Chapter 2

Data Analysis

Objectives: •Define SI base units for time, length, mass and temperature•Use prefixes with units •Compare the derived units for volume and density

•Use scientific notation •Use dimensional analysis to convert between units

and ………

Objectives (cont’d) • Define and compare accuracy and

precision • Use significant figures and rounding to

reflect the certainty of data • Use % error to describe the accuracy of

experimental data• Correctly create, read and interpret graphs

AL COS Objectives 4, 7, 9, 10, and 16

Terms to know….Base unit kilogramDerived unit densityMeter second Length accuracyLiter precisionKelvin conversion factor

Terms to know…. (Cont’d)

Random errorSystematic errorIndependent variable Dependent variable

Units of Measurement SIQuantity Base

unit symbo

lTime Second s

Length Meter mMass Kilogram kg

Temperature

Kelvin K

Amt of a substance

Mole mol

You can change any base unit with a prefix

Some prefixes are for large numbers: giga (G) 109

gigameter mega (M) 106 megameter kilo (k) 1000 kilometer

And some for smaller ones…deci 0.1 10-1

centi 0.01 10-2

milli 0.001 10-3

micro 0.000 001 10-6

nano 0.000 000 001

10-9

pico 0.000 000 000 001

10-12

Metric StairsMetric Stairs You should be comfortable with converting from You should be comfortable with converting from [cm] [cm] toto

[m], [mm] [m], [mm] toto [km], [km], and so on.and so on.

Convert: 1527 centigrams into hectograms: going four steps up means you move the decimal 4 places to the left. Therefore:

1527 centigrams = .1527 hectograms&

9.8712345 kg = (steps to the right) 9871234.5 mg

So, how much is….A milliliter? ________A centigram? ________A kilowatt? ________A nanometer? ________A picosecond? ________

Base vs derived units•A base unit is a single unit and is measured directly--- Example: - time in seconds with a stopwatch - mass in grams with a balance - length in cm with a ruler

A derived unit is a combination of two or more base units

Example: speed meters/seckm/hr

density g/mL or g/cm3

and …….

Volume is a derived unit = L x w x h 1 liter = 10 cm x 10 cm x 10 cm

= 1000 cm3

or = 1 dm* x 1 dm x 1 dm = 1 dm3

* 10 cm = 1 decimeter

Density is the amount of mass packed in to a same volume

Low density High density

Density = mass volume

Example #1: A 121.5 g sample of aluminum has a volume of 45 cm3. What is the density of Al?

Solution:

D = m/v

= 121.5g 45 cm3

= 2.7 g/cm3

Another density problem:

Example #2:

What would be the mass of 1000 mL of liquid mercury, density 13.5 g/mL?

Solution:

D = m v

13.5 g/mL = m 1000 mL (13.5g/mL) (1000 mL) = 13 500 g = 13.5 kg

Scientific notation• Used to simplify reading numbers that are very large or very small

• In normal scientific notation there is one number to the left of the decimal Ex.: 5.66 x 1013

Another version of scientific notation is called

Engineering mode

• In Engineering mode two numbers are left in front of the decimal Ex: 56.6 x 10 12

You will see both in yourScientific calculator

Chemists typically use normal scientific notation

Ex: 1.66 x 10-19 6.022 x 1023

-is a way of solving problems that focuses on the units rather than the numbers to solve the problem

-solves problems by using a conversion factor

Dimensional analysis

A conversion factor is ----a ratio of equivalent values expressed in different units

-a conversion factor is like multiplying a number by 1

such as……

12 inches1 foot

1 hour 3600 seconds

= a conversion factor

= a conversion factor

Same thingSame thing

= 1

Ex. #1:

How many seconds are in 9 weeks?

9 wks x 7 days x 24hrs x 60min x 60 sec 1 1 wk 1 day 1 hr 1 min

Answer

Notice how the units canceled out:

9 wks x 7 days x 24hrs x 60min x 60 sec 1 1 wk 1 day 1 hr 1 min = 5 443 200 sec

Answer:

In Europe, the posted speed limit is 90 km/hr – what would that be in miles/hr?

Ex.#2:

1 km = 0.62 miles

Ex. #2

90 km x 0.62 miles = 55.8 miles1 hr 1 km hr

Ex. #3: If a baseball player hit 70 home runs in a single season, and it’s 90 feet between each of the four bases, howmany miles did he run that year in justhome runs? Try this on your own paper.

1 mile = 5280 feet

Answer

70 home runs x 360 ft x 1 mile 1season 1 homerun 5280 ft

= 4.77 miles/season

Answer #3

How reliable are measurements?

Scientists are concerned with• the accuracy of a measurement• the precision of a measurement

Accuracy= how close your answer comesto the “real answer,” the acceptedvalue

Poor accuracy Good accuracy

Precision

• Precision refers to how close a series of measurements are to each other

• Precision is the “repeatability” of a measurement

…accuracy is typically determined by the equipment you use

Did you use a good analytical balance or a old triple-beam balance?

Remember…

… determined by how good YOU are with the equipment provided.

