D imensional Analysis Conversions

Dimensional Analysis ConversionsWith Significant Digits!

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Metric Prefixes ReviewPREFIX ABBREVI

ATIONMEANS ONE LARGER

EQUALSMANY SMALLER

Pico p 1 X 10-12 1 gram = 1,000,000,000,000 picograms

Nano n 1 X 10-9 1 gram = 1,000,000,000 nanogramsMicro μ 1 X 10-6 1 gram = 1,000,000 microgramsMilli m 1 X 10-3 1 gram = 1,000 milligramsCenti c 1 X 10-2 1 gram = 100 centigramsDeci d 1 X 10-1 1 gram = 10 decigramsBASE UNIT m, g, L,

etc.1

Deka da 1 X 101 1 dekagram = 10 gramsHecto h 1 X 102 1 hectogram = 100 gramsKilo k 1 X 103 1 kilogram = 1000 gramsMega M 1 X 106 1 megagram = 1,000,000 grams

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Other Metric Mentionables• For ALL substances: 1 mL = 1cm3

• Water is special:• 1 gram of has a volume of 1 ml or 1 cm3 • 1 g water = 1mL water• 1 g water =1 cm3 water

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Let’s Do It!!!

• Write the numbers 1-5 on your paper.• Write which of these values is larger:• 1. A microliter or a liter• 2. A kilogram or a gram• 3. A centiliter or a liter• 4. A megasecond or a second• 5. A decigram or a gram

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Let’s Do It!!!

• Write the numbers 1-5 on your paper.• Write which of these values is larger:• 1. A microliter or a liter• 2. A kilogram or a gram• 3. A centiliter or a liter• 4. A megasecond or a second• 5. A decigram or a gram

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Let’s Do It!!!

• Write the numbers 1-5 on your paper.• How many ____________ are in a

____________:• 1. micrograms in a gram• 2. meters in a kilometer• 3. centiliters in a liter• 4. seconds in a megasecond• 5. decigrams in a gram

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Let’s Do It!!!

• Write the numbers 1-5 on your paper.• How many ____________ are in a

____________:• 1. micrograms in a gram 1,000,000 or 1X106

• 2. kilometers in a meter 1000 or 1X10-3

• 3. centiliters in a liter 100 or 1X102

• 4. megaseconds in a second 1,000,000 or 1X10-6

• 5. decigrams in a gram 10 or 1X101

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Conversions

• How many kilometers are in 50 meters?• This can get a little hairy!

• How many inches in a meter?• Now we are trying to relate English units and

metric units! Even more confusing!• This is where we will use a technique called ……. (drum roll please)• DIMENSIONAL ANALYSIS!!!!!!!!

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Dimensional Analysis What is it?• A problem solving method• A series of multiplications and divisions• Units are also multiplied and divided (units

can cancel)•We multiply by fractions called

“conversion factors”• Conversion factor- a statement of fact

expressed as a fraction equaling one that helps us convert from one unit to another

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Conversion Factors• All conversion factors are equal to 1• Conversion factors can be made from• English-English relationships • 1 foot = 12 inches

• Metric-Metric relationships• 1 km = 1x103m

• English-Metric relationships• 1.00 in = 2.54 cm

• Conversion factors can be made from information given in a problem

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Equal to 1???

122

Conversion factors are made from equivalents

2 = 2 X = X

1xx

4 - 2 = 1 + 1

11124

1 foot = 12 inches 1inches 12foot 1

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Equivalent

1 foot = 12 inches

1 km = 1 x 103 m

Conversion Factor

inches 12foot 1

foot 1inches 21

m310 x 1km 1

km 110 x 1 3 m

1 cm = 1 x 10-2 m m2-10 x 1cm 1

cm 110 x 1 -2 m

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Metric Conversions• Fact: 1 kilometer = 1000 meters from our table• This is a conversion factor and we can write it 2 ways: a. 1 km or b. 1000 m 1000 m 1 km

What do these mean? The exact same thing!a. That there is 1 km PER 1000 m or 1 km = 1000mb. That there are 1000 m PER 1 km or 1000m = 1 kmBoth the top and bottom are equal so the whole conversion factor is equal to one so we can use it without randomly changing values

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Metric Conversions• So…..• How many kilometers ARE there in 50 meters?• We can use a conversion factor that relates km to

meters, but which one? a. 1 km or b. 1000 m 1000 m 1 km

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Question Mark Format-Reference

• We follow some steps and the UNITS WILL TELL US WHAT TO DO.• Follow this QUESTION MARK FORMAT, when

doing dimensional analysis! Even if you can solve it in your head, these questions will get a LOT tougher!• FOLLOW THE METHOD!!!!• DID I SAY FOLLOW THE METHOD??????

