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### Dimensional Analysis

### Dimensional Analysis

### How many minutes in 2.5 hours?

23°C

23°C

The number of SFs in a measured value is equal to the number of known digits plus one uncertain digit.

22°C

22°C

21°C

21°C

you record 21.6°C

you record 21.68°C

Example B

Example A

1. If the glassware is marked every 10 mLs, the volume you record should be in mLs. (Example A)

2. If the glassware is marked every 1 mL, the volume you record should be in tenths of mLs.

3. If the glassware is marked every 0.1 mL, the volume you record should be in hundredths of mLs. (Example B)

0 mL

30 mL

20 mL

1 mL

10 mL

30-mL beaker:

the volume you write in your lab report should be 13 mL

2 mL

Buret marked in 0.1 mL: you record volume as 0.67 mL

Cheap balance measurements are trustworthy to the nearest gram. Measurement = 25 g, so implied

precision is +/-1g.

Standard lab balance are trustworthy to

the nearest milligram (0.001g)

measurement: 25.000g, so

implied precision is +/-0.001g

The analytical balance is very precise. Measurements are trustworthy to the nearest 0.1mg. Measurement:25.0000

implied precision: +/-0001g

Reporting Measurements

- Using significant figures
- Report what is known with certainty
- Add ONE digit of uncertainty (estimation)

Davis, Metcalfe, Williams, Castka, Modern Chemistry, 1999, page 46

How good are the measurements? (that’s where sig fig’s come in!)

- Scientists use two word to describe how good the measurements are
- Accuracy- how close the measurement is to the actual value
- Precision- how well can the measurement be repeated
- In short, when you plug these three numbers into your calculator, remember your calculator neither knows nor cares about how good (significant) the numbers it’s working with are. However, to you, the taker of data, these three numbers tell you whether or not your data is good enough to pay attention to.

Significant Figures are concerned with Accuracy vs. Precision in measurement

Poor accuracy

Poor precision

Good accuracy

Good precision

Poor accuracy

Good precision

Random errors:

reduce precision

Systematic errors:

reduce accuracy

Differences

- Accuracy can be true of an individual measurement or the average of several
- Precision requires several measurements before anything can be said about it

accurate, not precise

not accurate, precise

accurate and precise

accurate, low resolution

3

2

1

time offset [arbitrary units]

0

-1

-2

-3

Accuracy Precision Resolutionsubsequent samples

Significant figures

- Using proper significant figures in measured and calculated values conveys a sense of precision to the reader and defines a limit of error in the value.

Significant figures are all the digits in a measurement that are known with certainty plus a last digit that must be estimated

2

3

4

5

0

cm

1

2

3

4

5

0

cm

1

2

3

4

5

0

cm

Practice Measuring4.5 cm

4.54 cm

3.0 cm

Timberlake, Chemistry 7th Edition, page 7

Using Significant Figures reflects precision by estimating the last digit

- What is the certain measurement?
- What is the estimated measurement?

The instrument determines the amount of precision of the data.

- What is the certain measurement here?
- What is the estimated measurement here?

Error vs. Mistakes

ERROR

MISTAKES

Mistakes are caused by PEOPLE

Misreading, dropping, or other human mistakes are NOT error

- Scientific errors are caused by INSTRUMENTS
- Scientific measurements vary in their level of certainty

Significant Figures

- What is the smallest mark on the ruler that measures 142.15 cm?
- 142 cm?
- 140 cm?
- Here there’s a problem, does the zero count or not?
- They needed a set of rules to decide which zeros count. (Non-zeros always count)

SIG FIG Rules

- Rule one
- All non zero numbers are significant!
- PRACTICE
- 245.36 g
- 2 g
- 6.4 g
- 428999999 g

SIG FIG Rules

- Rule two
- All zeros between significant figures are significant
- So if the zeros are sandwiched or embedded in the number they ARE significant
- PRACTICE
- 5004 g
- 62.0003 g
- 20.03 g
- 2.06004 g

SIG FIG Rules

- Rule three
- All trailing zeros to the right of the decimal ONLY are significant
- Those at the end of a number without the decimal point don’t count
- Practice
- 42.00000 g
- 0.3300 g
- 50000 g
- 50.0000 g
- 200.002200 g
- 0.0000040 g

Which zeros count?

