# Measurements and Calculations Chapter 2 2.

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Measurements and Calculations Chapter 2 2

Measurement Quantitative Observation
Comparison Based on an Accepted Scale e.g. Meter Stick Has 2 Parts – the Number and the Unit Number Tells Comparison Unit Tells Scale 2

Scientific Notation Technique Used to Express Very Large or Very Small Numbers Based on Powers of 10 3

Writing Numbers in Scientific Notation
1. Move the decimal point so there is only one non-zero number to the left of it. The new number is now between 1 and 9 2. Multiply the new number by 10n where n is the number of places you moved the decimal point 3. Determine the sign on the exponent n If the decimal point was moved left, n is + If the decimal point was moved right, n is – If the decimal point was not moved, n is 0 4

Writing Numbers in Standard Form
Determine the sign of n of 10n If n is + the decimal point will move to the right If n is – the decimal point will move to the left Determine the value of the exponent of 10 Tells the number of places to move the decimal point Move the decimal point and rewrite the number 5

Standard to Scientific Notation
75,000,000 8,031,000,000

Scientific to Standard Notation
2.75 x 10-7 5.22 x 104 7.10 x 10-5 9.38 x 1012

More practice Change to scientific notation 41080.642 1.8732
Change to standard notation x 106 391 x 10-2

Related Units in the Metric System
All units in the metric system are related to the fundamental unit by a power of 10 The power of 10 is indicated by a prefix The prefixes are always the same, regardless of the fundamental or basic unit 6

Length SI unit = meter (m) About 3½ inches longer than a yard
1 meter = one ten-millionth the distance from the North Pole to the Equator = distance between marks on standard metal rod in a Paris vault = distance covered by a certain number of wavelengths of a special color of light Commonly use centimeters (cm) 1 m = 100 cm 1 cm = 0.01 m = 10 mm 1 inch = 2.54 cm (exactly) 7

Figure 2.1: Comparison of English and metric units for length on a ruler.

Volume Measure of the amount of three-dimensional space occupied by a substance SI unit = cubic meter (m3) Commonly measure solid volume in cubic centimeters (cm3 (cm x cm x cm)) 1 m3 = 106 cm3 1 cm3 = 10-6 m3 = m3 Commonly measure liquid or gas volume in milliliters (mL) 1 L is slightly larger than 1 quart 1 L = 1 dL3 = 1000 mL = 103 mL 1 mL = L = 10-3 L 1 mL = 1 cm3 8

Figure 2.3: A 100-mL graduated cylinder.

Mass Measure of the amount of matter present in an object
SI unit = kilogram (kg) Commonly measure mass in grams (g) or milligrams (mg) 1 kg = pounds, 1 lbs.. = g 1 kg = 1000 g = 103 g, 1 g = 1000 mg = 103 mg 1 g = kg = 10-3 kg, 1 mg = g = g 9

Figure 2.4: An electronic analytical balance used in chemistry labs.

Metric conversions 250 mL to Liters 1.75 kg to grams 88 daL to mL

Metric conversions 475 cg to mg 328 hm to Mm nL to cL

Uncertainty in Measured Numbers
A measurement always has some amount of uncertainty Uncertainty comes from limitations of the techniques used for comparison To understand how reliable a measurement is, we need to understand the limitations of the measurement 10

Reporting Measurements
To indicate the uncertainty of a single measurement scientists use a system called significant figures The last digit written in a measurement is the number that is considered to be uncertain Unless stated otherwise, the uncertainty in the last digit is ±1 11

Rules for Counting Significant Figures
Nonzero integers are always significant How many significant figures are in the following examples: 2753 89.659 0.281 12

Significant Figures Zeros
Captive zeros are always significant How many significant figures are in the following examples: 1001.4

Significant Figures Zeros
Leading zeros never count as significant figures How many significant figures are in the following examples:

Significant Figures Zeros
Trailing zeros are significant if the number has a decimal point How many significant figures are in the following examples: 22,000 63,850. 100,000

Significant Figures Scientific Notation
All numbers before the “x” are significant. Don’t worry about any other rules. 7.0 x 10-4 g has 2 significant figures 2.010 x 108 m has 4 significant figures

Rules for Rounding Off Round 87.482 to 4 sig figs.
If the digit to be removed is less than 5, the preceding digit stays the same Round to 4 sig figs. is equal to or greater than 5, the preceding digit is increased by 1 Round to 3 sig figs. 13

Rules for Rounding Off In a series of calculations, carry the extra digits to the final result and then round off Ex: Convert 80,150,000 seconds to years Don’t forget to add place-holding zeros if necessary to keep value the same!! Round 80,150,000 to 3 sig figs.

