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### Dimensional Analysis data.

### Dimensional Analysis data.

### How many minutes in 2.5 hours? bill

Measurements in the Lab:

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

Measurements in the Lab:

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 gram. Measurement = 25 g, so implied

- 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 Precision in measurement

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

Let’s use a golf analogy Precision in measurement

not accurate, not precise Precision in measurement

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 Precision in measurement

- 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

1 are known with certainty

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 are known with certainty 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 data.

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 data.

- 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 data.FIG Rules

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

- PRACTICE

- All non zero numbers are significant!

SIG FIG Rules data.

- 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

- PRACTICE

- So if the zeros are sandwiched or embedded in the number they ARE significant

- All zeros between significant figures are significant

SIG FIG Rules data.

- 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

- Practice

- Those at the end of a number without the decimal point don’t count

- All trailing zeros to the right of the decimal ONLY are significant

Which zeros count? data.

- 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 data.

- 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

- PRACTICE:

- That is the first significant figure in the number, and all digits after (as you move toward the right) ARE significant

- Find the first non zero from the LEFT

- If data. 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

- Practice

- That is the first significant figure in the number, and all digits after (as you move toward the left) ARE significant.

- Find the first non zero from the RIGHT

Sig Figs data.

- Only measurements have sig figs.
- Counted numbers are exact
- Ex:
- A dozen is exactly 12
- Number of students in this classroom

- Ex:
- 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

- Ex:

Sig figs. data.

- 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. data.

- 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 data.

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

Calculations data.

- 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 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!

For example data.

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 data.

- 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 data.

- 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 data.

- 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 data.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 data.

- 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 data.

- Described with a value (number) & a unit (reference scale)
- Both the value and unit are of equal importance!!
NO NAKED NUMBERS!!!!!!!!!!

Problems data.

- 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 data.

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 data.NOTATION

Numbers that are very small

The electrical charge on one electron:

0.0000000000000000001602 = 1.602 X 10-19 C

Or numbers that are data. very big!!

The mass of the moon:

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

SCIENTIFIC NOTATION data.

- 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 data.

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

SCIENTIFIC NOTATION data.

- 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 data.

- 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 data.

- -“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

- Measured – acquired directly from a measuring instrument.

Quantitative Measurements data.

- 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 data. 4) Volume – 3-D space that matter occupies UNITS OF VOLUME??????????????

- 3) Time
- Fundamental unit is …..SECONDS

- Liquids are defined (measured) or derived
- solids
-- regular shapes – formulas (derived value)

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

- gases – formulas (derived values)

- SI UNIT IS 1 LITER data.
- Meters X Meters X Meters = 1 m3 = 1000 liters
OR

- .1m X.1m X .1m = .001 m3 1 = 1000ml = 1 liter

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

- Mass/Volume

- Kilo- data. 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)

Unit Symbol Meter Equivalent data.

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 data.SI Units

1 inch

2.54 cm

1 inch = 2.54 cm

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

Converting measurements data.

- 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 data.

- 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 data.

- 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

9.3 cups data.

=

?quarts

Draw the dimensional “jumps”.9.3 cups

x

* Use charts or tables to find relationships

Insert relationship so units cancel. data.

quart

1

9.3 cups

x

4

cups

*units of known in denominator (bottom) first

*** units of unknowns in numerator (top

1 data.

quart

9.3 cups

x

4

cups

Do Math- Follow order of operations!
- Multiply values in numerator
- If necessary multiply values in denominator
- Divide.

How do we record what we did? bill

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? quarters do you have?

- 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? quarters do you have?

- 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? quarters do you have?

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 your problem.
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? your problem.

- 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 your problem.

- 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 your problem.

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 your problem.

- 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???? your problem.

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

Lead floats on liquid mercury! your problem.

Archimedes and the Kings Crown your problem.

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

FACTORS AFFECTING DENSITY your problem.

- ATomS!!!!

- How are Submarines like fish…. your problem.
- 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 your problem.

- 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 your problem.

- 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

- .55kg X your problem. 1000g X cm3 = 69.620cm3
1kg 7.9g

Your answer should be 2 sig figs so

7.0 X 101 cm3

One more density problem your problem.

- How heavy in pounds is a gold bar?

- The dimensions of your gold bar: your problem.
- 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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