# Measurements and Their Uncertainty

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Measurements and Their Uncertainty
3.1 On January 4, 2004, the Mars Exploration Rover Spirit landed on Mars. Each day of its mission, Spirit recorded measurements for analysis. In the chemistry laboratory, you must strive for accuracy and precision in your measurements.

Using and Expressing Measurements
3.1 Using and Expressing Measurements Using and Expressing Measurements How do measurements relate to science?

Using and Expressing Measurements
3.1 Using and Expressing Measurements A measurement is a quantity that has both a number and a unit. Measurements are fundamental to the experimental sciences. For that reason, it is important to be able to make measurements and to decide whether a measurement is correct.

Using and Expressing Measurements
3.1 Using and Expressing Measurements In scientific notation, a given number is written as the product of two numbers: a coefficient and 10 raised to a power. The number of stars in a galaxy is an example of an estimate that should be expressed in scientific notation. Expressing very large numbers, such as the estimated number of stars in a galaxy, is easier if scientific notation is used.

Accuracy, Precision, and Error
3.1 Accuracy, Precision, and Error Accuracy, Precision, and Error How do you evaluate accuracy and precision?

Accuracy, Precision, and Error
3.1 Accuracy, Precision, and Error Accuracy and Precision Accuracy is a measure of how close a measurement comes to the actual or true value of whatever is measured. Precision is a measure of how close a series of measurements are to one another.

Accuracy, Precision, and Error
3.1 Accuracy, Precision, and Error To evaluate the accuracy of a measurement, the measured value must be compared to the correct value. To evaluate the precision of a measurement, you must compare the values of two or more repeated measurements.

Accuracy, Precision, and Error
3.1 Accuracy, Precision, and Error The distribution of darts illustrates the difference between accuracy and precision. a) Good accuracy and good precision: The darts are close to the bull’s-eye and to one another. b) Poor accuracy and good precision: The darts are far from the bull’s-eye but close to one another. c) Poor accuracy and poor precision: The darts are far from the bull’s-eye and from one another.

Accuracy, Precision, and Error
3.1 Accuracy, Precision, and Error Determining Error The accepted value is the correct value based on reliable references. The experimental value is the value measured in the lab. The difference between the experimental value and the accepted value is called the error.

Accuracy, Precision, and Error
3.1 Accuracy, Precision, and Error The percent error is the absolute value of the error divided by the accepted value, multiplied by 100%. Expressing very large numbers, such as the estimated number of stars in a galaxy, is easier if scientific notation is used.

Accuracy, Precision, and Error
3.1 Accuracy, Precision, and Error

Accuracy, Precision, and Error
3.1 Accuracy, Precision, and Error Just because a measuring device works, you cannot assume it is accurate. The scale below has not been properly zeroed, so the reading obtained for the person’s weight is inaccurate. The scale below has not been properly zeroed, so the reading obtained for the person’s weight is inaccurate. There is a difference between the person’s correct weight and the measured value. Calculating What is the percent error of a measured value of 114 lb if the person’s actual weight is 107 lb?

Significant Figures in Measurements
3.1 Significant Figures in Measurements Significant Figures in Measurements Why must measurements be reported to the correct number of significant figures?

Significant Figures in Measurements
3.1 Significant Figures in Measurements Suppose you estimate a weight that is between 2.4 lb and 2.5 lb to be 2.46 lb. The first two digits (2 and 4) are known. The last digit (6) is an estimate and involves some uncertainty. All three digits convey useful information, however, and are called significant figures. The significant figures in a measurement include all of the digits that are known, plus a last digit that is estimated.

Significant Figures in Measurements
3.1 Significant Figures in Measurements Measurements must always be reported to the correct number of significant figures because calculated answers often depend on the number of significant figures in the values used in the calculation.

Significant Figures in Measurements
3.1 Significant Figures in Measurements

Significant Figures in Measurements
3.1 Significant Figures in Measurements

Significant Figures in Measurements
3.1 Significant Figures in Measurements Three differently calibrated meter sticks are used to measure the length of a board. a) A meter stick calibrated in a 1-m interval. b) A meter stick calibrated in 0.1-m intervals. c) A meter stick calibrated in 0.01-m intervals. Measuring How many significant figures are reported in each measurement?

for Conceptual Problem 3.1

Significant Figures in Calculations
3.1 Significant Figures in Calculations Significant Figures in Calculations How does the precision of a calculated answer compare to the precision of the measurements used to obtain it?