Can you do over and over and get similar answers?

While precision is

Poor precisionPoor accuracy

Good precisionPoor accuracy

Good precisionGood accuracy

Reporting measurements• Every measurement has some reported error or uncertainty• Uncertainty is assumed to

be ± one unit in the last

reported place

Example: You record 1.24g as the mass of a sample.

This is automatically assumed to be between 1.23 and 1.25 or ± 0.01 grams.

When reading normal laboratoryequipment, avoid parallax errors.

Take measurementsfrom the bottom of the meniscuswhen observing at eyeLevel.

Read the bottom of the meniscusin a graduated cylinder- while holding it ateye level.

What about errors?• All experiments have some errors! - human error

- mechanical error

• An error can be a random error (because it is equally likely to be high or low)

Error (cont’d) • Random errors are sometimes called indeterminate errors, because it is largely due to the limitations of the lab equipment you use

•You can also have systematic errors (determinate errors). These are often caused by faulty measurements or problems with your measuring device.

Error (cont’d)

(Did you zero your balance before using it?)

Finding a % error

% error is the ratio of your error (How much you were off)divided by the actual acceptedanswer.

Ex. #1Donna calculated the density of aluminum to be 2.52 g/mL. The accepted density of aluminum is 2.70 g/mL. What is her % error? Error x 100 = % errorAccepted (2.70-2.52) x 100= 6.67% error 2.70

Ex. #2Bill did the same lab as Donna, But he found the density of Al to be 2.95 g/mL. What is his% error? Error* x 100 = % errorAccepted(2.95-2.70) x 100= 9.26% 2.70

* Error is always considered positive

Ex. #3: Your turn Juan calculated the density of Copper to be 9.34 g/mL. What was his % error?

(Find density using the chart on pg 914 in your textbook.)

Answer

Ex. #3 Answer

Error x 100 = % errorAccepted

(9.34-8.92) x 100 =

4.71% 8.92

Significant figures

When you measure using anyscientific instrument, your answer’s precision depends upon the equipment you used.

•The digits that are reported when reading a scientific instrument called significant figures.

• Significant figures include all KNOWN digits plus one ESTIMATED digit.

2 3 4

•This ruler would read 2.5_ cm We would probably say 2.56 cm or maybe even 2.57 cm. •The 2.5 are certain digits the last digit is an estimate.

Rules for significant figures:

1.Non-zero numbers are always significant.

Example: 957 = 3 SF 1893 = 4 SF

2. Zeroes between non-zero numbers are always significant. Example: 209 = 3 SF

34007 = 5 SF

•3. All final zeroes to the right of the decimal are significant.

Example: 256.90 = 5 SF 6,220,000.780 = 10 SF

240.519000 = 9 SF

4. Zeroes that act as placeholders are NOT significant.

Example: 0.06 = 1 SF290 = 2 SF982 400 = 4 SF0.0045100 = 5 SF

•5. Counting numbers and defined constants have an infinite number of significant figures.

Rounding off numbers…• Suppose you have to find the density of a sample of Al with amass of 382.15 g and a volume of 145ml.Your calculator would give you an answer of 2.63551724.......• Your answer can not have any more significant figures than the data with the least # of SF. So then

382.15 = 5 SF145 = 3 SF So our answer can only have 3 SF

2.63551724....... Must be reported as 2.64 g/mL

Addition and subtraction with SF

When you add or subtract, youranswer is based on the number thathas the fewest number of digits tothe right of the decimal

Ex. #1

14.63 + 9.009+135.2158.839 158.8

Ex. #2

195.336+ 46.7788242.1148 242.115

Ex. #3: Your turn

1200.03- 125.669

And the answer

is….

Answer:

1074.361 1074.36

1200.03- 125.669

Pg. 41, Prac. Problems35.a.) 142.9 cm

b.) 768 kgc.) 0.1119 mg

36.a.) 12.12 cmb.) 2.10 cmc.) 2.7 x 103 cm

Multiplying and dividing SF

Remember that old rule about no chain is any stronger than it’s weakest link? When you multiply or divide numbers, your answer must have the same number of sig. figs as the measurement with the fewest sig. figs.!!!!!!

Ex. #1:

6.220 x 12 = 74.64(4 SF x 2SF = 2 SF)

So 74.64 (4 SF) 75 (2 SF)

Ex. #2:

9.67 x 12 000 x 0.9007= ?

Answer

Answer9.67 x 12 000 x 0.9007= ? 3SF 2SF 4 SF = 2 SF

104 517.228 becomes100 000*

* Bar shows that the zero is significant and not just a place holder!

Ex. #3

1.46 x 106 x 14.566 x 0.0230 63.289

Answer

Answer #3

1.46 x 106 x 14.566 x 0.0230 63.2893SF x 5 SF x 3 SF/5 SF = 3SF= 7 728.4564............(etc)= 7 730

Pg. 4237. a. 78 b. 12 c. 2.5 d. 81.138. a. 2.0 c. 2.00 b. 3.00 d. 2.9

Insignificant

End Chapter 2