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**Scientific Measurement**

Chapter 3

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**Measurements and Their Uncertainty**

3.1

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**Using and Expressing Measurements**

3.1 Using and Expressing Measurements Measurements are fundamental to the experimental sciences. For that reason, it is important to be able to make measurements, and to use them correctly when calculating answers. A measurement is a quantity that has both a number and a unit.

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**Accuracy, Precision, and Error**

Good measurements are accurate, precise, and have low error.

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**Accuracy, Precision, and Error**

3.1 Accuracy, Precision, and Error 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. The more sensitive the instrument (smaller unit), the more precise will be the measurements. Good measurements need to be BOTH accurate and precise.

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**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.

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**Accuracy, Precision, and Error**

3.1 Accuracy, Precision, and Error Error is the difference between the experimental value and the accepted value. The accepted value is the correct value based on reliable references (reference tables). The experimental value is the value measured in the lab.

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**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%. See reference table T. Expressing very large numbers, such as the estimated number of stars in a galaxy, is easier if scientific notation is used.

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Try this: What is a student’s percent error if she found the boiling point of water to be 99.1°C at STP?

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**Significant Figures in Measurements**

Numbers that are part of a measurement are called significant figures. Measurements should always be reported to the correct number of significant figures. Suppose you estimate a weight between 2.4 lb and 2.5 lb to be 2.46 lb. The last digit (6) is an estimate and involves some uncertainty. However, all three digits convey useful information and are called significant figures.

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**Significant Figures in Measurements**

The significant figures in a measurement include all of the digits that are known, plus a last digit that is estimated. Good measurements always estimate one decimal place beyond the calibration of the instrument. In other words, read between the lines!

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**Significant Figures in Measurements**

Sometimes zeroes are just place holders and are used to locate the decimal point. They are NOT a part of the measurement, and are therefore insignificant.

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**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?

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**Significant Figures in Measurements**

When doing calculations with measurements, it becomes necessary to round off the answer. This must be done correctly so as not to imply better instrumentation than was actually used. To round off correctly, you must be able to determine the precision (number of significant figures) in a measurement.

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**Significant Figures in Measurements**

What if you can’t see the instrument used? How do you determine the number of significant figures in a measurement (its precision)?

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**Rules for determining Significant Figures in Measurements**

Nonzero digits are always significant. Leading zeros (occur before any nonzero digit) are never significant. Embedded zeros (between nonzero digits) are always significant. Trailing zeros (after the last nonzero digit) are only significant if the measurement has a decimal point. Counted quantities, physical constants and conversion factors have unlimited significant figures.

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**Significant Figures in Calculations**

When calculating using measurements, the answer should never be more precise than the least precise measurement from which it was calculated. The calculated answer must be rounded to make it consistent with the measurements from which it was calculated.

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**Significant Figures in Calculations**

Rounding calculations depends upon (1) the number of significant figures in the measurements and (2) the mathematical process used to arrive at the answer.

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**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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**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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**The International System of Units**

3.2

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**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.

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**Units and Quantities 3.2 What SI units are commonly used in Chemistry?**

See reference table D

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**Units and Quantities 3.2 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 is more convenient to use a unit of length that has a prefix. See reference table C

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

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**Units and Quantities 3.2 Units of Volume**

The SI unit of volume is the cubic meter (m)3. A more convenient unit of volume for everyday use is the liter, a non-SI unit. 1 mL = 1 cm3 There are 1000 mL in a liter

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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?

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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?

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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?

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**Units and Quantities 3.2 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. A penny has a mass of around 5 grams.

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**Units and Quantities 3.2 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.

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

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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 273 Kelvin (K), and the boiling point is 373 K. The Kelvin scale starts at absolute zero. “Standard Temperature” equals the freezing point of water, 0°C or 273K See reference table A

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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.

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**Units and Quantities 3.2 Units of Energy**

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

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3.2 Units and Quantities The joule (J) is the SI unit of energy. We will use this unit this year. It takes 4.18 joules of heat (or I calorie) to raise the temperature of 1 g of pure water by 1°C. See reference table B

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Conversion Problems 3.3

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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.

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Conversion Factors Chemists often need to convert from one unit of measurement to another. To do this they use conversion factors.

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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.

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

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3.3 Dimensional Analysis Dimensional analysis is a way to analyze and solve problems using the units, or dimensions, of the measurements. Dimensional analysis uses conversion factors to solve problems. Given quantities are multiplied by conversion factors so that units cancel out until the desired unit remains.

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Try this: An exchange student lands on Mars and finds the chemical stockroom. She needs 14 grams of silver (Ag) for a research project. The attendant asks her “How many zooms of silver do you need?” Whoops! Different mass units! The student finds that on Mars the following units are used: woofs, zings, warps and zooms. It turns out 9 woofs = 1 gram. Also, 2 zings = 7 warps. 8 woofs = 3 warps and 4 zooms = 3 zings. How many zooms does the student need?

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Density 3.4

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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.

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**Determining Density Determining Density**

What determines the density of a substance?

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Determining Density Density is the ratio of the mass of an object to its volume. See reference table T

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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?

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

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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.

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**Density and Temperature**

How does a change in temperature affect density?

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**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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