# 2 Standards for Measurement with Tables Careful and 2 Standards for Measurement with Tables Careful and accurate measurements of ingredients are important both when cooking and in the chemistry laboratory! Foundations of College Chemistry, 15th Ed. Morris Hein, Susan Arena, and Cary Willard & Schultz Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Chapter Outline 2.1Scientific Notation 2.2 Measurement and Uncertainty 2.3 Significant Figures A. Rounding Off Numbers 2.4 Significant Figures in Calculations 2.5

A. Addition or Subtraction B. Multiplication or Division The Metric System A. Measurement of Length B. Measurement of Mass C. Measurement of Volume 2.6 Dimensional Analysis: A Problem Solving Method 2.7 Percent 2.8Measurement of Temperature 2.9 Density Copyright 2016 John Wiley & Sons, Inc. All rights reserved. 2.1 Scientific Notation Scientific Notation: A way to write very large or small numbers (measurements) in a compact form. 2.468 x

108 Number written from 110 Raised to a power (-/+ or fractional) Method for Writing a Number in Scientific Notation 1. Move the decimal point in the original number so that it is located after the first nonzero digit. 2. Multiply this number by 10 raised to the number of places the decimal point was moved. 3. Exponent sign indicates which direction the decimal was moved. Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Scientific Notation Practice Write 0.000423 in scientific notation. Place the decimal between the 4 and 2. 4.23 The decimal was moved 4 places so the

exponent should be a 4. The decimal was moved to the right so the exponent should be negative. 4.23 x 104 Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Scientific Notation Practice What is the correct scientific notation for the number 353,000 (to 3 significant figures)? a. 35.3 x 104 b. 3.53 x 105 c. 0.353 x 106 d. 3.53 x 10-5 e. 3.5 x 105 Copyright 2016 John Wiley & Sons, Inc. All rights reserved. 2.2 Measurement and Uncertainty Measurement: A quantitative observation. Examples: 1 cup, 3 eggs, 5 molecules, etc.

Measurements are expressed by 1. a numerical value and 2. a unit of the measurement. Example: 50 kilometers Numerical Value Unit A measurement always requires a unit. Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Measurement and Uncertainty Every measurement made with an instrument requires estimation. Uncertainty exists in the last digit of the measurement because this portion of the numerical value is estimated. The other two digits are certain. These digits would not change in

readings made by one person to another. Numerical values obtained from measurement are never exact values. 21.2 C Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Measurement and Uncertainty Some degree of uncertainty exists in all measurements. By convention, a measurement typically includes all certain digits plus one digit that is estimated. Because of this level of uncertainty, any measurement is expressed by a limited number of digits. These digits are called significant figures. Copyright 2016 John Wiley & Sons, Inc. All rights reserved.

Measurement and Uncertainty a. Recorded as 22.0 C (3 significant figures with uncertainty in the last digit) b. Recorded as 22.11 C (4 significant figures with uncertainty in the last digit) 22.0 C 22.11 C Copyright 2016 John Wiley & Sons, Inc. All rights reserved. 2.3 Significant Figures Because all measurements involve uncertainty, we must be careful to use the correct number of significant figures in calculations. Copyright 2016 John Wiley & Sons, Inc. All rights reserved.

Rules for Counting Significant Figures 1. All nonzero digits are significant. 2. Some numbers have an infinite number of sig figs Ex. 12 inches are always in 1 foot Exact numbers have no uncertainty. 3. Zeroes are significant when: a. They are in between non zero digits Ex. 75.04 has 4 significant figures (7,5,0 and 4) b. They are at the end of a number after a decimal point. Ex. 32.410 has five significant figures (3,2,4,1 and 0) Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Significant Figures Rules for Counting Significant Figures 4. Zeroes are not significant when: a. They appear before the first nonzero digit. Ex. 0.00321 has three significant figures (3,2

and 1) b. They appear at the end of a number without a decimal point. Ex. 6920 has three significant figures (6,9 and 2) When in doubt if zeroes are significant, use scientific notation! Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! How many significant figures are in the following measurements? 3.2 inches 2 significant figures 25.0 grams 3 significant figures

