institute of Standards and Technology (NIST) in Maryland. (credit: National Institutes of Standards and Technology)

    Temperature

    Temperature is an intensive property. The SI unit of temperature is the kelvin (K). The IUPAC convention is to use kelvin (all lowercase) for the word, K (uppercase) for the unit symbol, and neither the word “degree” nor the degree symbol (°). The degree Celsius (°C) is also allowed in the SI system, with both the word “degree” and the degree symbol used for Celsius measurements. Celsius degrees are the same magnitude as those of kelvin, but the two scales place their zeros in different places. Water freezes at 273.15 K (0 °C) and boils at 373.15 K (100 °C) by definition, and normal human body temperature is approximately 310 K (37 °C). The conversion between these two units and the Fahrenheit scale will be discussed later in this chapter.

    Time

    The SI base unit of time is the second (s). Small and large time intervals can be expressed with the appropriate prefixes; for example, 3 microseconds = 0.000003 s = 3 × 10−6 and 5 megaseconds = 5,000,000 s = 5 × 106 s. Alternatively, hours, days, and years can be used.

    Derived SI Units

    We can derive many units from the seven SI base units. For example, we can use the base unit of length to define a unit of volume, and the base units of mass and length to define a unit of density.

    Volume

    Volume is the measure of the amount of space occupied by an object. The standard SI unit of volume is defined by the base unit of length (Figure 2.3). The standard volume is a cubic meter (m3), a cube with an edge length of exactly one meter. To dispense a cubic meter of water, we could build a cubic box with edge lengths of exactly one meter. This box would hold a cubic meter of water or any other substance.

    A more commonly used unit of volume is derived from the decimeter (0.1 m, or 10 cm). A cube with edge lengths of exactly one decimeter contains a volume of one cubic decimeter (dm3). A liter (L) is the more common name for the cubic decimeter. One liter is about 1.06 quarts.

    A cubic centimeter (cm3) is the volume of a cube with an edge length of exactly one centimeter. The abbreviation cc (for cubic centimeter) is often used by health professionals. A cubic centimeter is also called a milliliter (mL) and is 1/1000 of a liter.

    CNX_Chem_01_04_Volume.jpg
    Figure 2.3
    (a) The relative volumes are shown for cubes of 1 m3, 1 dm3 (1 L), and 1 cm3 (1 mL) (not to scale). (b) The diameter of a dime is compared relative to the edge length of a 1-cm3 (1-mL) cube.

    Density

    We use the mass and volume of a substance to determine its density. Thus, the units of density are defined by the base units of mass and length.

    The density of a substance is the ratio of the mass of a sample of the substance to its volume. The SI unit for density is the kilogram per cubic meter (kg/m3). For many situations, however, this as an inconvenient unit, and we often use grams per cubic centimeter (g/cm3) for the densities of solids and liquids, and grams per liter (g/L) for gases. Although there are exceptions, most liquids and solids have densities that range from about 0.7 g/cm3 (the density of gasoline) to 19 g/cm3 (the density of gold). The density of air is about 1.2 g/L. Table 2.3 shows the densities of some common substances.

    Table 2.3.
    Densities of Common Substances
    SolidsLiquidsGases (at 25 °C and 1 atm)
    ice (at 0 °C) 0.92 g/cm3water 1.0 g/cm3dry air 1.20 g/L
    oak (wood) 0.60–0.90 g/cm3ethanol 0.79 g/cm3oxygen 1.31 g/L
    iron 7.9 g/cm3acetone 0.79 g/cm3nitrogen 1.14 g/L
    copper 9.0 g/cm3glycerin 1.26 g/cm3carbon dioxide 1.80 g/L
    lead 11.3 g/cm3olive oil 0.92 g/cm3helium 0.16 g/L
    silver 10.5 g/cm3gasoline 0.70–0.77 g/cm3neon 0.83 g/L
    gold 19.3 g/cm3mercury 13.6 g/cm3radon 9.1 g/L

    While there are many ways to determine the density of an object, perhaps the most straightforward method involves separately finding the mass and volume of the object, and then dividing the mass of the sample by its volume. In the following example, the mass is found directly by weighing, but the volume is found indirectly through length measurements.

    (2.1)
    _autogen-svg2png-0016.png
    Example 2.1

    Calculation of Density

    Gold—in bricks, bars, and coins—has been a form of currency for centuries. In order to swindle people into paying for a brick of gold without actually investing in a brick of gold, people have considered filling the centers of hollow gold bricks with lead to fool buyers into thinking that the entire brick is gold. It does not work: Lead is a dense substance, but its density is not as great as that of gold, 19.3 g/cm3. What is the density of lead if a cube of lead has an edge length of 2.00 cm and a mass of 90.7 g?