Significant Figures in Calculations
3.1 Significant Figures in Calculations In general, a calculated answer cannot be more precise than the least precise measurement from which it was calculated. The calculated value must be rounded to make it consistent with the measurements from which it was calculated.

Significant Figures in Calculations
3.1 Significant Figures in Calculations Rounding To round a number, you must first decide how many significant figures your answer should have. The answer depends on the given measurements and on the mathematical process used to arrive at the answer.

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for Sample Problem 3.1

Significant Figures in Calculations
3.1 Significant Figures in Calculations Addition and Subtraction The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places.

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for Sample Problem 3.2

Significant Figures in Calculations
3.1 Significant Figures in Calculations Multiplication and Division In calculations involving multiplication and division, you need to round the answer to the same number of significant figures as the measurement with the least number of significant figures. The position of the decimal point has nothing to do with the rounding process when multiplying and dividing measurements.

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for Sample Problem 3.3

3.1 Section Quiz 1. In which of the following expressions is the number on the left NOT equal to the number on the right? X 10–8 = 4.56 X 10–11 454 X 10–8 = 4.54 X 10–6 842.6 X 104 = X 106 X 106 = 4.52 X 109

3.1 Section Quiz 2. Which set of measurements of a 2.00-g standard is the most precise? 2.00 g, 2.01 g, 1.98 g 2.10 g, 2.00 g, 2.20 g 2.02 g, 2.03 g, 2.04 g 1.50 g, 2.00 g, 2.50 g

3.1 Section Quiz 3. A student reports the volume of a liquid as L. How many significant figures are in this measurement? 2 3 4 5

The International System of Units
3.2 In the signs shown here, the distances are listed as numbers with no units attached. Without the units, it is impossible to communicate the measurement to others. When you make a measurement, you must assign the correct units to the numerical value.

Measuring with SI Units
3.2 Measuring with SI Units Measuring with SI Units Which five SI base units do chemists commonly use?

Measuring with SI Units
3.2 Measuring with SI Units All measurements depend on units that serve as reference standards. The standards of measurement used in science are those of the metric system. The International System of Units (abbreviated SI, after the French name, Le Système International d’Unités) is a revised version of the metric system.

Measuring with SI Units
3.2 Measuring with SI Units The five SI base units commonly used by chemists are the meter, the kilogram, the kelvin, the second, and the mole.

3.2 Units and Quantities Units and Quantities What metric units are commonly used to measure length, volume, mass, temperature and energy?

3.2 Units and Quantities Units of Length In SI, the basic unit of length, or linear measure, is the meter (m). For very large or and very small lengths, it may be more convenient to use a unit of length that has a prefix.

3.2 Units and Quantities Common metric units of length include the centimeter, meter, and kilometer.

3.2 Units and Quantities Units of Volume The SI unit of volume is the amount of space occupied by a cube that is 1 m along each edge. This volume is the cubic meter (m)3. A more convenient unit of volume for everyday use is the liter, a non- SI unit. A liter (L) is the volume of a cube that is 10 centimeters (10 cm) along each edge (10 cm  10 cm  10 cm = 1000 cm3 = 1 L).

3.2 Units and Quantities Common metric units of volume include the liter, milliliter, cubic centimeter, and microliter.

3.2 Units and Quantities The volume of 20 drops of liquid from a medicine dropper is approximately 1 mL. These photographs above give you some idea of the relative sizes of some different units of volume. a) The volume of 20 drops of liquid from a medicine dropper is approximately 1 mL. b) A sugar cube is 1 cm on each edge and has a volume of 1 cm3. Note that 1 mL is the same as 1 cm3. c) A gallon of milk has about twice the volume of a 2-L bottle of soda. Calculating How many cubic centimeters are in 2 liters?

A sugar cube has a volume of 1 cm3. 1 mL is the same as 1 cm3.
3.2 Units and Quantities A sugar cube has a volume of 1 cm3. 1 mL is the same as 1 cm3. These photographs above give you some idea of the relative sizes of some different units of volume. a) The volume of 20 drops of liquid from a medicine dropper is approximately 1 mL. b) A sugar cube is 1 cm on each edge and has a volume of 1 cm3. Note that 1 mL is the same as 1 cm3. c) A gallon of milk has about twice the volume of a 2-L bottle of soda. Calculating How many cubic centimeters are in 2 liters?