103 people Exact number ( number of sig figs) 0.003 kilometers 1 significant figure Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Rounding Off Numbers With a calculator, answers are often expressed with more digits than the proper number of significant figures. These extra digits are omitted from the reported number, and the value of the last digit is determined by Rules rounding for Rounding off. Off

If the first digit after the number that will be retained is: 1. < 5, the digit retained does not change. Ex. 53.2305 = 53.2 (other digits dropped) digit retained 2. 5, the digit retained is increased by one. Ex. 11.789 = 11.8 (other digits dropped) digit rounded up to 8 Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! Round off the following numbers to the given number of significant figures. 79.137 (four) 79.14 0.04345 (three)

0.0435 136.2 (three) 136 0.1790 (two) 0.18 Copyright 2016 John Wiley & Sons, Inc. All rights reserved. 2.4 Significant Figures in Calculations The results of a calculation are only as precise as the least precise measurement. Calculations Involving Multiplication or Division The significant figures of the answer are based on the measurement with the least number of significant Example figures.

79.2 x 1.1 = 87.12 The answer should contain two significant figures, as 1.1 contains only two significant figures. 79.2 x 1.1 = 87 Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! Round the following calculation to the correct number of significant figures. (12.18) = 4.872 (5.2) 13 a. 4.9 b. 4.87 c. 4.8 d. 4.872 The answer is rounded to 2 sig figs. (5.2 and 13 each contain only 2 sig.

figures) e. 5.0 Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Significant Figures in Calculations The results of a calculation are only as precise as the least precise measurement. Calculations Involving Addition or Subtraction The significant figures of the answer are based on the precision of the least precise measurement. Add 136.23, 79, and 31.7. Example 136.23 79 31.7 246.93 The least precise number is 79, so the answer should be rounded to 247.

Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! Round the following calculation to the correct number of significant figures. 142.57 - 13.0 a. 129.57 b. 129.6 c. 130 d. 129.5 e. 129 142.57 - 13.0 129.57 The answer is rounded to the tenths place. Copyright 2016 John Wiley & Sons, Inc. All rights reserved.

Lets Practice! Round the following calculation to the correct number of significant figures. 12.18 - 5.2 10.1 a. 0.69109 12.18 - 5.2 b. 0.70 6.98 c. 0.693 d. 0.69 The numerator must be rounded to the tenths place.

7.0 = 0.693069 10.1 Final answer is now rounded to 2 significant figures. Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! How many significant figures should the answer to the following calculation contain? 1.6 + 23 0.005 a. 1 b. 2 c. 3 d. 4 1.6 23 0.005 24.595

Round to least precise number (23). Round to the ones place (25). Copyright 2016 John Wiley & Sons, Inc. All rights reserved. 2.5 The Metric System Metric or International System (SI): Standard system of measurements for mass, length, and other quantities. Based ontime standard units physical that change based on factors ofare 10.used to indicate multiples of 10. Prefixes

This makes the metric system a decimal system. Quantity Unit Name Abbreviation Length Meter m Mass Kilogram kg Temperature

Kelvin K Time Second s Amount of Substance Mole mol Electric current Ampere

A Copyright 2016 John Wiley & Sons, Inc. All rights reserved. The Metric System Common Prefixes and Numerical Values for SI Units Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Measurements of Length Meter (m): standard unit of length of the metric system. Definition: the distance light travels in a vacuum during 1/299,792,458 of a second. Common Length Relationships: 1 meter (m) = 100 centimeters (cm) = 1000 millimeters (mm) 1 kilometer (km) = 1000 meters Relationship Between the Metric and English System: 1 inch (in.) = 2.54 cm

Copyright 2016 John Wiley & Sons, Inc. All rights reserved. How to solve a problems just using units A. You must write the following steps in order to get full credit. 1. Write what you know. 2. Write what you dont know. 3. Write a plan on how to get from the known to the unknown. 4. Write the conversion(s) you are going to use. How to solve a problems just using units A. You must write the following steps in order to get full credit. 5. Complete the table a. Draw a table based on the below: 1 column for known 1 column for each conversion