    Solution

    The density of a substance can be calculated by dividing its mass by its volume. The volume of a cube is calculated by cubing the edge length.

    (2.2)
    _autogen-svg2png-0017.png
    (2.3)
    _autogen-svg2png-0018.png

    (We will discuss the reason for rounding to the first decimal place in the next section.)

    Check Your Learning

    (a) To three decimal places, what is the volume of a cube (cm3) with an edge length of 0.843 cm?

    (b) If the cube in part (a) is copper and has a mass of 5.34 g, what is the density of copper to two decimal places?

    Answer:

    (a) 0.599 cm3; (b) 8.91 g/cm3

    Example 2.2

    Using Displacement of Water to Determine Density

    This PhET simulation illustrates another way to determine density, using displacement of water. Determine the density of the red and yellow blocks.

    Solution

    When you open the density simulation and select Same Mass, you can choose from several 5.00-kg colored blocks that you can drop into a tank containing 100.00 L water. The yellow block floats (it is less dense than water), and the water level rises to 105.00 L. While floating, the yellow block displaces 5.00 L water, an amount equal to the weight of the block. The red block sinks (it is more dense than water, which has density = 1.00 kg/L), and the water level rises to 101.25 L.

    The red block therefore displaces 1.25 L water, an amount equal to the volume of the block. The density of the red block is:

    (2.4)
    _autogen-svg2png-0019.png

    Note that since the yellow block is not completely submerged, you cannot determine its density from this information. But if you hold the yellow block on the bottom of the tank, the water level rises to 110.00 L, which means that it now displaces 10.00 L water, and its density can be found:

    (2.5)
    _autogen-svg2png-0020.png

    Check Your Learning

    Remove all of the blocks from the water and add the green block to the tank of water, placing it approximately in the middle of the tank. Determine the density of the green block.

    Answer:

    2.00 kg/L

    Key Concepts and Summary

    Measurements provide quantitative information that is critical in studying and practicing chemistry. Each measurement has an amount, a unit for comparison, and an uncertainty. Measurements can be represented in either decimal or scientific notation. Scientists primarily use the SI (International System) or metric systems. We use base SI units such as meters, seconds, and kilograms, as well as derived units, such as liters (for volume) and g/cm3 (for density). In many cases, we find it convenient to use unit prefixes that yield fractional and multiple units, such as microseconds (10−6 seconds) and megahertz (106 hertz), respectively.

    Key Equations

    • _autogen-svg2png-0021.png

    Chemistry End of Chapter Exercises

    Exercise 1.

    Is one liter about an ounce, a pint, a quart, or a gallon?

    Is a meter about an inch, a foot, a yard, or a mile?

    about a yard
    Exercise 3.

    Indicate the SI base units or derived units that are appropriate for the following measurements:

    (a) the length of a marathon race (26 miles 385 yards)

    (b) the mass of an automobile

    (c) the volume of a swimming pool

    (d) the speed of an airplane

    (e) the density of gold

    (f) the area of a football field

    (g) the maximum temperature at the South Pole on April 1, 1913

    Indicate the SI base units or derived units that are appropriate for the following measurements:

    (a) the mass of the moon

    (b) the distance from Dallas to Oklahoma City

    (c) the speed of sound

    (d) the density of air

    (e) the temperature at which alcohol boils

    (f) the area of the state of Delaware

    (g) the volume of a flu shot or a measles vaccination

    (a) kilograms; (b) meters; (c) kilometers/second; (d) kilograms/cubic meter; (e) kelvin; (f) square meters; (g) cubic meters
    Exercise 5.

    Give the name and symbol of the prefixes used with SI units to indicate multiplication by the following exact quantities.

    (a) 103

    (b) 10−2

    (c) 0.1

    (d) 10−3

    (e) 1,000,000

    (f) 0.000001

    Give the name of the prefix and the quantity indicated by the following symbols that are used with SI base units.

    (a) c

    (b) d

    (c) G

    (d) k

    (e) m

    (f) n

    (g) p

    (h) T

    (a) centi-, × 10−2; (b) deci-, × 10−1; (c) Giga-, × 109; (d) kilo-, × 103; (e) milli-, × 10−3; (f) nano-, × 10−9; (g) pico-, × 10−12; (h) tera-, × 1012
    Exercise 7.