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Question Mark Format-Reference

• PSSSSTTTT. FOLLOW THE METHOD• 1. Start every problem with the QUESTION MARK

FORMAT (the “?” means “how many?”)• 2. Cancel the units• 3. Insert fact• 4. Do the math (multiply across and divide top by

bottom just like fractions)How many kilometers in 50 meters?• ? km = 50 m 16

Metric Conversions



bottom just like fractions)How many kilometers in 50 meters?• ? km = 50 m x ____km__ m

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Metric Conversions



bottom just like fractions)How many kilometers in 50 meters?• ? km = 50 m x _____ km__ m

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Metric Conversions



bottom just like fractions)How many kilometers in 50 meters?• ? km = 50 m x __1__km__ 1000 m

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Metric Conversions



bottom just like fractions)How many kilometers in 50 meters?• ? km = 50 m x __1__km_ = 50 X 1 km = ? 1000 m 1000

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• Convert 7.0 feet to inches1 ft = 12 in

12*1

inft

84in7.0 ft

121

inft

112

ftin

? in =

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Reference

• Convert 2.89 hg to g

2.89hg

1hg=1*102g

289g21*10*

1g

hg? g =

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Reference

• Convert 2.89 Mg to cg

2.89Mg 289,000,000cg2

1*1*10

cgg

61*10*1

gMg

1Mg=1*106g1cg=1*10-2g

? cg =

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Reference

Let’s Do It!!!!

• 1. How many kilograms are in a 350 grams?

• 2. I have 45 milliliters. How liters is this?

• 3. Superchallenge!!! How many kilometers are in a millimeter? You have to use two conversion factors!

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Metric ConversionsMetric video 8 min• One last important step:• CHECK SIGNIFICANT DIGITS!• Any fact that is found in a conversion factor is

not a measurement and so is ignored when calculating the number of significant digits.

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http://education-portal.com/academy/lesson/the-metric-system-units-and-conversion.html

• Convert 4.756 dm to yd

.521yd1*12

ftin

39.4*1.00

inm

11*10*1

mdm

4.756dm

measured definition measured definition definition 4 SD NO SD 3 SD NO SD NO SD

1dm=1*10-1m1.00m=39.4in

12in=1ft3ft=1yd

1*3ydft

? yd =

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Reference

How many pounds are in a gallon of water?

1 g water = 1 mL

8.33lb1.00*454

lbg

1*1

gmL

946*1.00

mLqt

4*1

qtgal

1.00gal

? lb =

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Dimensional Analysis with Multiple Units- Reference

Strategy:1. Convert each unit in the fraction separately

2. Make sure units cancel!

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Example 1:Convert 70.0 mi/hr to km/sec

.0313seckm

1min*60sec

1*60min

hr3

1*1 10

kmx m

1.00*39.4

min

12*1

inft

5280*1

ftmi

70.0mihr

1 mi = 5280 ft1 ft = 12 in

39.4in = 1.00m1x103m = 1km1hr = 60 min

1 min = 60 sec? km/sec =

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Reference

Example 2:• Convert 15.3 g/L to lb/gal

4*1

qtgal

.128 lbgal

.946*1.00

Lqt

15.3gL

1.00*454

lbg

454g = 1.00lb.946L = 1.00qt

4qt = 1gal

? lb/gal =

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Let’s Do It!!!!

• 1. How many Megagrams are in a pound?

• 2. I have 45 liters. How many cm3 is this?

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ReviewDimensional Analysis video 9min• Number your paper from 1-5 and answer the

following questions. • 1. Which of these has 3 significant digits?• a. 0.000000312• b. 3.5 X 104

• c. 0.4506• d. 910

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http://education-portal.com/academy/lesson/unit-conversion-and-dimensional-analysis.html

Review

• A• 2. Which of these is a qualitative observation?• a. the sky is dark• b. his head is 45 cm in diameter• c. she has 20 freckles• d. the candle was 17 cm long

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Review

• A• 3. Which of these is a conversion factor?• a. 3 cm• b. 12 inches = 1 foot• c. 3 ft 1 yard

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Review

• C• 4. What’s the first step in setting up a

dimensional analysis problem?• a. cancel the units• b. write down all the conversion factors• c. cry• d. set up the question mark format

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Review

• D• 5. What does this conversion factor mean? 14 frizzles drizzle • a. There are drizzles that are frizzles• b. There are 14 frizzles per drizzle• c. A frizzle and a drizzle are equal• d. There are 14 drizzles per frizzle 36

Review• BWhat observations can you make about this picture?