1 column for the unknown Every table has 2 rows How to solve a problems just using units A. You must write the following steps in order to get full credit. 5. Complete the table Known_Unit (Given) Unknown_Units Conversion (Answer) Known_Units Conversion (Given) Unknown_Unit (Answer) How to solve a problems just using units

6. How do you know what goes on top in the conversion column? The units you start with go on the bottom. The units you end with go on the top. Unknown_Units Known_Unit Unknown_Unit Conversion (Answer) (Given) (Answer) Known_Units Conversion (Given) How to solve a problems just using units 7. The first column is always what you start with. 8. The last column is always what you need to end with. Known_Unit

(Given) Unknown_Units Conversion (Answer) Known_Units Conversion (Given) Unknown_Unit (Answer) How to solve a problems just using units 9. Notice the red units drop out, Z*X/Z = X and you are left with the answer. Known_Unit (Given) Unknown_Units Conversion (Answer) Known_Units

Conversion (Given) Unknown_Unit (Answer) How to solve a problems just using units 10. IE: How many inches are in X feet? Known: X feet ?: inches Plan: feet inches Conversion: 1_ft = 12_in How to solve a problems just using units 10: How many inches are in X feet? Known: X feet ?: inches Plan: feet inches Conversion: 1_ft = 12_in

X feet 12 inches 1 feet inches How to solve a problems just using units 10: How many inches are in X feet? The vertical lines mean multiplication The horizontal lines mean division Since its either multiplication or division there is no order of operations! X feet 12 inches 1 feet

(X * 12) inches 2.6 Dimensional Analysis: A Problem Solving Method Dimensional analysis: converts one unit of measure to another by using conversion factors. Conversion factor: A ratio of equivalent quantities. Example: 1 km = 1000 m 1 km Conversion factor: 1000 m or 1000 m 1 km Conversion factors can always be written two ways. Both ratios are equivalent quantities and will equal

1. Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Dimensional Analysis: A Problem Solving Method Any unit can be converted to another unit by multiplying the quantity by a conversion factor. Unit1 x conversion factor = Unit2 Example 2m 1 km x = 0.002 km 1000 m Units are treated like numbers and can cancel. A conversion factor must cancel the original unit and behind only the

new unit. and The leave original unit must be in the(desired) denominator new unit Copyrightnumerator 2016 John Wiley & Sons, Inc.to All rights reserved. must be in the cancel correctly. Dimensional Analysis: A Problem Solving Method Many chemical principles or problems are illustrated mathematically.

A systematic method to solve these types of numerical problems is key. Our approach: the dimensional analysis method Create solution maps to solve problems. Overall outline for a calculation/conversion progressing from known to desired quantities. Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Dimensional Analysis: A Problem Solving Practice Convert 215 centimeters to meters. Solution Map: cm m desired quantity known quantity 1m 215 cm x = 2.15 100 cm

m Convert 125 meters to kilometers. Solution Map: known m km desired quantity quantity 1 km = 0.125 km 125 m x 1000 m Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! How many micrometers are in 0.03 meters? a. 30,000 b. 300,000 c. 300 d. 3000

Solution Map: known quantity 0.03 m x m mm desired quantity 1,000,000 m = 30,000 1m m Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Dimensional Analysis: A Problem Solving Method

Some problems require a series of conversions to get to the desired unit. Each arrow in the solution map corresponds to the use of a conversion factor. Example Convert from days to seconds. Solution Map: days hours minutes seconds 24 60 60 1 day x x x

= 8.64 x 104 sec hour second 1 day 1minute hour 1 minute s s reserved. Copyright s 2016 John Wiley & Sons, Inc. All rights Dimensional Analysis: A Problem Solving Practice Metric to English Conversions How many feet are in 250 centimeters? Solution Map: cm inches

ft 1 foot 1 inch 250 cm x = 8.20 ft x 2.54 cm 12 inches Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! Metric to English Conversions How many meters are in 5 yards? a. 9.14 b. 457 c. 45.7 d. 4.57 yards 5 yardsx