    A large piece of jewelry has a mass of 132.6 g. A graduated cylinder initially contains 48.6 mL water. When the jewelry is submerged in the graduated cylinder, the total volume increases to 61.2 mL.

    (a) Determine the density of this piece of jewelry.

    (b) Assuming that the jewelry is made from only one substance, what substance is it likely to be? Explain.

    Visit this PhET density simulation and select the Same Volume Blocks.

    (a) What are the mass, volume, and density of the yellow block?

    (b) What are the mass, volume and density of the red block?

    (c) List the block colors in order from smallest to largest mass.

    (d) List the block colors in order from lowest to highest density.

    (e) How are mass and density related for blocks of the same volume?

    (a) 8.00 kg, 5.00 L, 1.60 kg/L; (b) 2.00 kg, 5.00 L, 0.400 kg/L; (c) red < green < blue < yellow; (d) If the volumes are the same, then the density is directly proportional to the mass.
    Exercise 9.

    Visit this PhET density simulation and select Custom Blocks and then My Block.

    (a) Enter mass and volume values for the block such that the mass in kg is less than the volume in L. What does the block do? Why? Is this always the case when mass < volume?

    (b) Enter mass and volume values for the block such that the mass in kg is more than the volume in L. What does the block do? Why? Is this always the case when mass > volume?

    (c) How would (a) and (b) be different if the liquid in the tank were ethanol instead of water?

    (d) How would (a) and (b) be different if the liquid in the tank were mercury instead of water?

    Visit this PhET density simulation and select Mystery Blocks.

    (a) Pick one of the Mystery Blocks and determine its mass, volume, density, and its likely identity.

    (b) Pick a different Mystery Block and determine its mass, volume, density, and its likely identity.

    (c) Order the Mystery Blocks from least dense to most dense. Explain.

    (a) (b) Answer is one of the following. A/yellow: mass = 65.14 kg, volume = 3.38 L, density = 19.3 kg/L, likely identity = gold. B/blue: mass = 0.64 kg, volume = 1.00 L, density = 0.64 kg/L, likely identity = apple. C/green: mass = 4.08 kg, volume = 5.83 L, density = 0.700 kg/L, likely identity = gasoline. D/red: mass = 3.10 kg, volume = 3.38 L, density = 0.920 kg/L, likely identity = ice; and E/purple: mass = 3.53 kg, volume = 1.00 L, density = 3.53 kg/L, likely identity = diamond. (c) B/blue/apple (0.64 kg/L) < C/green/gasoline (0.700 kg/L) < D/red/ice (0.920 kg/L) < E/purple/diamond (3.53 kg/L) < A/yellow/gold (19.3 kg/L)

    2.2Mathematical Treatment of Measurement Results*

    By the end of this section, you will be able to:

    • Explain the dimensional analysis (factor label) approach to mathematical calculations involving quantities

    • Use dimensional analysis to carry out unit conversions for a given property and computations involving two or more properties

    It is often the case that a quantity of interest may not be easy (or even possible) to measure directly but instead must be calculated from other directly measured properties and appropriate mathematical relationships. For example, consider measuring the average speed of an athlete running sprints. This is typically accomplished by measuring the time required for the athlete to run from the starting line to the finish line, and the distance between these two lines, and then computing speed from the equation that relates these three properties:

    (2.6)
    _autogen-svg2png-0001.png

    An Olympic-quality sprinter can run 100 m in approximately 10 s, corresponding to an average speed of

    (2.7)
    _autogen-svg2png-0002.png

    Note that this simple arithmetic involves dividing the numbers of each measured quantity to yield the number of the computed quantity (100/10 = 10) and likewise dividing the units of each measured quantity to yield the unit of the computed quantity (m/s = m/s). Now, consider using this same relation to predict the time required for a person running at this speed to travel a distance of 25 m. The same relation between the three properties is used, but in this case, the two quantities provided are a speed (10 m/s) and a distance (25 m). To yield the sought property, time, the equation must be rearranged appropriately:

    (2.8)
    _autogen-svg2png-0003.png

    The time can then be computed as:

    (2.9)
    _autogen-svg2png-0004.png

    Again, arithmetic on the numbers (25/10 = 2.5) was accompanied by the same arithmetic on the units (m/m/s = s) to yield the number and unit of the result, 2.5 s. Note that, just as for numbers, when a unit is divided by an identical unit (in this case, m/m), the result is “1”—or, as commonly phrased, the units “cancel.”