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Density • Density = mass per (divided by) unit

volume• D = mass/volume• D= m/V• Common units of density: g/mL,

g/cm3

• Remember that 1 mL=1cm3 !• Suppose we have an object with a

mass of 5.0 grams and a volume of 2.0 mL. What would be the density? • D = 5.0g/2.0mL = 2.5g/mL (2 sig figs)

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Let’s Do It!!!

• 1. Given a mass of 12g and a volume of 3 mL, calculate the density.

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Let’s Do It!!!

• 1. Given a mass of 12g and a volume of 3 mL, calculate the density.• Need density• D = m/V• D = 12g/3mL• D = 4m/mL (1 sig fig since following multiplication

rules

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Let’s Do It!!!• 2. Given that the density of iron is 7.9m/cm3

what would be the volume of a 15.8 g piece of iron?

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Let’s Do It!!!• 2. Given that the density of iron is 7.9g/cm3 what

would be the volume of a 15.8 g piece of iron?• D = m and asking for volume so solve for V V• Multiply both sides by V: V x D = m x V VSo V x D = m• Divide both sides by D: V x D = m D DSo V = m = 15.8g = 2.00 cm3 (3 sig figs) D 7.9g/cm3

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Let’s Do It!!!

• 3. Find the density of a block with a length of 4.0cm, width of 3.0cm, and a height of 2.0cm and a mass of 36g

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Let’s Do It!!!

• 3. Find the density of a block with a length of 4.0cm, width of 3.0cm, and a height of 2.0cm and a mass of 36g• Need density• D = m/V • V = l x w x h = 4.0cm x 3.0cm x 2.0cm = 24cm3

• D = 36g/24cm3 or 1.5g/cm3

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Review

• Number your paper from 1-5 and answer the questions that follow• 1. Which of these is correct?• A. 1 gram = 1 x 10-3 milligrams• B. 1 gram = 1 x 106 kilograms• C. 1 gram = 1 x 103 milligrams• D. 1 gram = 1 x 10-6 micrograms

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Review

• C• Round to the appropriate significant digit• 2. If I dropped a toy car into a glass and the toy

car displaced 20mL of water, what is the volume of the toy?• A. 20cm• B. 2mL• C. 20mm• D. 20cm3

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Review

• D• 3. What is the mass of an object with a density of

4.0g/cm3 that displaces 3.0 cm3 of water?• A. 7 g• B. 12 g• C. 0.75 g• D. 1 g

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Review

• B• 4. What is the mass of 50mL of water?

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Review

• 1 g = 1 mL = 1 cm3 of water, so 50mL = 50g• D = m/V not needed!• 5. Super challenge! You have a piece of silver

with a mass of 31.5 g. Silver has a density of 10.5g/cm3 . What would be the new level of water if this piece of silver is placed into 15mL of water?• A. 12.0mL• B. 5.0mL• C. 18.0mL• D. 21.0 mL

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Review

• C• Solve D = m/V • So V = m = 31.5g = 3.00cm3 = 3.00mL D 10.5g/cm3

So add 15.0 mL + 3.00mL = 18.0mL

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Experimental and Percentage Error

•ALL measurements are subject to some uncertainty because an estimate is involved•Accuracy - the nearness of a measurement to its accepted value•Accuracy can be expressed in either absolute or relative terms called experimental error or percentage error respectively.

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Experimental and Percentage Error-Reference

•Experimental error (absolute)=observed - accepted experimental error•Percentage error (relative) = --------------------X 100 % accepted value

An accepted value is a standard or known value 1. 12 inches in a foot2. Water freezes at 0 oC

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•Sample Problem:•A student is asked to weigh out 5.00 grams of salt the balance. When checked by the instructor, it is found that the student has weighed out 4.942 g. Calculate the student’s experimental error (EE) and percentage error (PE).