- If the number is smaller than one, zeros before the first number don’t count
- 0.045 g
- Zeros between other sig figs do.
- 1002 g
- zeroes at the end of a number after the decimal point do count
- 45.8300 g
- If they are holding places, they don’t.
- 0.0006 g
- If measured (or estimated) they do count

Another way to figure it out

- If the number has a decimal in it:
- Find the first non zero from the LEFT
- That is the first significant figure in the number, and all digits after (as you move toward the right) ARE significant
- PRACTICE:
- 3.4000000 g
- 0.00003420 g
- 404.0 g
- 0.1100000 g
- 30000 g

If the number does NOT have a decimal in it:

- Find the first non zero from the RIGHT
- That is the first significant figure in the number, and all digits after (as you move toward the left) ARE significant.
- Practice
- 560000 g
- 20000020 g
- 2002.00 g
- 3030300 g

Sig Figs

- Only measurements have sig figs.
- Counted numbers are exact
- Ex:
- A dozen is exactly 12
- Number of students in this classroom
- Unit conversions in most cases will be considered exact numbers
- Ex:
- 1 milliliter is exactly .001 of a liter
- 100 centimeters is exactly 1 meter

Sig figs.

- How many sig figs in the following measurements?
- 458 g
- 4000085 g
- 4850 g
- 0.0485 g
- 0.0040850 g
- 40.004085 g

Sig Figs.

- 405.0 g
- 4050 g
- 0.450 g
- 4050.05 g
- 0.0500060 g
- Next we learn the rules for calculations

Calculations with sig figs

- The answer can’t be more precise than the question

Calculations

- Addition/Subtraction
- The answer is based on the number with the fewest decimal points

- Multiplication/Division
- The answer is based on the number with the fewest significant digits

Calculations Involving Measured Data

- Addition/Subtraction:
- The answer contains the same number of digits to the right of the decimal as that of the measurement with the fewest number of decimal places.

3.14159 g

+ 25.2 g

28.34159 g

28.3 g

3 SFs

33.14159 g

- 33.04 g

0.10159 g

0.10g

2 SFs

- Calculators do NOT know these rules. It’s up to you to apply them!

27.93

+

6.4

27.93

27.93

+

6.4

6.4

1. First line up the decimal places

2. Then do the adding

3. Count the sig figs in the decimal portion of each addend.

34.33

4. Round your answer to the place value of the addend with the least number of decimal places

Rounding rules

- look at the number to the right of place you are rounding to.
- If it is 0 to 4 don’t change it
- If it is 5 to 9 make it one bigger
- round 45.462 to four sig figs
- to three sig figs
- to two sig figs
- to one sig fig

Practice

- 4.8 g+ 6.8765 g
- 520 cm + 94.98 cm
- 0.0045 m+ 2.113 m
- 6.0 x102 L - 3.8 x 103 L
- 5.4 ml - 3.28 ml
- 6.7 g- .542 g
- 500 cm -126 cm
- 6.0 x 10-2 mg - 3.8 x 10-3 mg

Multiplication and Division

- Same number of sig figs in the answer as the least in the question
- 3.6 gx653 g
- 2350.8 g is the answer you would get in your calculator
- 3.6 has 2 s.f. 653 has 3 s.f.
- So your answer can only have 2 s.f.
- You must round to 2 sig figs and your answer is 2400 g

Multiplying or Dividing Measured Data

- answer contains the same number of SFs as the measurement with the fewest SFs.

25.2 mx6.1 m= 153.72m (on my calculator)

= 1.5 x102 m (correct answer)

25.2 g

------------ = 7.3122535 g

3.44627 g

= 7.31g (correct answer)

(6.626 x 10-34)(3 x 108)

------------------------------- = 3.06759 x 10-2 (on my calculator)

6.48 x 10-24

= 0.03 (correct answer)

Multiplication and Division

- Same rules for division
- practice
- 4.5 g/ 6.245 g
- 4.5 mx6.245 m
- 9.8764 L x .043 L
- 3.876 mg / 1983 mg
- 16547 g/ 714 g

QUANTITATIVE MEASUREMENTS

- Described with a value (number) & a unit (reference scale)
- Both the value and unit are of equal importance!!

NO NAKED NUMBERS!!!!!!!!!!

Problems

- 50 is only 1 significant figure, but if I actually measured or estimated the ones place value to be 0, it really has two significant figures. How can I write it?
- A zero at the end only counts after the decimal place
- So, I can use Scientific notation
- 5.0 x 101
- now the zero counts.

SCIENTIFIC NOTATION

Based on Powers of 10

Technique Used to

1. Express Very Large or Very Small Numbers

2. Reduce likely-hood of errors

3. Compare Numbers Written in Scientific Notation

- First Compare Exponents of 10 (order of magnitude)
- Then Compare Numbers

SCIENTIFIC NOTATION

Numbers that are very small

The electrical charge on one electron:

0.0000000000000000001602 = 1.602 X 10-19 C

Or numbers that are very big!!