    These calculations are examples of a versatile mathematical approach known as dimensional analysis (or the factor-label method). Dimensional analysis is based on this premise: the units of quantities must be subjected to the same mathematical operations as their associated numbers. This method can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities.

    Conversion Factors and Dimensional Analysis

    A ratio of two equivalent quantities expressed with different measurement units can be used as a unit conversion factor. For example, the lengths of 2.54 cm and 1 in. are equivalent (by definition), and so a unit conversion factor may be derived from the ratio,

    (2.10)
    _autogen-svg2png-0005.png

    Several other commonly used conversion factors are given in Table 2.4.

    Table 2.4.
    Common Conversion Factors
    LengthVolumeMass
    1 m = 1.0936 yd1 L = 1.0567 qt1 kg = 2.2046 lb
    1 in. = 2.54 cm (exact)1 qt = 0.94635 L1 lb = 453.59 g
    1 km = 0.62137 mi1 ft3 = 28.317 L1 (avoirdupois) oz = 28.349 g
    1 mi = 1609.3 m1 tbsp = 14.787 mL1 (troy) oz = 31.103 g

    When we multiply a quantity (such as distance given in inches) by an appropriate unit conversion factor, we convert the quantity to an equivalent value with different units (such as distance in centimeters). For example, a basketball player’s vertical jump of 34 inches can be converted to centimeters by:

    (2.11)
    _autogen-svg2png-0006.png

    Since this simple arithmetic involves quantities, the premise of dimensional analysis requires that we multiply both numbers and units. The numbers of these two quantities are multiplied to yield the number of the product quantity, 86, whereas the units are multiplied to yield _autogen-svg2png-0007.png. Just as for numbers, a ratio of identical units is also numerically equal to one, _autogen-svg2png-0008.png and the unit product thus simplifies to cm. (When identical units divide to yield a factor of 1, they are said to “cancel.”) Using dimensional analysis, we can determine that a unit conversion factor has been set up correctly by checking to confirm that the original unit will cancel, and the result will contain the sought (converted) unit.

    Example 2.3

    Using a Unit Conversion Factor

    The mass of a competition frisbee is 125 g. Convert its mass to ounces using the unit conversion factor derived from the relationship 1 oz = 28.349 g (Table 2.4).

    Solution

    If we have the conversion factor, we can determine the mass in kilograms using an equation similar the one used for converting length from inches to centimeters.

    (2.12)
    _autogen-svg2png-0009.png

    We write the unit conversion factor in its two forms:

    (2.13)
    _autogen-svg2png-0010.png

    The correct unit conversion factor is the ratio that cancels the units of grams and leaves ounces.

    (2.14)
    _autogen-svg2png-0011.png

    Check Your Learning

    Convert a volume of 9.345 qt to liters.

    Answer:

    8.844 L

    Beyond simple unit conversions, the factor-label method can be used to solve more complex problems involving computations. Regardless of the details, the basic approach is the same—all the factors involved in the calculation must be appropriately oriented to insure that their labels (units) will appropriately cancel and/or combine to yield the desired unit in the result. This is why it is referred to as the factor-label method. As your study of chemistry continues, you will encounter many opportunities to apply this approach.

    Example 2.4

    Computing Quantities from Measurement Results and Known Mathematical Relations

    What is the density of common antifreeze in units of g/mL? A 4.00-qt sample of the antifreeze weighs 9.26 lb.

    Solution

    Since _autogen-svg2png-0012.png, we need to divide the mass in grams by the volume in milliliters. In general: the number of units of B = the number of units of A × unit conversion factor. The necessary conversion factors are given in Table 2.4: 1 lb = 453.59 g; 1 L = 1.0567 qt; 1 L = 1,000 mL. We can convert mass from pounds to grams in one step:

    (2.15)
    _autogen-svg2png-0014.png

    We need to use two steps to convert volume from quarts to milliliters.

    1. Convert quarts to liters.

      (2.16)
      _autogen-svg2png-0015.png
    2. Convert liters to milliliters.

      (2.17)
      _autogen-svg2png-0016.png

    Then,

    (2.18)
    _autogen-svg2png-0017.png

    Alternatively, the calculation could be set up in a way that uses three unit conversion factors sequentially as follows:

    (2.19)
    _autogen-svg2png-0018.png

    Check Your Learning

    What is the volume in liters of 1.000 oz, given that 1 L = 1.0567 qt and 1 qt = 32 oz (exactly)?

    Answer:

    _autogen-svg2png-0019.png

    Example 2.5

    Computing Quantities from Measurement Results and Known Mathematical Relations

    While being driven from Philadelphia to Atlanta, a distance of about 1250 km, a 2014 Lamborghini Aventador Roadster uses 213 L gasoline.

