Measurement Measurement Systems Accuracy vs. Precision

Percent Error Significant Digits

Scientific Notation Dimensional Analysis

Measurements Involve QuantitiesQuantity = number + unit

UNITS DO MATTER!!

If you are going “100” could you get a ticket?

What does it depend upon?

SI Units are Universally Accepted

Measure Base Unit

Abbrev.

Length

Mass

Time

Temp

meter

kilogram

second

kelvin

m

kg

s

K

Amount mole mol

Symboll

m

t

T

n

CONVERSION FACTORSIn the real world, it is necessary to convert from one unit to another.

To do this, we must know some equivalents. These are called

“conversion factors!”A conversion factor expresses an equal value for a

measurement but in different units.Example: 12 inches = 1 foot

Metric System Conversion Factors

1012 meters = 1 terameter (Tm)109 meters = 1 gigometer (Gm)106 meters = 1 megameter (Mm)

103 meters = 1 kilometer (km)102 meters = 1 hectometer (hm)10 meters = 1 dekameter (dam)

 

1 meter = 10 decimeters (dm)1 meter = 102 centimeters (cm)1 meter = 103 millimeters (mm)

1 meter = 106 micrometers (mm)1 meter = 109 nanometers (nm)

1 meter = 1012 picometers (pm) 

EASY – ALWAYS BASED ON A FACTOR OF TEN!!!!!!!!

D ISTANCE

Metric System Conversion Factors1012 grams = 1 teragram (Tg)

109 grams = 1 gigogram(Gg)106 grams = 1 megagram (Mg)

103 grams = 1 kilogram (kg)102 grams = 1 hectogram (hg)10 grams = 1 dekagram (dag)

 

1 gram= 10 decigrams (dg)1 gram= 102 centigrams (cg)1 gram = 103 milligrams (mg)

1 gram = 106 micrograms (mg)1 gram = 109 nanograms (ng)

1 gram = 1012 picograms (pg) 

EASY – ALWAYS BASED ON A FACTOR OF TEN!!!!!!!!

MASS

Metric System Conversion Factors1012 liters = 1 teraliter (Tl)109 liters = 1 gigoliter (Gl)106 liters = 1 megaliter (Ml)

103 liters = 1 kiloliter (kl)102 liters = 1 hectoliter (hl)10 liters = 1 dekaliter (dal)

 

1 liter = 10 deciliters (dl)1 liter = 102 centiliters (cl)1 liter = 103 milliliters (ml)

1 liter = 106 microliters (ml)1 liter = 109 nanoliters (nl)

1 liter = 1012 picoliters (pl) 

EASY – ALWAYS BASED ON A FACTOR OF TEN!!!!!!!!

VOLUME

METRICS – EASY!!!BASED ON MULTIPLES OF TEN!!!!!!

KNOW THE PREFIXES AND YOU KNOW THE POWER!!

WHY HAVEN’T AMERICANS SWITCHED???????

WHO KNOWS???? – WE HAVE BEEN TALKING ABOUT IT FOR ABOUT FOUR DECADES, SINCEI WAS A KID!!!

SO THAT WE CAN KEEP OUR SYSTEM AND USE THE ONE THE REST OF THE WORLD USES, WE MUST ALSO KNOW METRIC – ENGLISH CONVERSIONS!!!!!

English to Metric Conversion FactorsLENGTH:1 inch = 2.54 centimeters1 meter = 39.37 inches1 mile = 1.609 kilometers 

MASS:1 ounce = 28.35 grams1 pound = 453.6 grams

1 kilogram = 2.205 pounds1 gram = 15.43 grains

VOLUME:1 quart = 0.946 liters

1 gallon = 3.785 liters1 liter = 33.81 fluid ounces

1 fluid ounce = 29.57 milliliters1 liter = 1.057 quarts

NOT ALWAYS EASY – NOT A FACTOR OF TEN, not as precise!!!

SIGNS OF A GOOD MEASUREMENT:Accuracy - how close a measurement is to the

accepted value

Precision - how close a series of measurements are to each other

ACCURATE = CORRECT

PRECISE = CONSISTENT

ACCURACY VS. PRECISION

Percent Error: A SIGN OF ACCURACYIndicates accuracy of a measurement, how close

you come to an accepted value

100literature

literaturealexperimenterror %

your value

accepted value

Percent Error Sample CalculationProblem: A student determines the

density of a substance to be 1.400 g/mL. Find the % error if the accepted value of the density is 1.360 g/mL.

100g/mL 1.360

g/mL 1.360g/mL 1.400error %

% error = 2.9 %

Significant Figures: A Sign of Precision

Indicate precision of a measurement.

When making scientific measurements, you are allowed to estimate only one digit – THE LAST DIGIT!!!!!!!! All others you must know for certain!

2.35 cm

Examples of Scientific Measurements

ONLY ONE ESTIMATED DIGIT!!!!!

Examples of Scientific Measurements

AGAIN - ONLY ONE ESTIMATED DIGIT!!!!!