The mass of the moon:

73,600,000,000,000,000,000,000 kg = 7.36 X 1022kg

SCIENTIFIC NOTATION

- Writing Numbers in Scientific Notation
- Locate the Decimal Point
- Move the decimal point just to the right of the non-zero digit in the largest place
- The new number is now greater than one but less than ten
- Multiply the new number by 10n
- where n is the number of places you moved the decimal point
- 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

Practice writing in scientific notation

- 46600000 = 4.66 x 107
- 0.00053 = 5.3 x 10-4
- 123,000,000,000 = 1.23 x 1011

SCIENTIFIC NOTATION

- 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

Practice writing in standard form

- 4.50 X 10 7 mm = 45,000,000 mm
- 8.08 X 10-11g= 0.0000000000808 g
- 4.0 X 101 m = 40 m
- 1.200 X 100 L= 1.200 L

QUANTITATIVE MEASUREMENTS

- -“Derived vs. Measured”
- Measured – acquired directly from a measuring instrument.
- EX: ruler, balance, scale, graduated cylinder
- Derived – calculated or determined from Measured values using formulas
- EX: area of a square or circle, volume of a cylinder, density of an object

Quantitative Measurements

- 2) Temperature
- Celcius (Centigrade) -- 00 C – Freezing Point of water
- Fahrenheit -- 320 F – Freezing Point of water
- Kelvin-------- SI Unit of temperature with 00K absolute zero, and 273.16 0 K equal to the triple point of water (the point at which all three phases of water are at equilibrium)
- Kelvin = 0C + 273
- Celcius = K -273
- Farenheit = 0C(1.8 F/1 0C) + 32
- 0C = F - 32 (1 0C/1.8 F)

Quantitative Measurements

- 3) Time
- Fundamental unit is …..SECONDS
- 4) Volume – 3-D space that matter occupies
- Liquids are defined (measured) or derived
- solids

-- regular shapes – formulas (derived value)

-- irregular shapes – displacement of another liquid, usually water

- gases – formulas (derived values)
- UNITS OF VOLUME??????????????

SI UNIT IS 1 LITER

- Meters X Meters X Meters = 1 m3 = 1000 liters

OR

- .1m X.1m X .1m = .001 m3 1 = 1000ml = 1 liter
- 5) Mass
- matter
- weight – the gravitational pull on an object (related to its mass)
- SI Units
- UNITS OF MASS?????????????
- 6) Density
- Mass/Volume
- A derived quantity
- Grams/cm3

Kilo- means 1000 of that unit

- 1 kilometer (km) = 1000 meters (m)
- Centi- means 1/100 of that unit
- 1 meter (m) = 100 centimeters (cm)
- 1 dollar = 100 cents
- Milli- means 1/1000 of that unit
- 1 Liter (L) = 1000 milliliters (mL)

kilometer km 1,000 m or 103 m

meter m 1 m or 100 m

decimeter dm 0.1 m or 10-1 m

centimeter cm 0.01 m or 10-2 m

millimeter mm 0.001 m or 10-3 m

micrometer mm 0.000001 m or 10-6 m

nanometer nm 0.000000001 m or 10-9 m

SI System for Measuring LengthComparison of English and SI Units

1 inch

2.54 cm

1 inch = 2.54 cm

Zumdahl, Zumdahl, DeCoste, World of Chemistry2002, page 119

Converting measurements

- We use a method called dimensional analysis, also referred to as factor label method or “train tracks”

Whether you call it Dimensional Analysis, the Unit Method, or the Factor-Label Method, this is one of the most useful techniques you will learn.

Use it and you will never ask,

“Do I multiply or divide?” when you are converting from one set of units to another.

Measurements include two parts:

a number and the units.

Reporting the distance as 20.0 gives no information at all until the units are included.

Is that 20.0 kilometers, meters, miles, feet, or light-years?

Dimensional Analysis

- Dimensional Analysis is based on two simple algebraic concepts:
- 1. If x = y, then x/y = 1 and y/x = 1
- 2. Any number times 1 is unchanged, (N x 1 = N)
- If you can understand and remember these, you can use Dimensional Analysis!

PROBLEM SOLVING STEPS

- Read problem
- Identify data
- Make a unit plan from the initial unit to the desired unit
- Select conversion factors
- Change initial unit to desired unit
- Cancel units and check
- Do math on calculator
- Give an answer using significant figures

Insert relationship so units cancel.

quart

1

9.3 cups

x

4

cups

*units of known in denominator (bottom) first

*** units of unknowns in numerator (top

How do we record what we did?