A gallon of milk has about twice the volume of a 2-L bottle of soda.
3.2 Units and Quantities A gallon of milk has about twice the volume of a 2-L bottle of soda. These photographs above give you some idea of the relative sizes of some different units of volume. a) The volume of 20 drops of liquid from a medicine dropper is approximately 1 mL. b) A sugar cube is 1 cm on each edge and has a volume of 1 cm3. Note that 1 mL is the same as 1 cm3. c) A gallon of milk has about twice the volume of a 2-L bottle of soda. Calculating How many cubic centimeters are in 2 liters?

3.2 Units and Quantities Units of Mass The mass of an object is measured in comparison to a standard mass of 1 kilogram (kg), which is the basic SI unit of mass. A gram (g) is 1/1000 of a kilogram; the mass of 1 cm3 of water at 4°C is 1 g.

3.2 Units and Quantities Common metric units of mass include kilogram, gram, milligram, and microgram.

Weight is a force that measures the pull on a given mass by gravity.
3.2 Units and Quantities Weight is a force that measures the pull on a given mass by gravity. The astronaut shown on the surface of the moon weighs one sixth of what he weighs on Earth. An astronaut’s weight on the moon is one sixth as much as it is on Earth. Earth exerts six times the force of gravity as the moon. Inferring How does the astronaut’s mass on the moon compare to his mass on Earth?

Temperature is a measure of how hot or cold an object is.
3.2 Units and Quantities Units of Temperature Temperature is a measure of how hot or cold an object is. Thermometers are used to measure temperature. Thermometers are used to measure temperature. a) A liquid-in-glass thermometer contains alcohol or mineral spirits. b) A dial thermometer contains a coiled bimetallic strip. c) A Galileo thermometer contains several glass bulbs that are calibrated to sink or float depending on the temperature. The Galileo thermometer shown uses the Fahrenheit scale, which sets the freezing point of water at 32°F and the boiling point of water at 212°F.

3.2 Units and Quantities Scientists commonly use two equivalent units of temperature, the degree Celsius and the kelvin.

3.2 Units and Quantities On the Celsius scale, the freezing point of water is 0°C and the boiling point is 100°C. On the Kelvin scale, the freezing point of water is kelvins (K), and the boiling point is K. The zero point on the Kelvin scale, 0 K, or absolute zero, is equal to °C.

3.2 Units and Quantities Because one degree on the Celsius scale is equivalent to one kelvin on the Kelvin scale, converting from one temperature to another is easy. You simply add or subtract 273, as shown in the following equations.

Conversions Between the Celsius and Kelvin Scales
3.2 Units and Quantities Conversions Between the Celsius and Kelvin Scales These thermometers show a comparison of the Celsius and Kelvin temperature scales. Note that a 1°C change on the Celsius scale is equal to a 1 K change on the Kelvin scale. Interpreting Diagrams What is a change of 10 K equivalent to on the Celsius scale?

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for Sample Problem 3.4

3.2 Units and Quantities Units of Energy Energy is the capacity to do work or to produce heat. The joule and the calorie are common units of energy.

3.2 Units and Quantities The joule (J) is the SI unit of energy. One calorie (cal) is the quantity of heat that raises the temperature of 1 g of pure water by 1°C.

3.2 Units and Quantities This house is equipped with solar panels. The solar panels convert the radiant energy from the sun into electrical energy that can be used to heat water and power appliances. Photoelectric panels convert solar energy into electricity.

3.2 Section Quiz. 1. Which of the following is not a base SI unit?
meter gram second mole

3.2 Section Quiz. 2. If you measured both the mass and weight of an object on Earth and on the moon, you would find that both the mass and the weight do not change. both the mass and the weight change. the mass remains the same, but the weight changes. the mass changes, but the weight remains the same.

3.2 Section Quiz. 3. A temperature of 30 degrees Celsius is equivalent to 303 K. 300 K. 243 K. 247 K.

3.33 Conversion Problems Because each country’s currency compares differently with the U.S. dollar, knowing how to convert currency units correctly is very important. Conversion problems are readily solved by a problem- solving approach called dimensional analysis.

