Chapter 3

Scientific Measurement

MeasurementIn chemistry, #’s are either very small or very

large

1 gram of hydrogen =

602,000,000,000,000,000,000,000 atoms

Mass of an atom of gold =

0.000 000 000 000 000 000 000 327 gram

Scientific Notation

• Condensed form of writing large or small numbers• When a given number is written as the product of 2 numbers• M x 10n

M must be:• greater than or equal to 1 • less than 10

n must be:• whole number • positive or negative

Find M by moving the decimal point over in the original number to the left or right so that only one non-zero number is to the left of the decimal.

Find n by counting the number of

places you moved the decimal:

To the left (+) or

To the right (-)

Scientific Notation Examples

20 =

200 =

501 =

2000 =

2.0 x 101

2.0 x 102

5.01 x 102

2.000 x 103

More examples…

0.3 =

0.21 =

0.06 =

0.0002 =

0.000314 =

Rule:If a number starts out as < 1, the exponent is always negative.

3 x 10-1

2.1 x 10-1

6 x 10-2

2 x 10-4

3.14 x 10-4

Scientific NotationAdding & Subtracting:

• if they have the same n, just add or subtract the M values and keep same n

• if they don’t have the same n, change them so they do

Scientific Notation

Multiplying:• the M values are multiplied• the n values are added

Scientific Notation Division:

• the M values are divided• the n values are subtracted

Accuracy & Precision

‘How close you are really counts!’

Accuracy• Accuracy – a measure of how close a

measurement comes to the actual or true value of what is measured

To evaluate… the measured

value must be compared

to the correct value

Precision

• Precision – a measure of how close a series of measurements are to one another

To evaluate… you must compare the values of 2 or more repeated measurements

Accuracy vs. Precision

Errors are Unavoidable

• Measuring instruments have limitations

• Hence, there will always be errors in measurement.

Not All Errors are Equal• Consider the following two errors:• You fly from NY to San Francisco

• Your plane is blown off course by 3cm

• You are an eye surgeon

• Your scalpel misses the mark by 3cm

The errors sound equal… but are they?

Absolute Error• The error in each of the previous examples is 3cm• But the error in each is not equivalent!• This type of error is the absolute error.

Absolute error = | measured value – accepted value |

Accepted value is the most probable value or the value based on references

Only the size of the error matters, not the sign

Significance of an Error• The absolute error tells you how far you are from

the accepted value• It does not tell you how significant the error is.

o Being 3cm off course on a trip to San Francisco is insignificant because the city of San Francisco is very large.

o Being 3cm off if you are an eye surgeon means your operating on the wrong eye!

• It is necessary to compare the size of the error to the size of what is being measured to understand the significance of the error.

Percentage Error

• The percentage error compares the absolute error to the size of what is being measured.

% error = |measured value – accepted value| x 100%

accepted value

• Example: Measuring the boiling point of H2O

Thermometer reads – 99.1OCYou know it should read – 100OC

Error = measured value – accepted value

% error = |error| x 100% accepted value

Sample Problem

% error = |99.1oC – 100.0oC| x 100%

100oC

= 0.9o x 100%

100o

= 0.009 x 100%

= 0.9%

C

C

Significant Figures• Used as a way to express which numbers

are known with certainty and which are estimated

What are significant figures?

Significant Figures –

all the digits that are known, plus

a last digit that is estimated

Rules1) All digits 1-9 are significant Example: 129

2) Embedded zeros between significant digits are always significant Example: 5,007

3) Trailing zeros in a number are significant only if the number contains a decimal point Example: 100.0

3600

3 sig figs

4 sig figs

4 sig figs

2 sig figs

4) Leading zeros at the beginning of a number are never significant

Example: 0.0025

5) Zeros following a decimal significant

figure are always significant

Example: 0.000470

0.47000

6) Exceptions to the rule are numbers with an

unlimited number of sig figs

Example = Counting – 25 students

Exact quantities – 1hr = 60min, 100cm = 1m

2 sig figs

3 sig figs

5 sig figs

Significant Figure Examples 123m = 9.8000 x 104m = 0.070 80 = 40, 506 = 22 meter sticks = 98, 000 = 143 grams = 0.000 73m = 8.750 x 10-2g =

35

4

5

unlimited

2

32

4

Calculations Using Significant Figures

• Rounding1st determine the number of sig figs

Then, count from the left, & round

If the digit < 5, the value remains the same.

If the digit is ≥ 5, the value of the last sig fig is increased by 1.

Try your hand at rounding…

Round each measurement to 3 sig figs.

87.073 meters = 4.3621 x 108 meters = 0.01552 meter = 9009 meters = 1.7777 x 10-3 meter = 629.55 meters =

87.1m

0.0155m or 1.55 x 10-2m

4.36 x 108 m

9010m

1.78 x 10-3m

630. m or 6.30 x 102m

• Multiplying and Dividing

Limit and round to the least number of significant figures in any of the factors.

23.0cm x 432cm x 19cm =

Answer =

Because 19 only has 2 sig figs

190,000cm3 or 1.9 x 103cm3

188,784cm3

• Addition and SubtractionLimit and round your answer to least number of decimal places in any of the numbers that make up your answer.