    (a) What (average) fuel economy, in miles per gallon, did the Roadster get during this trip?

    (b) If gasoline costs $3.80 per gallon, what was the fuel cost for this trip?

    Solution

    (a) We first convert distance from kilometers to miles:

    (2.20)
    _autogen-svg2png-0020.png

    and then convert volume from liters to gallons:

    (2.21)
    _autogen-svg2png-0021.png

    Then,

    (2.22)
    _autogen-svg2png-0022.png

    Alternatively, the calculation could be set up in a way that uses all the conversion factors sequentially, as follows:

    (2.23)
    _autogen-svg2png-0023.png

    (b) Using the previously calculated volume in gallons, we find:

    (2.24)
    _autogen-svg2png-0024.png

    Check Your Learning

    A Toyota Prius Hybrid uses 59.7 L gasoline to drive from San Francisco to Seattle, a distance of 1300 km (two significant digits).

    (a) What (average) fuel economy, in miles per gallon, did the Prius get during this trip?

    (b) If gasoline costs $3.90 per gallon, what was the fuel cost for this trip?

    Answer:

    (a) 51 mpg; (b) $62

    Conversion of Temperature Units

    We use the word temperature to refer to the hotness or coldness of a substance. One way we measure a change in temperature is to use the fact that most substances expand when their temperature increases and contract when their temperature decreases. The mercury or alcohol in a common glass thermometer changes its volume as the temperature changes. Because the volume of the liquid changes more than the volume of the glass, we can see the liquid expand when it gets warmer and contract when it gets cooler.

    To mark a scale on a thermometer, we need a set of reference values: Two of the most commonly used are the freezing and boiling temperatures of water at a specified atmospheric pressure. On the Celsius scale, 0 °C is defined as the freezing temperature of water and 100 °C as the boiling temperature of water. The space between the two temperatures is divided into 100 equal intervals, which we call degrees. On the Fahrenheit scale, the freezing point of water is defined as 32 °F and the boiling temperature as 212 °F. The space between these two points on a Fahrenheit thermometer is divided into 180 equal parts (degrees).

    Defining the Celsius and Fahrenheit temperature scales as described in the previous paragraph results in a slightly more complex relationship between temperature values on these two scales than for different units of measure for other properties. Most measurement units for a given property are directly proportional to one another (y = mx). Using familiar length units as one example:

    (2.25)
    _autogen-svg2png-0025.png

    where y = length in feet, x = length in inches, and the proportionality constant, m, is the conversion factor. The Celsius and Fahrenheit temperature scales, however, do not share a common zero point, and so the relationship between these two scales is a linear one rather than a proportional one (y = mx + b). Consequently, converting a temperature from one of these scales into the other requires more than simple multiplication by a conversion factor, m, it also must take into account differences in the scales’ zero points (b).

    The linear equation relating Celsius and Fahrenheit temperatures is easily derived from the two temperatures used to define each scale. Representing the Celsius temperature as x and the Fahrenheit temperature as y, the slope, m, is computed to be:

    (2.26)
    _autogen-svg2png-0026.png

    The y-intercept of the equation, b, is then calculated using either of the equivalent temperature pairs, (100 °C, 212 °F) or (0 °C, 32 °F), as:

    (2.27)
    _autogen-svg2png-0027.png

    The equation relating the temperature scales is then:

    (2.28)
    _autogen-svg2png-0028.png

    An abbreviated form of this equation that omits the measurement units is:

    (2.29)
    _autogen-svg2png-0029.png

    Rearrangement of this equation yields the form useful for converting from Fahrenheit to Celsius:

    (2.30)
    _autogen-svg2png-0030.png

    As mentioned earlier in this chapter, the SI unit of temperature is the kelvin (K). Unlike the Celsius and Fahrenheit scales, the kelvin scale is an absolute temperature scale in which 0 (zero) K corresponds to the lowest temperature that can theoretically be achieved. The early 19th-century discovery of the relationship between a gas's volume and temperature suggested that the volume of a gas would be zero at −273.15 °C. In 1848, British physicist William Thompson, who later adopted the title of Lord Kelvin, proposed an absolute temperature scale based on this concept (further treatment of this topic is provided in this text’s chapter on gases).