3.3 Conversion Factors Conversion Factors What happens when a measurement is multiplied by a conversion factor?

A conversion factor is a ratio of equivalent measurements.
3.3 Conversion Factors A conversion factor is a ratio of equivalent measurements. The ratios 100 cm/1 m and 1 m/100 cm are examples of conversion factors. The two parts of a conversion factor, the numerator and the denominator, are equal.

3.3 Conversion Factors When a measurement is multiplied by a conversion factor, the numerical value is generally changed, but the actual size of the quantity measured remains the same.

3.3 Conversion Factors The scale of the micrograph is in nanometers. Using the relationship 109 nm = 1 m, you can write the following conversion factors. In this computer image of atoms, distance is marked off in nanometers (nm). Inferring What conversion factor would you use to convert nanometers to meters?

3.3 Dimensional Analysis Dimensional Analysis Why is dimensional analysis useful?

3.3 Dimensional Analysis Dimensional analysis is a way to analyze and solve problems using the units, or dimensions, of the measurements. Dimensional analysis provides you with an alternative approach to problem solving.

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for Sample Problem 3.5

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for Sample Problem 3.6 For Sample Problem 3.6

Converting Between Units
3.3 Converting Between Units Converting Between Units What types of problems are easily solved by using dimensional analysis?

Converting Between Units
3.3 Converting Between Units Problems in which a measurement with one unit is converted to an equivalent measurement with another unit are easily solved using dimensional analysis.

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for Sample Problem 3.7

Converting Between Units
3.3 Converting Between Units Multistep Problems When converting between units, it is often necessary to use more than one conversion factor. Sample problem 3.8 illustrates the use of multiple conversion factors.

3.8

for Sample Problem 3.8

Converting Between Units
3.3 Converting Between Units Converting Complex Units Many common measurements are expressed as a ratio of two units. If you use dimensional analysis, converting these complex units is just as easy as converting single units. It will just take multiple steps to arrive at an answer.

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for Sample Problem 3.9

3.3 Section Quiz 1. 1 Mg = 1000 kg. Which of the following would be a correct conversion factor for this relationship? x 1000. x 1/1000. ÷ 1000. 1000 kg/1Mg.

3.3 Section Quiz 2. The conversion factor used to convert joules to calories changes the quantity of energy measured but not the numerical value of the measurement. neither the numerical value of the measurement nor the quantity of energy measured. the numerical value of the measurement but not the quantity of energy measured. both the numerical value of the measurement and the quantity of energy measured.

3.3 Section Quiz 3. Express the density 5.6 g/cm3 in kg/m3. 5.6 x 106kg/m3 5.6 x 103kg/m3 0.56 kg/m3 kg/m3

3.4 Density If you think that these lily pads float because they are lightweight, you are only partially correct. The ratio of the mass of an object to its volume can be used to determine whether an object floats or sinks in water.

3.4 Determining Density Determining Density What determines the density of a substance?

3.4 Determining Density Density is the ratio of the mass of an object to its volume.

3.4 Determining Density Each of these 10-g samples has a different volume because the densities vary. A 10-g sample of pure water has less volume than 10 g of lithium, but more volume than 10 g of lead. The faces of the cubes are shown actual size. Inferring Which substance has the highest ratio of mass to volume?

3.4 Determining Density Density is an intensive property that depends only on the composition of a substance, not on the size of the sample.

3.4 Determining Density The density of corn oil is less than the density of corn syrup. For that reason, the oil floats on top of the syrup. Because of differences in density, corn oil floats on top of corn syrup.

Density and Temperature
3.4 Density and Temperature Density and Temperature How does a change in temperature affect density?

Density and Temperature
3.4 Density and Temperature Experiments show that the volume of most substances increases as the temperature increases. Meanwhile, the mass remains the same. Thus, the density must change. The density of a substance generally decreases as its temperature increases.

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for Sample Problem 3.10

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for Sample Problem 3.11

3.4 Section Quiz 1. If 50.0 mL of corn syrup have a mass of g, the density of the corn syrup is 0.737 g/mL. 0.727 g/mL. 1.36 g/mL. 1.37 g/mL.

3.4 Section Quiz 2. What is the volume of a pure gold coin that has a mass of 38.6 g? The density of gold is g/cm3. 0.500 cm3 2.00 cm3 38.6 cm3 745 cm3

3.4 Section Quiz 3. As the temperature increases, the density of most substances increases. decreases. remains the same. increases at first and then decreases.

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