Solution Map: feet inches cm m 3 feet 1m 12 inches 2.54 x x x = 4.57 1 yard 1 foot 100 cm 1cm

inch m Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! Metric to English Conversions How many cm3 are in a box that measures 2.20 x 4.00 x 6.00 inches? Solution Map: (in cm)3 = 52.8 2.20 in x 4.00 in x 6.00 in in3 52.8 in3 x 2.54 cm 3 = 865 1 in

cm3 Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Measurement of Mass Mass: amount of matter in an object Mass is measured on a balance. Weight: effect of gravity on an object. Weight is measured on a scale, which measures force against a spring. Mass is independent of location, but weight is not. Mass is the standard measurement of the metric system. The SI unit of mass is the kilogram. (The gram is too small a unit of mass to be the standard unit.) Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Measurement of Mass 1 kilogram (kg) is the mass of a Pt-Ir cylinder standard.

Metric to English Conversions 1 kg = 2.2015 pounds (lbs) Metric Units of Mass Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! Convert 343 grams to kilograms. Solution Map: g kg Use the new conversion factor: 1 kg 1000 g 343 g x or 1 kg

1000 g 1000 g 1 kg = 0.343 kg Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! How many centigrams are in 0.12 kilograms? a. 120 b. 1.2 x 104 c. 1200 d. 1.2 Solution Map: kg 0.12 kg x

1000 g x 1 kg g 100 cg 1g cg = 1.2 x 104 cg Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Measurement of Volume Volume: the amount of space occupied by matter. The SI unit of volume is the cubic meter (m 3) The metric volume more typically used is the liter (L) or milliliter (mL). A liter is a cubic decimeter of water (1 kg) at 4 C. Volume can be

measured with several laboratory devices. Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Measurement of Volume Common Volume Relationships 1 L = 1000 mL = 1000 cm3 1 mL = 1 cm3 1 L = 1.057 quarts (qt) Volume Problem Convert 0.345 liters to milliliters. Solution Map: L 0.345 L x mL 1000 mL = 345 1L

mL Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! How many milliliters are in a cube with sides measuring 13.1 inches each? a. 3690 Solution Map: b. 3.69 in. cm cm3 Convert from mL inches to cm: c. 369

2.54 cm 1 in. = 33.3 cm Determine the volume of the cube: d. 3.69 x 10 4 13.1 in. x Volume = (33.3 cm) x (33.3 cm) x (33.3 = 3.69 x 104 cm3 cm) Convert to the proper units: 1 mL 4 3.69 x 104 cm3

= 3.69 x 10 x 1 cm3 mL Copyright 2016 John Wiley & Sons, Inc. All rights reserved. 2.7 Percent The composition of many mixtures is often given in percent. Percent can be defined as parts per 100 percent x parts = x

100 total parts where x equals 100 If we do not have 100 parts then you must convert to parts per 100 Use the formula percent = parts x 100% total parts Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Percent In a genetics experiment there are 25 red flowers, 33 yellow flowers and 22 white flowers. What is the percentage of red

flowers? Solve for percent Use the formula 100% percent = parts x total parts What is the total number flowers? 25 + 33 + 22 = 75 total Percent red flowers = 33% 25 red x 100% =

75 total Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Mass percent In chemistry we often use mass percent Use the formula mass percent = mass part mass total x 100% Since the same units cancel out any mass units can be used in the formula Copyright 2016 John Wiley & Sons, Inc. All rights reserved.

Mass percent A sample of nickel oxide is composed of 14.00g nickel and 7.64g oxygen. Calculate the percentage of nickel and oxygen. Use the formula mass percent = mass part x 100% mass total a. 14%O 7.64%Ni b. 65%O 35%Ni Total mass = 14.00g + 7.64g = 21.64g c. 35% O 65%Ni d. 53%O 47%Ni