    The freezing temperature of water on this scale is 273.15 K and its boiling temperature 373.15 K. Notice the numerical difference in these two reference temperatures is 100, the same as for the Celsius scale, and so the linear relation between these two temperature scales will exhibit a slope of _autogen-svg2png-0031.png. Following the same approach, the equations for converting between the kelvin and Celsius temperature scales are derived to be:

    (2.31)TK=T°C+273.15
    (2.32)T°C=TK−273.15

    The 273.15 in these equations has been determined experimentally, so it is not exact. Figure 2.4 shows the relationship among the three temperature scales. Recall that we do not use the degree sign with temperatures on the kelvin scale.

    CNX_Chem_01_06_TempScales.jpg
    Figure 2.4
    The Fahrenheit, Celsius, and kelvin temperature scales are compared.

    Although the kelvin (absolute) temperature scale is the official SI temperature scale, Celsius is commonly used in many scientific contexts and is the scale of choice for nonscience contexts in almost all areas of the world. Very few countries (the U.S. and its territories, the Bahamas, Belize, Cayman Islands, and Palau) still use Fahrenheit for weather, medicine, and cooking.

    Example 2.6

    Conversion from Celsius

    Normal body temperature has been commonly accepted as 37.0 °C (although it varies depending on time of day and method of measurement, as well as among individuals). What is this temperature on the kelvin scale and on the Fahrenheit scale?

    Solution

    (2.33)K=°C+273.15=37.0+273.2=310.2 K
    (2.34)
    _autogen-svg2png-0035.png

    Check Your Learning

    Convert 80.92 °C to K and °F.

    Answer:

    354.07 K, 177.7 °F

    Example 2.7

    Conversion from Fahrenheit

    Baking a ready-made pizza calls for an oven temperature of 450 °F. If you are in Europe, and your oven thermometer uses the Celsius scale, what is the setting? What is the kelvin temperature?

    Solution

    (2.35)
    _autogen-svg2png-0036.png
    (2.36)
    _autogen-svg2png-0037.png

    Check Your Learning

    Convert 50 °F to °C and K.

    Answer:

    10 °C, 280 K

    Key Concepts and Summary

    Measurements are made using a variety of units. It is often useful or necessary to convert a measured quantity from one unit into another. These conversions are accomplished using unit conversion factors, which are derived by simple applications of a mathematical approach called the factor-label method or dimensional analysis. This strategy is also employed to calculate sought quantities using measured quantities and appropriate mathematical relations.

    Key Equations

    • _autogen-svg2png-0038.png

    • _autogen-svg2png-0039.png

    • TK=°C+273.15

    • T°C=K−273.15

    Chemistry End of Chapter Exercises

    Write conversion factors (as ratios) for the number of:

    (a) yards in 1 meter

    (b) liters in 1 liquid quart

    (c) pounds in 1 kilogram

    (a) _autogen-svg2png-0042.png; (b) _autogen-svg2png-0043.png; (c) _autogen-svg2png-0044.png
    Exercise 12.

    Write conversion factors (as ratios) for the number of:

    (a) kilometers in 1 mile

    (b) liters in 1 cubic foot

    (c) grams in 1 ounce

    The label on a soft drink bottle gives the volume in two units: 2.0 L and 67.6 fl oz. Use this information to derive a conversion factor between the English and metric units. How many significant figures can you justify in your conversion factor?

    _autogen-svg2png-0045.png
    Only two significant figures are justified.
    Exercise 14.

    The label on a box of cereal gives the mass of cereal in two units: 978 grams and 34.5 oz. Use this information to find a conversion factor between the English and metric units. How many significant figures can you justify in your conversion factor?

    Soccer is played with a round ball having a circumference between 27 and 28 in. and a weight between 14 and 16 oz. What are these specifications in units of centimeters and grams?

    68–71 cm; 400–450 g
    Exercise 16.

    A woman's basketball has a circumference between 28.5 and 29.0 inches and a maximum weight of 20 ounces (two significant figures). What are these specifications in units of centimeters and grams?

    How many milliliters of a soft drink are contained in a 12.0-oz can?

    Exercise 18.

    A barrel of oil is exactly 42 gal. How many liters of oil are in a barrel?

    The diameter of a red blood cell is about 3 × 10−4 in. What is its diameter in centimeters?

    8 × 10−4 cm
    Exercise 20.

    The distance between the centers of the two oxygen atoms in an oxygen molecule is 1.21 × 10−8 cm. What is this distance in inches?

    Is a 197-lb weight lifter light enough to compete in a class limited to those weighing 90 kg or less?

    yes; weight = 89.4 kg
    Exercise 22.

    A very good 197-lb weight lifter lifted 192 kg in a move called the clean and jerk. What was the mass of the weight lifted in pounds?