Multiplication/Division with Significant Figures
Count the number of significant figures in each measurement Round the result so it has the same number of significant figures as the measurement with the smallest number of significant figures cm x cm = 3.7 x 103 x = 16

Calculations with Significant Figures
Calculators/computers do not know about significant figures!!! Exact numbers do not affect the number of significant figures in an answer Answers to calculations must be rounded to the proper number of significant figures round at the end of the calculation 15

Exact Numbers Exact Numbers are numbers known with certainty
Unlimited number of significant figures They are either counting numbers number of sides on a square or defined 100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm 1 kg = 1000 g, 1 LB = 16 oz 1000 mL = 1 L; 1 gal = 4 qts. 1 minute = 60 seconds 14

Problem Solving and Dimensional Analysis
Many problems in chemistry involve using equivalence statements to convert one unit of measurement to another Conversion factors are relationships between two units Conversion factors generated from equivalence statements e.g. 1 inch = 2.54 cm can give or 18

Problem Solving and Dimensional Analysis
Arrange conversion factor so starting unit is on the bottom of the conversion factor You may string conversion factors together for problems that involve more than one conversion factor. 19

Converting One Unit to Another
Find the relationship(s) between the starting and final units. Write an equivalence statement and a conversion factor for each relationship. Arrange the conversion factor(s) to cancel starting unit and result in goal unit. 20

Converting One Unit to Another
Check that the units cancel properly Multiply all the numbers across the top and divide by each number on the bottom to give the answer with the proper unit. Round your answer to the correct number of significant figures. Check that your answer makes sense! 21

English Units Conversions
28.5 inches to feet 4.0 gallons to quarts 48.39 minutes to hours 155.0 pounds to grams 2.00 x 108 seconds to hours

More Difficult Conversions
682 mg to pounds 3.5 x 10-4 L to cm3 0.091 ft2 to inches2 47.1 mm3 to kL

Complex Conversion Problems
25 miles per hour to feet per second 4.70 gallons per minute to mL per year 5.6 x 10-6 centiliters per square meter (cL/m2) to cubic meters per square foot (m3/ft2)

Temperature Scales Fahrenheit Scale, °F Celsius Scale, °C
Water’s freezing point = 32°F, boiling point = 212°F Celsius Scale, °C Temperature unit larger than the Fahrenheit Water’s freezing point = 0°C, boiling point = 100°C Kelvin Scale, K Temperature unit same size as Celsius Water’s freezing point = 273 K, boiling point = 373 K 22

Temperature Conversions
Fahrenheit to Celsius oC = 5/9(oF -32) Celsius to Fahrenheit oF = 1.8(oC) +32 Celsius to Kelvin K = oC + 273 Kelvin to Celsius oC = K – 273

Figure 2.6: Thermometers based on the three temperature scales in (a) ice water and (b) boiling water.

Figure 2.7: The three major temperature scales.

Figure 2.8: Converting 70. 8C to units measured on the Kelvin scale.

Figure 2.9: Comparison of the Celsius and Fahrenheit scales.

Temperature Conversion Examples
86oF to oC -5.0oC to oF 352 K to oC 12oC to K 248 K to oF 98.6oF to K

Density Density is a property of matter representing the mass per unit volume For equal volumes, denser object has larger mass For equal masses, denser object has small volume Solids = g/cm3 1 cm3 = 1 mL Liquids = g/mL Gases = g/L Volume of a solid can be determined by water displacement Density : solids > liquids >>> gases In a heterogeneous mixture, denser object sinks 23

Using Density in Calculations
24

Spherical droplets of mercury, a very dense liquid.

Density Example Problems
What is the density of a metal with a mass of g whose volume occupies cm3? What volume, in mL, of ethanol (density = g/mL) has a mass of 2.04 lbs? What is the mass (in mg) of a gas that has a density of g/L in a 500. mL container?

Figure 2. 10: (a) Tank of water
Figure 2.10: (a) Tank of water. (b) Person submerged in the tank, raising the level of the water.

Volume by displacement
To determine the volume to insert into the density equation, you must find out the difference between the initial volume and the final volume. A student attempting to find the density of copper records a mass of 75.2 g. When the copper is inserted into a graduated cylinder, the volume of the cylinder increases from mL to 58.5 mL. What is the density of the copper in g/mL?

Percent Error Percent error – absolute value of the error divided by the accepted value, multiplied by 100%. % error = measured value – accepted value x 100% accepted value Accepted value – correct value based on reliable sources. Experimental (measured) value – value physically measured in the lab.

Percent Error Example In the lab, you determined the density of ethanol to be 1.04 g/mL. The accepted density of ethanol is g/mL. What is the percent error? The accepted value for the density of lead is g/cm3. When you experimentally determined the density of a sample of lead, you found that a 85.2 gram sample of lead displaced 7.35 mL of water. What is the percent error in this experiment?

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