123.25mL + 46.0mL + 86.257mL =

Answer =

Because 46.0 has only 1 decimal place

255.5mL

255.507mL

The International System of Units

• Based on the #10

• Makes conversions easier

• Old name = metric system

Units and Quantities• Length – the distance between 2 points or

objects

Base unit = meter

• Volume – the space occupied by any sample of matter

V = length x width x height

Base unit = liter

Based on a 10cm cube (10cm x 10cm x 10cm = 1000cm3)

1 liter = 1000cm3

• Mass – the amount of matter contained in an object

Base unit = gram

Different than weight…

Weight - a force that measures the pull of gravity

MMeettrriicc CCoonnvveerrssiioonn CChhaarrtt

1000 100 10 1 10-1 10-2 10-3

10-6 10-9

KILO – HECTA – DEKA – [BASE] – DECI – CENTI – MILLI – MICRO - NANO Meter Liter Gram

IF YOU ARE MOVING THE DECIMAL POINT: IF YOU ARE MULTIPLYING OR DIVIDING:

1. Start with the unit given to you 1. Start with the unit given to you. 2. Count how many times you need to move to get to the new unit 2. If moving to the Right Multiply: x10 for 1 jump, 3. Move the decimal in the number that many spaces and x100 for 2 jumps, x1000 for 3 jumps, etc.

in the same direction. 3. If moving to the Left Divide: /10 for 1 jump, 4. Re-write the number with the new units. /100 for 2 jumps, /1000 for 3 jumps, etc.

4. Rewrite the number with the new units.

Move Decimal Left OR Divide

Move Decimal Right OR Multiply

• Temperature – a measure of the energy of motion

How fast are the molecules moving?

When 2 objects are at different temperatures heat is always transferred from the warmer → the colder object

Temperature Scales• Celsius scale –

Freezing point of H2O = 0oCBoiling point of H2O = 100oC

• Kelvin scale –Freezing point of H2O = 273.15KBoiling point of H2O = 373.15K

K = C + 273

C = K - 273

Temperature Scale Conversions

Conversion Factorsand

Unit Cancellation

A physical quantity must include:A physical quantity must include:

NumberNumber + Unit+ Unit

1 foot = 12 inches1 foot = 12 inches

1 foot = 12 inches1 foot = 12 inches

1 foot1 foot

12 inches12 inches= 1= 1

1 foot = 12 inches1 foot = 12 inches

1 foot1 foot

12 inches12 inches= 1= 1

12 inches12 inches

1 foot1 foot= 1= 1

1 foot1 foot

12 inches12 inches

12 inches12 inches

1 foot1 foot

“Conversion factors”“Conversion factors”

1 foot1 foot

12 inches12 inches

12 inches12 inches

1 foot1 foot

“Conversion factors”“Conversion factors”

3 feet3 feet12 inches12 inches

1 foot1 foot= 36 inches= 36 inches(( ))(( ))

How many inches are in 3 feet?

How many cm are in 1.32 meters?

conversion factors:

equality:

or

X cm = 1.32 m =

1 m = 100 cm

______1 m100 cm

We use the idea of unit cancellation

to decide upon which one of the two

conversion factors we choose.

______1 m

100 cm

( )______1 m

100 cm 132 cm

How many meters is 8.72 cm?

conversion factors:

equality:

or

X m = 8.72 cm =

1 m = 100 cm

______1 m100 cm

Again, the units must cancel.

______1 m

100 cm

( )______ 0.0872 m1 m100 cm

How many feet is 39.37 inches?

conversion factors:

equality:

or

X ft = 39.37 in =

1 ft = 12 in

______1 ft 12 in

Again, the units must cancel.

( )____ 3.28 ft1 ft12 in

______1 ft

12 in

How many kilometers is 15,000 decimeters?

X km = 15,000 dm = 1.5 km( )____1,000 m

1 km10 dm

1 m ( )______

How many seconds is 4.38 days?

=

1 h60 min24 h

1 d 1 min60 s____( ) ( )____( )_____X s = 4.38 d

378,432 s

3.78 x 105 sIf we are accounting for significant figures, we would change this to…

Why do some objects float in water while others sink?

• Need to know the ratio of the mass of an object to it’s volume

• Pure H2O at 4oC = 1.000g/cm3

• If an object has a lower ratio it will float

• If an object has a greater ratio it will sink

Density• The ratio of an object’s mass to it’s volume

Density = mass volume

Example: A 10.0cm3 piece of lead has a mass of 114g. What is the density of lead?

114g = 10.0cm3

11.4g/cm3

Recall…

What type of property is density?

Does the density of a material change in relation to the sample size?

NO… density is an Intensive property

it depends only on the composition of

the material

What might affect a substance’s density?

• Temperature The volume of most substances ↑ withan ↑ in

temperature the mass remains the same

If the volume increases… what affect does it have on a substance’s density?

The density decreases

*Exception –Water’s volume ↑ with a ↓ in temperatureIts density decreases & ice floats

H2O

Calculating Density

What is the volume of a pure silver coin that has a mass of 14g, and a density of 10.5g/cm3?

D = 10.5g/cm3

M = 14gV = ?

Rearrange the density formula to solve for V

D

M

V

V = M

D

V = 14g = 14 x 1 cm3 =

10.5g/cm3 10.5g

g 1.3cm3

What is the mass of mercury that has a density of 13.5g/cm3 and a volume of 0.324cm3?

Once again, rearrange the

density formula… and solve

for M.

D

M

VM = D x V

M = 13.5g x 0.342 cm3

cm3

= 4.62g