    Many medical laboratory tests are run using 5.0 μL blood serum. What is this volume in milliliters?

    5.0 × 10−3 mL
    Exercise 24.

    If an aspirin tablet contains 325 mg aspirin, how many grams of aspirin does it contain?

    Use scientific (exponential) notation to express the following quantities in terms of the SI base units in Table 2.2:

    (a) 0.13 g

    (b) 232 Gg

    (c) 5.23 pm

    (d) 86.3 mg

    (e) 37.6 cm

    (f) 54 μm

    (g) 1 Ts

    (h) 27 ps

    (i) 0.15 mK

    (a) 1.3 × 10−4 kg; (b) 2.32 × 108 kg; (c) 5.23 × 10−12 m; (d) 8.63 × 10−5 kg; (e) 3.76 × 10−1 m; (f) 5.4 × 10−5 m; (g) 1 × 1012 s; (h) 2.7 × 10−11 s; (i) 1.5 × 10−4 K
    Exercise 26.

    Complete the following conversions between SI units.

    (a) 612 g = ________ mg

    (b) 8.160 m = ________ cm

    (c) 3779 μg = ________ g

    (d) 781 mL = ________ L

    (e) 4.18 kg = ________ g

    (f) 27.8 m = ________ km

    (g) 0.13 mL = ________ L

    (h) 1738 km = ________ m

    (i) 1.9 Gg = ________ g

    Gasoline is sold by the liter in many countries. How many liters are required to fill a 12.0-gal gas tank?

    Exercise 28.

    Milk is sold by the liter in many countries. What is the volume of exactly 1/2 gal of milk in liters?

    A long ton is defined as exactly 2240 lb. What is this mass in kilograms?

    1.0160 × 103 kg
    Exercise 30.

    Make the conversion indicated in each of the following:

    (a) the men’s world record long jump, 29 ft 4¼ in., to meters

    (b) the greatest depth of the ocean, about 6.5 mi, to kilometers

    (c) the area of the state of Oregon, 96,981 mi2, to square kilometers

    (d) the volume of 1 gill (exactly 4 oz) to milliliters

    (e) the estimated volume of the oceans, 330,000,000 mi3, to cubic kilometers.

    (f) the mass of a 3525-lb car to kilograms

    (g) the mass of a 2.3-oz egg to grams

    Make the conversion indicated in each of the following:

    (a) the length of a soccer field, 120 m (three significant figures), to feet

    (b) the height of Mt. Kilimanjaro, at 19,565 ft the highest mountain in Africa, to kilometers

    (c) the area of an 8.5 × 11-inch sheet of paper in cm2

    (d) the displacement volume of an automobile engine, 161 in.3, to liters

    (e) the estimated mass of the atmosphere, 5.6 × 1015 tons, to kilograms

    (f) the mass of a bushel of rye, 32.0 lb, to kilograms

    (g) the mass of a 5.00-grain aspirin tablet to milligrams (1 grain = 0.00229 oz)

    (a) 394 ft
    (b) 5.9634 km
    (c) 6.0 × 102
    (d) 2.64 L
    (e) 5.1 × 1018 kg
    (f) 14.5 kg
    (g) 324 mg
    Exercise 32.

    Many chemistry conferences have held a 50-Trillion Angstrom Run (two significant figures). How long is this run in kilometers and in miles? (1 Å = 1 × 10−10 m)

    A chemist’s 50-Trillion Angstrom Run (see Exercise 32.) would be an archeologist’s 10,900 cubit run. How long is one cubit in meters and in feet? (1 Å = 1 × 10−8 cm)

    0.46 m; 1.5 ft/cubit
    Exercise 34.

    The gas tank of a certain luxury automobile holds 22.3 gallons according to the owner’s manual. If the density of gasoline is 0.8206 g/mL, determine the mass in kilograms and pounds of the fuel in a full tank.

    As an instructor is preparing for an experiment, he requires 225 g phosphoric acid. The only container readily available is a 150-mL Erlenmeyer flask. Is it large enough to contain the acid, whose density is 1.83 g/mL?

    Yes, the acid's volume is 123 mL.
    Exercise 36.

    To prepare for a laboratory period, a student lab assistant needs 125 g of a compound. A bottle containing 1/4 lb is available. Did the student have enough of the compound?

    A chemistry student is 159 cm tall and weighs 45.8 kg. What is her height in inches and weight in pounds?

    62.6 in (about 5 ft 3 in.) and 101 lb
    Exercise 38.