X 100

X 4

X 5

$100 bill X 100 $1 bills X 4 quarters X 5 nickles= 2000 nickles

1 $100 bill 1 $1 bill 1 quarter

2.5 hr X 60min/1 hr = 150 min

By using dimensional analysis / factor-label method, the UNITS ensure that you have the conversion right side up, and the UNITS are calculated as well as the numbers!

2.5 hr 60min = 150min

1hr

Conversion factor

You have $7.25 in your pocket in quarters. How many quarters do you have?

- 7.25 $ X 4 quarters/1 $ = 29 quarters
- Set up railroad tracks

7.25$ 4 quarters = 29 quarters

1$

Conversion Factor

WHAT’S WRONG WITH THIS?

- How many seconds in 1.4 days?
- 1.4 day X 1 day/24 hr X 60 min/1 hr X 60 sec/1 min
- Set up the railroad tracks

1.4day 1day 60min 60sec =

24hr 1hr 1 min

Can you figure it out?

- You\'re throwing a pizza party for 15. Each person may eat 4 slices. How much is the pizza going to cost you? You call the pizza place and learn that each pizza will cost $14.78 and will be cut into 12 slices. You have budgeted $70 for your party. Do you have enough money?

What units do you begin with?

I know I have 15 attendees at the party

What do I need to know?

How much will the party cost me?

- So I am beginning with 15 people and need to know how much money the party will cost SO
- Begin with 15 people and I need to find dollars

party party

Multiply by factors equivalent to ONE until you have solved your problem.

- Think about how you converted fractions in elementary school…..
- ½ a pie is equivalent to how many 1/8s of a pie
- ½ X 4/4 (equivalent to one) = 4/8

15 people 4 slices 1 pizza $14.78

party 1 people 12 slices 1 pizza

- Now cancel the units
- Multiply across the top of the train tracks $886.80
- Multipy across the bottom of the train tracks
- 12 parties
- Now divide = $73.90/party
- YOUR OVERBUDGET!

What about Square and Cubic units?

- Use the conversion factors you already know, but when you square or cube the unit, don’t forget to cube the number also!
- Best way: Square or cube the ENTIRE conversion factor
- Example: Convert 4.3 cm3 to mm3
- 4.3 cm3 (10 mm/ 1 cm)3 = 4.3 cm3 X 103 mm3/ 13 cm3 = 4300 mm3

DENSITY

- Depends on:
- Mass
- Volume

D = m/v (g/cm3)

Mass usually expressed in grams

Volume usually expressed in cm3 or liters, etc.

The proximity of like atoms or molecules

More than just the “heaviness” of a substance, density includes how much space an object takes up!!

All substances have density including liquids, solids, and gases

Density is the measure of the “compactness” of a materialBalloon and liquid nitrogen

- http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/balloon.html#c1
- http://paer.rutgers.edu/pt3/movies/TVrhoandFb.mov

www.dkimages.com

What would happen????

- Mercury density = 13600kg/m3
- Lead density = 11340kg/m3

Archimedes and the Kings Crown

http://3quarksdaily.blogs.com/3quarksdaily/images/2007/07/18/archimedes.jpg

FACTORS AFFECTING DENSITY

- ATomS!!!!

How are Submarines like fish….

- The swim bladder in bony fish control their relative density in order to rise or dive in the water….buoyancy
- When air is added to the swim bladder, by diffusion through the blood vessels in the bladder walls, the fish becomes less dense overall
- when air is removed fish become more dense
- By changing the volume of air in the bladder, the fish’s density can be made equal to that of the surrounding water at a given depth.

DETERMINING DENSITY

- Regular Shapes – mass, then determine the volume by formula

EX: cubes, cylinders, spheres, cones, etc.

- Irregular shapes – mass, then measure displacement of a liquid (usually water) by that irregularly shaped object

Density and Dimensional analysis

- You are given a .55kg piece of iron.
- The density of iron is 7.9g/cm3
- What is the volume of your piece of iron

One more density problem

- How heavy in pounds is a gold bar?

The dimensions of your gold bar:

- 7 X 3.625 X 1.75 inches
- The Density of gold is 19.3g/cm3
- One pound is 454 g
- SOLVE
- The volume comes to 44.40625 inches
- 44.40625 in3 X 2.54 cmX 2.54 cm X 2.54 cm = 727.6880608 cm3

1 in 1 in 1 in

727.6880608 cm3 X 19.3g X 1 pound = 30.93475677 pounds

1 cm3 454 g

According to significant figures, 30 pounds

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