%Ni = 14.00gNi x 100% = 65%Ni 21.64g total %O 7.64gO x 100% = 35%O 21.64g total = The total of the masses should equal 100% Copyright 2016 John Wiley & Sons, Inc. All rights reserved. 2.8 Measurement of Temperature Thermal energy: A form of energy involving the

motion of small particles of matter. Temperature: measure of the intensity of thermal energy of a system (i.e. how hot or cold). Heat: flow of energy due to a temperature difference. Heat flows from regions of higher to lower temperature. The SI unit of temperature is the Kelvin (K). Temperature is measured using a thermometer. Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Different Temperature Scales Temperature can be expressed in 3 commonly used scales. Celsius (C), Fahrenheit (F), and Kelvin (K). Celsius and Fahrenheit are both measured in degrees, but the scales are different. H2O

C F K Freezing Point 0 C 32 F 273.15 K Boiling Point 100 C 212 F

373.15 K The Fahrenheit scale has a range of 180 between freezing and boiling. The lowest temperature possible on the Kelvin scale is absolute zero (-273.15 C). Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Converting Between Temperature Scales Mathematical Relationships Between Temperature Scales K = C + 273.15 F = 9/5(C) + 32 Temperature Problem Convert 723 C to temperature in both K and F. Solution Map: C K

K = 723 + 273.15 = 996 K C F F = 9/5(723) + 32 = 1333 F Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! What is the temperature if 98.6 F is converted to C? Solution Map: a. 37 F C b. 371 98.6 = 9/5(C) + 32

c. 210 98.6 - 32 = 9/5(C) d. 175 66.6 = 9/5(C) C = (5/9)(66.6) = 37 C Copyright 2016 John Wiley & Sons, Inc. All rights reserved. 2.9 Density Density (d): the ratio of the mass of a substance to the volume occupied by that mass. mass d = volum e Density is a physical property of a substance. The units of density are generally expressed as g/mL or 3

g/cm for solids and liquids andas g/L for gases. The volume of a liquid changes a function of temp, so density specified for g/mL a given Ex. The densitymust of H2be O at 4 C is 1.0

while temperature. the density is 0.97 g/mL at 80 C. Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Density Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Density: Specific Gravity Specific gravity (sp gr): ratio of the density of a substance to the density of another substance (usually H 2O at Specific gravity is unit-less 4 C).(in the ratio all units cancel). An important measurement of proper kidney function is the kidneys ability to concentrate urine as measured by specific gravity. What is the SG of a sample of urine with a density of 1.031g/ mL?

Specific Gravity = density of sample_____ density of water at 4oC Solve = 1.031 Specific Gravity = 1.031g/mL 1.000g/mL Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! Calculate the density of a substance if 323 g occupy a volume of 53.0 mL. Solution: mass d = volum e 323 g = 6.09 53.0 mL g/mL

Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! The density of gold is 19.3 g/mL. What is the volume of 25.0 g of gold? Solution Map: Use density as a conversion factor! g Au 25.0 g x mL Au 1 mL 19.3 g = 1.30 mL Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Lets Practice! What is the mass of 1.50 mL of ethyl alcohol? (d = 0.789 g/mL at 4 C)

a. 1.90 g Solution Map: b. 1.18 g c. 0.526 g d. 2.32 g e. 1.50 g mL 1.50 mL x g 0.789 g 1 mL Copyright 2016 John Wiley & Sons, Inc. All rights reserved. = 1.18 g Learning Objectives 2.1 Scientific Notation

Write decimal numbers in scientific notation. 2.2 Measurement and Uncertainty Explain the significance of uncertainty in measurements in chemistry and how significant figures are used to indicate a measurement. 2.3 Significant Figures Determine the number of significant figures in a given measurement and round measurements to a specific number Copyright 2016 John Wiley & Sons, figures. Inc. All rights reserved. of significant Learning Objectives 2.4 Significant Figures in Calculations Apply the rules for significant figures in calculations

involving addition, subtraction, multiplication, and division. 2.5 The Metric System Name the units for mass, length, and volume in the metric system and convert from one unit to another. 2.6 Dimensional Analysis: A Problem Solving Method Use dimensional analysis to solve problems involving unit conversions. Copyright 2016 John Wiley & Sons, Inc. All rights reserved. Learning Objectives 2.7 Percent Solve problems involving percent. 2.8 Measurement of Temperature Convert measurements among the Fahrenheit, Celsius and Kelvin temperature scales. 2.9 Density