Measurements and their uncertainty

Objectives

Convert measurements to scientific notation

Distinguish among accuracy, precision and error of measurement

Determine the number of significant figures in a measurement and in a calculated answer

Measurements

Definition – quantity that has both a number and a unit

Measurements are made every day:Buying productsSports activitiesCooking

Measurement types

2 types of measurementsQualitative – measurements are words, not numbers – hot, heavy

Quantitative – measurements involve numbers and depends on: Reliability of measuring instrument

The care with which it is read

Scientific Notation

Number is written as the product of two numbers:A coefficient10 raise to a power

Example:602,000,000,000,0000,0000,000,000

6.02 x 1023

Accuracy, Precision, and Error

Accuracy – how close measurement is to true valueMeasured value must be compared to correct value

Precision – how close the measurements are to each otherCompare values of two or more repeated measurements (reproducible)

Precision vs. Accuracy

Determining Error

Accepted value – correct value based on reliable referencesExample: Density table pg 90

Experimental value – value measure in the lab

Determining Error

Error – difference between experimental value and accepted valueError = exp. value – accepted value

Error can be either positive or negative

Percent Error

Percent error – absolute value of error divided by the accepted value multiplied by 100

Percent error = |error| accepted value

x 100

Sample Problem

For a boiling point measurement of water in the lab you read 99.1 oC. What is the percent error?

Sample Problem

For a boiling point measurement of water in the lab you read 99.1 oC. What is the percent error?Answer: 0.9%

Why Is There Uncertainty?

Measurements are performed with instrumentsNo instrument can be read to an infinite number of decimal places

Why Is There Uncertainty?

Which of these has the greatest uncertainty?

Significant Figures

Significant digits – in a measurement it includes all of the digits that are unknown plus a last digit that is estimated

Measurements are reported in correct sig figs b/c calculated answers depend on number of sig figs in the values used in the calculations

Figure 3.5 Significant Figures - Page 67

Which measurement is the best?

Sig Fig Rules

Non – zeros always count as significant figures:

3456

4 sig figs

Sig Fig Rules

ZerosLeading zeros do not count as significant figures

0.0486

3 sig figs

Sig Fig Rules

ZerosCaptive zeros always count as significant figures

16.07

4 sig figs

Sig Fig Rules

ZerosTrailing zeros are significant only if the number contains a decimal point

9.300 9300

4 sig figs 2 sig figs

Sig Fig Rules

Two special situations – have unlimited number of sig figsCounted items

23 people, 435 thumbtacksExactly defined quantities

60 minutes = 1 hour

Sig Fig Practice

How many sig figs in the following?1.0070 m17.10 kg100890 L3.29 x 103 s0.0054 cm3,200,000 mL5 dogs

Sig Figs in Calculations

A calculated answer cannot be more precise than the least precise measurement from which it was calculatedThe chain is only as strong as its weakest link

Calculated values need to be rounded

Rounding

Decide how many sig figs are needed

Round to that many digits counting from the left!Is the next digit <5? Drop itIs the next digit >5? Increase by 1

Problems

Round off each measurement to the number of sig figs shown in ( ). Write the answers in scientific notation.314.721 (four)0.001755 (two)8792 (two)

Rounding – Addition and Subtraction

Answer should be rounded to same number of decimal places as the LEAST number of decimal places in problem

Problem:Calculate the sum:

12.52 m + 349.0 m + 8.24 m

Rounding – Addition and Subtraction

Answer should be rounded to same number of decimal places as the LEAST number of decimal places in problem

Problem:Calculate the sum:

12.52 m + 349.0 m + 8.24 m369.8 m

Rounding – Multiplication and Division

Round the answer to the same number of significant figures as the LEAST number of sig figs in the problem

How many sig figs for the following operations:7.55 x 0.34 meters2.10 x 0.70 meters2.4526/8.4

Rounding – Multiplication and Division

Round the answer to the same number of significant figures as the LEAST number of sig figs in the problem

How many sig figs for the following operations:7.55 x 0.34 meters 2 sig figs2.10 x 0.70 meters 2 sig figs2.4526/8.40 3 sig figs

Section Assessment

A technician experimentally determined the boiling point of octane to be 124.1 oC. The actual BP is 125.7 oC. Calculate the error and percent error.