•Experimental error = observed value - accepted value•Experimental error = 4.942 g - 5.00 g = -.06 g (sig digits!)

experimental error - .06 g•% error = ------------------------ X 100 % = ------------ X 100 % = -1 % accepted value 5.00 g sig digits!!! 53


•Reminders:• Experimental Error • 1. Always put the observed (student) value first • 2. Answer can be positive or negative with same units

as what you subtracted• 3. Significant digits are done by place value because

it’s addition• Percentage Error• 1. When calculating, do not use the % key on your

calculator • 2. Answer will have the same sign as experimental

error and units will always be %• 3. Significant digits are done by counting because it’s

multiplication

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Let’s Do It!!!!

• As the result of experimental work, a student finds the density of a liquid to be .145 g/mL. The known density of the liquid is .146 g/mL. Find the student’s experimental error (EE) and percentage error (PE).

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Let’s Do It!!!!

• Experimental error = observed value - accepted value•Experimental error = 0.145 g/mL – 0.146 g/mL = -0.010 g (sig digits!)

•Percentage error experimental error - .010 g•% error = ------------------------ X100% = ------------ X100% = -.07 % accepted value 0.146g sig digits!!!

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Temperature Scales-Reference• There are 3

commonly used temperature scales in science. • 1. Fahrenheit, • 2. Celsius (or

centigrade)• 3. Kelvin (absolute)

scale.

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Temperature Scales-Reference

• Conversions between temperature scales:• **Celsius to Kelvin conversion: TK = T°C + 273 273 has no SD’s because it’s a known number• Example: convert 25 °C to Kelvin

TK = T°C + 273TK = 25 + 273TK = 298 K notice the unit is K not °K

•

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• **Kelvin to Celsius conversion: T°C = TK - 273• Example: convert 250 K to Celsius

T°C = TK - 273T°C = 250 - 273T°C = - 20 °C rounded to the tens place

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• **Fahrenheit to oC conversion: T°C = (T°F - 32) ÷ 1.8

32 and 1.8 have no SD’s• Example: convert 98.7 °F to Celsius

T°C = (T°F - 32) ÷ 1.8T°C = (98.6 - 32) ÷ 1.8T°C = 66.7 ÷ 1.8T°C = 37.0 °C

•

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• **Celsius to Fahrenheit conversion: T°F=(1.8 • T°C) + 32 1.8 and 32 have no SD’s• Example: convert 36 °C to Fahrenheit

T°F = (1.8 • T°C) + 32T°F = (1.8 • 36) + 32T°F = 65 + 32T°F = 97 °F

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Let’s Do It!!!

• 1. Convert 25.0 °C to Kelvin

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Let’s Do It!!!!

• TK = T°C + 273

• TK = 25.0 + 273

• TK = 298.0 K the 273 does not affect SD’s because it is not measured

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Let’s Do It!!!!

• 2. Convert 303 K to Celsius

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Let’s Do It!!!!

• T°C = TK - 273

• T°C = 303 - 273

• T°C = 30 °C rounded to the ones place

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Let’s Do It!!!!

• 3. Convert - 25 °F to Celsius

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Let’s Do It!!!!

• T°C = (T°F - 32) ÷ 1.8

• T°C = (- 25 - 32) ÷ 1.8

• T°C = - 57 ÷ 1.8

• T°C = - 32 °C

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ReviewTemperature video 5 min• Number your paper from 1-5 and answer the

following questions• 1. How many sig figs should the product of 1.304

and 000.025 have?• A. 1• B. 2• C. 3• D. 4

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http://education-portal.com/academy/lesson/temperature-units-converting-between-kelvins-and-celsius.html

Review

• B• 2. Which of these is true?• A. density = mass X time• B. density = volume X mass• C. density = volume X time• D. density = mass/volume

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Review

• D• 3. What are the three systems of measuring

temperature?• A. Celsius, grams, liters• B. Celsius, second, liter• C. Celsius, Fahrenheit, Kelvin• D. Hot, Hotter, Hottest

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Review

• C• 4. You are in Paris and the weatherman says it’s

going to be 35 oC today. What kind of clothes should you wear?• A. A coat• B. A light jacket• C. A pair of shorts and a t-shirt• D. A heavy coat and mittens and hat. That’s way

too cold!72

Review

• C• 5. What is the temperature in Kelvin at which

water freezes?• A. 273.15 K• B. 273.15 o K• C. 0 K• D. -170 K

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Review

• A

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D imensional Analysis Conversions

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