RULES FOR MEASURING:REPORT

ALL OF THE DIGITS KNOWN FOR CERTAIN

ESTIMATE ONE DIGIT, THE LAST DIGIT

GIVE THE NUMBERS A LABEL

Rules for Estimating

IF EACH LINE ON THE INSTRUMENT EQUALS

10, YOU CAN ESTIMATE TO THE 11, YOU CAN ESTIMATE TO THE 0.10.1, YOU CAN ESTIMATE TO THE 0.010.01, YOU CAN ESTIMATE TO THE 0.001

Make the MeasurementFOR THIS VOLUME:2.7 IS KNOWN FOR CERTAIN,

NOW ESTIMATE THE NEXT DIGIT,

MAYBE 2.76 mlMAYBE 2.77 ml

BOTH ARE CORRECT!!!!!

Significant Figures: How do we know what digits are significant when looking at a

quantity?Count all digits EXCEPT:

Leading zeros -- 0.0025

Trailing zeros without a decimal point -- 2,500

4. 0.080

3. 5,280

2. 402

1. 23.50

Counting Sig Fig Examples

1. 23.50

2. 402

3. 5,280

4. 0.080100

4 sig figs

3 sig figs

3 sig figs

5 sig figs

Adding and Subtracting Sig FigsThe rule is:

The answer must be rounded to the place value of the least precise measurement.

Example:21.5 ml + 2.03 ml = 23.5 ml

This answer can only have one digit after the decimal point!!!

Multiplying and Dividing Sig FigsThe rule is:

The answer must have no more significant figures than there are in the measurement with

the fewest number of significant figures.Example:

15.82 mL x 2.4 g/mL = 38 g

4 2 2

This answer can only have 2 sig figs.

Practice Problems:23.6 + 43.27 =

45.165 – 44.12 =

23.6 x 33.456 =

23,200 x 33.265 =

(2.315 – 1.72) x 12.22 =

66.87 = 66.9 one digit after the decimal

1.045 = 1.05 two digits after the decimal

789.5616 = 790. three sig figs

771,748 = 772,000 three sig figs

0.60 x 12.22 = 7.3 two sig figs

Scientific Notation: Needed for very large and very small quantities

Converting into Scientific Notation:

Move decimal until there’s one nonzero digit to its left. Places moved = exponent.

Large # (>1) positive exponentSmall # (<1) negative exponent

Only include sig figs.

65,000 kg 6.5 × 104 kg

Scientific Notation

1. 2,400,000 g

2. 0.00256 kg

3. 7 10-5 km

4. 6.2 104 mm

Practice Problems

2.4 x 106 g

2.56 x 10-3 kg

0.00007 km

62,000 mm

Scientific NotationUsing the calculator:

(5.44 × 107 g) ÷ (8.1 × 104 mol) =

5.44EXPEXP

EEEE÷÷

EXPEXP

EEEE ENTERENTER

EXEEXE7 8.1 4

= 671.6049383 = 670 g/mol = 6.7 × 102 g/mol

Type on your calculator:The answer must have two sig figs!!! Multiplication rules!

The Process of Dimensional AnalysisIdentify the known quantity and the unknown quantity.

Find equivalents or conversion factors which will act as ratios to find the unknown unit or units.

Decide on a pathway to solve the problem.

Set up the dimensional analysis grid.

Cancel out units appearing on both the top and bottom, leaving only the unit(s) for the answer.

Multiply the numbers on the top, divide by every number on the bottom.

Report the answer to the proper number of sig figs with the desired label.

Let’s try these!Convert the following:1. 3.53 yards to centimeters2. 0.25 miles to inches3. 4.333 gallons to milliliters4. 2.5 tons to kilograms5. 4500 milligrams to ounces6. 3,345 feet to kilometers7. 3.2 X 106 meters to feet8. 8.520 X 108 micrograms to pounds

Derived UnitsCombinations of base units are called derived units.

Volume (m3 or cm3) = length x length x length

Area (m2 or cm2) = length x length

Density kg/m3 or g/cm3 = mass/volume

Derived UnitsFollow the same protocol in conversion

problems with the following twists:1. If the known quantity is squared or cubed,

the conversion factors must be squared or cubed as well.

2. If the known quantity is a ratio like m/s or kg/l or oz/cm, the denominator unit is now placed in the bottom of the first part of the dimensional analysis grid.

3. To convert the bottom unit, the conversion factors are now flipped to cancel from bottom to top.

Try these!Convert the following:1. 21.3 ft2 to yd2

2. 345 in3 to cm3 3. 1,234 g/cm3 to lb/liter4. 4.355 m/s to km/hr5. 23.4 mg/cm to lb/inch6. 4.2 X 109 picograms/microliter to lb/gallon7. 3.22 X 1012 ml/s to gallon/hr8. 6.5 X 10-7 g/cm2 to ton/mile2

Density DeterminationA. Must measure Mass, Volume, and Possibly LengthMass -- Instrument? __________ Unit? ___________Length -- Instrument? _________ Unit? ___________Volume

OBJECT METHOD MEASURING INSTRUMENT

LIQUID

BLOCK OF WOOD

IRREGULAR SHAPED OBJECT

D. Density CalculationsProblem: An object has a volume of 825 cm3 and a

density of 13.6 g/cm3. Find its mass in milligrams.

GIVEN:

V = 825 cm3

D = 13.6 g/cm3

M = ? mg

WORK:

V

MD

E. More Density CalculationsProblem: A liquid has a density of 0.87 g/mL.

What volume in liters is occupied by 25 g of the liquid?

GIVEN:

D = 0.87 g/mL

V = ? L

M = 25 g

WORK:

V

MD

C. Density by Graphical AnalysisM

ass

(g)

Volume (cm3)

Δx

Δyslope D

V

M