Section Assessment

Determine the number of sig figs in the following:11 soccer players10,800 meters0.070020 meters5.00 cubic meters

The International System of Units

Objectives

List SI units of measurement and common SI prefixes

Distinguish between the mass and weight of an object

Convert between Celsius and Kelvin temperature scales

International System of Units

SI – from French nameMeasurements depend on units that are used as reference standards

In chemistry we use metric system

5 SI Base Units

SI units normally used in chemistry:Quantity SI base unit Symbol

Length Meter m

Mass Kilogram kg

Temperature Kelvin K

Time Seconds s

Amount of substance

Mole mol

Nature of Measurements

Measurement – quantitative observation consisting of 2 parts:NumberUnit

Example: 20 grams or 20 g

SI

Non-SI units are used in chemistry as well:Liter – volumeCelsius – temperatureCalorie - heat

Common SI Prefixes

Prefix Abbreviation Meaning ExponentMega- M Million 106

Kilo- k thousand 103

Deci- d tenth 10-1

Centi- c hundredth 10-2

Milli- m thousandth 10-3

Micro- millionth 10-6

Nano- n billionth 10-9

Pico- P trillionth 10-12

Units of Length

SI unit – meterMeasured using rulersMost common metric units of length are:Centimeter – cmMeter – mKilometer - km

Volume

Space occupied by sample of matter

Calculated from a solid by multiplying length x width x heightThus, SI is cubic meter or cm3

We normally use liters for volume1 mL = 1 cm3

Measuring Volume

Graduated cylinderPipetsBuretsVolumetric flaskSyringes

Volume Changes

With an increase in temperature, the volume also increasesSeen most in gases

Instruments are calibrated for 20 oC which is room temperature

Units of Mass

Mass – measure of the quantity of matter presentWeight – force that measures the pull by gravity – changes with location

Measuring Mass

SI unit for mass is kg – we commonly use grams in the laboratory

Mass is measured using a triple beam balance (or electric balance)

Temperature

Measure of how hot or cold something is

Heat moves from object of higher temperature to object of lower temp

2 units used:KelvinCelsius

Temperature

Celsius scale defined by two readily determined temperatures:Freezing point of water = 0 oCBoiling point of water = 100 oC

Kelvin scale does not use the degree sign, but is just represented by K• absolute zero = 0 K (thus no negative values)

• formula to convert: K = oC + 273

Sample Problem

Normal human body temperature is 37 oC. What is the temperature in Kelvin?

Energy

Energy is the capacity to do work or produce heat

Energy can be measured using:Joule (J) – the SI unit for energyCalorie (cal) – the heat needed to raise 1 gram of water by 1 oC

Energy

Conversions between Joules and Calories can be carried out by using the following relationship:

1 cal = 4.184 J

Section Assessment

Name the quantity measured by each of the SI base units and give the SI symbol of the unit.

Conversion Problems

Objectives

Construct conversion factors from equivalent measurements

Apply the technique of dimensional analysis to a variety of conversion problems

Solve problems by breaking the solution into steps

Convert complex units, using DA

Conversion Factors

A “ratio” of equivalent measurements

Start with 2 things that are the same:1 meter = 100 centimeters

Practice

Write 2 conversion factors for the following:Between kilograms and gramsBetween feet and inchesUsing 1.096 qt = 1.00 L

Why Use Conversion Factors?

We can multiply them to change the units

Question: 13 inches is how many yards?We know that 36 inches = 1 yard

Dimensional Analysis

A way to analyze and solve problems by using units of measurement

Dimension = a unit Analyze = to solve

Using units so solve the problems

Dimensional Analysis

DA provides alternative approach to problem solving, instead of with equation or algebra.

A ruler is 12.0 inches long. How long is it in cm? ( 1 inch = 2.54 cm)

How long is this in meters?A race is 10.0 km long. How far is this in miles, if: 1 mile = 1760 yards 1 meter = 1.094 yards

Problem

How many minutes are there in a week?

Problem

An experiment requires that each student use an 8.5-cm length of magnesium ribbon. How many students can do the experiment if there is a 570-cm length of magnesium ribbon available?

Problem

Convert:0.044 km to m4.6 mg to g0.107 g to cg

Complex Units

Complex units – units that are expressed as a ratio of two unitsSpeed – meters/hour

Sample problem: Change 15 m/hr to cm/s

Density

Objectives

Calculate the density of a material from experimental data

Describe how density varies with temperature

Density

Density – relationship between mass and volume

Formula Density = mass

volumeCommon units: g/mL or g/cm3

Density

If you recall, density is a physical property so it does not depend on sample size

Density and Temperature

What happens to density as the temperature of an object increases?Mass remains the sameIncrease in volume as temperature increases

Density DECREASES as temperature increases

Density and Water

Water is an exception to the density rule

Over certain temperatures, the volume of water increases as the temperature decreasesDoes ice float in liquid water?Why?

Section Assessment

A 68 g bar of gold is cut into 3 equal pieces. How does the density of each piece compare to the density of the original gold bar?

Section Assessment

What is the volume, in cubic centimeters, of a sample of cough syrup that has a mass of 50.0 g? The density of cough syrup is 0.950 g/cm3