    In a recent Grand Prix, the winner completed the race with an average speed of 229.8 km/h. What was his speed in miles per hour, meters per second, and feet per second?

    Solve these problems about lumber dimensions.

    (a) To describe to a European how houses are constructed in the US, the dimensions of “two-by-four” lumber must be converted into metric units. The thickness × width × length dimensions are 1.50 in. × 3.50 in. × 8.00 ft in the US. What are the dimensions in cm × cm × m?

    (b) This lumber can be used as vertical studs, which are typically placed 16.0 in. apart. What is that distance in centimeters?

    (a) 3.81 cm × 8.89 cm × 2.44 m; (b) 40.6 cm
    Exercise 40.

    The mercury content of a stream was believed to be above the minimum considered safe—1 part per billion (ppb) by weight. An analysis indicated that the concentration was 0.68 parts per billion. What quantity of mercury in grams was present in 15.0 L of the water, the density of which is 0.998 g/ml? _autogen-svg2png-0072.png

    Calculate the density of aluminum if 27.6 cm3 has a mass of 74.6 g.

    2.70 g/cm3
    Exercise 42.

    Osmium is one of the densest elements known. What is its density if 2.72 g has a volume of 0.121 cm3?

    Calculate these masses.

    (a) What is the mass of 6.00 cm3 of mercury, density = 13.5939 g/cm3?

    (b) What is the mass of 25.0 mL octane, density = 0.702 g/cm3?

    (a) 81.6 g; (b) 17.6 g
    Exercise 44.

    Calculate these masses.

    (a) What is the mass of 4.00 cm3 of sodium, density = 0.97 g/cm3 ?

    (b) What is the mass of 125 mL gaseous chlorine, density = 3.16 g/L?

    Calculate these volumes.

    (a) What is the volume of 25 g iodine, density = 4.93 g/cm3?

    (b) What is the volume of 3.28 g gaseous hydrogen, density = 0.089 g/L?

    (a) 5.1 mL; (b) 37 L
    Exercise 46.

    Calculate these volumes.

    (a) What is the volume of 11.3 g graphite, density = 2.25 g/cm3?

    (b) What is the volume of 39.657 g bromine, density = 2.928 g/cm3?

    Convert the boiling temperature of gold, 2966 °C, into degrees Fahrenheit and kelvin.

    5371 °F, 3239 K
    Exercise 48.

    Convert the temperature of scalding water, 54 °C, into degrees Fahrenheit and kelvin.

    Convert the temperature of the coldest area in a freezer, −10 °F, to degrees Celsius and kelvin.

    −23 °C, 250 K
    Exercise 50.

    Convert the temperature of dry ice, −77 °C, into degrees Fahrenheit and kelvin.

    Convert the boiling temperature of liquid ammonia, −28.1 °F, into degrees Celsius and kelvin.

    −33.4 °C, 239.8 K
    Exercise 52.

    The label on a pressurized can of spray disinfectant warns against heating the can above 130 °F. What are the corresponding temperatures on the Celsius and kelvin temperature scales?

    The weather in Europe was unusually warm during the summer of 1995. The TV news reported temperatures as high as 45 °C. What was the temperature on the Fahrenheit scale?

    Glossary

    Celsius (°C)

    unit of temperature; water freezes at 0 °C and boils at 100 °C on this scale

    cubic centimeter (cm3 or cc)

    volume of a cube with an edge length of exactly 1 cm

    cubic meter (m3)

    SI unit of volume

    density

    ratio of mass to volume for a substance or object

    dimensional analysis

    (also, factor-label method) versatile mathematical approach that can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities

    Fahrenheit

    unit of temperature; water freezes at 32 °F and boils at 212 °F on this scale

    kelvin (K)

    SI unit of temperature; 273.15 K = 0 ºC

    kilogram (kg)

    standard SI unit of mass; 1 kg = approximately 2.2 pounds

    length

    measure of one dimension of an object

    liter (L)

    (also, cubic decimeter) unit of volume; 1 L = 1,000 cm3

    meter (m)

    standard metric and SI unit of length; 1 m = approximately 1.094 yards

    milliliter (mL)

    1/1,000 of a liter; equal to 1 cm3

    second (s)

    SI unit of time

    SI units (International System of Units)

    standards fixed by international agreement in the International System of Units (Le Système International d’Unités)

    unit

    standard of comparison for measurements

    unit conversion factor

    ratio of equivalent quantities expressed with different units; used to convert from one unit to a different unit

    volume

    amount of space occupied by an object

    Solutions