unit 1 slides pt2 copy - laulima : gateway : welcome non-zero numbers are always signiﬁcant 2.38 g...

Significant figuresAll non-zero numbers are always significant

2.38 g — three significant figures25 mm — two significant figures

Are zeros significant? It depends on their position in the number.

A zero is significant when it is:

• between non-zero digits205 ml — three significant figures71.02 cm3 — four significant figures

• at the end of a number that includes a decimal point1.50 km — three significant figures30.00 kg — four significant figures

All non-zero numbers are always significant2.38 g — three significant figures25 mm — two significant figures

Are zeros significant? It depends on their position in the number.

A zero is not significant when it is:

• before the first non-zero digit (leading zeros)0.03 L — one significant figure0.0025 m — two significant figures

• at the end of a number without a decimal point2500 mg — two significant figures30 cm — one significant figures

Significant figures

Significant figures and scientific notationThe number of significant figures is always clear for numbers expressed in scientific notation• in some cases, expressing a number in scientific notation is the only way to

show the correct number of significant figures without ambiguity

Example: Suppose you weighed an object on a scale that measured weight precisely to the nearest ±10 lbs (i.e., hundreds place is exact, tens place is uncertain) and the scale reading was “2500 lbs”.

2500 lbs

Where is the uncertainty in this measurement?

2500 lbs

How many significant figures in this measurement?

3

How can you write the value for this measurement and show the correct number of significant figures?

2500 lbs 2500.0 lbs 2.50 x 103 lbs (2 sig figs) (5 sig figs) (3 sig figs)

How many significant figures in these measurements?

4.5 miles

!

3.025 gallons

!

125.0 °F

!

15 pencils

!

0.002 centimeters

1000 inches

!

1000.0 inches

!

1.0 x 103 inches

!

1.000 x 103 inches

!

1 x 103 inches

Rounding off numbers

When we do calculations, we often obtain answers that have excess digits -- i.e., more digits than the correct number of significant figures

We must drop the excess digits so the answer will have the correct number of significant figures

rounding off numbers -- the process of determining the value of the last digit that is retained after dropping the excess (non-significant) digits

= 1.03285714286 inches7.23 in

7Example:

Rule 1: If the first of the digits to be dropped is less than 5 ( i.e., 0, 1, 2, 3, or 4), drop the excess digits and don’t do anything to the last retained digit.

Example: Round off 5.38256 inches to 3 significant figures

5.38256 inches

Rules for rounding off numbers

drop these digitsThe first dropped digit is less than 5

Do not round up: 5.38 inches5.38256 is closer to 5.38 than it is to 5.39

Rule 2: If the first of the digits to be dropped is 5 or greater ( i.e., 5, 6, 7, 8, or 9), drop the excess digits and increase the last retained digit by one.


8.31956 inches

drop these digitsThe first dropped digit is greater than or equal to 5

Round up: 8.32 inches

Rules for rounding off numbers

8.31956 is closer to 8.32 than it is to 8.31

CAUTION: Common mistakes when rounding off numbers

When you are just putting a number into scientific notation, do NOT change the number of significant figures

62,301 (5 significant figures)

6.2301 x 104 (5 significant figures)

NOT



Example: Express the number 62,301 in scientific notation

After rounding off, do not drop zeroes that are significant


5.62981 inches


Round up: 5.630 inches (4 significant figures -- Correct)

5.63 inches (3 significant figures -- Incorrect)

CAUTION: Common mistakes involving zeroes when rounding off numbers

After rounding off, do not drop zeroes that are needed as place holders

Example: Round off 354 inches to 2 significant figures

354 inches

drop this digitThe first dropped digit is less than 5

Do not round up:

(2 significant figures -- Correct order of magnitude)

35 inches (2 significant figures -- Incorrect order of magnitude)

Even though the number of significant figures is correct, the value of the measurement is now off by a factor of 10

350 inches

CAUTION: Common mistakes involving zeroes when rounding off numbers

Round off $5268 to 2 significant figures ( i.e., to the hundreds place)

$5268

To help you remember this, think of an example involving money...


Your friend owes you exactly $5268, but you both agree to settle the debt to the nearest $100 (two significant figures in this case)

Round up: (2 significant figures -- Correct order of magnitude)

$53 (2 significant figures -- Incorrect order of magnitude)

Even though the number of significant figures is correct, the value of the measurement is now off by two factors of 10

$5300

Significant figures in calculations

The results of a calculation based on measurements can not be more precise than the least precise measurement.

For multiplication and division: The answer must contain the same number of significant figures as the measurement with the least number of significant figures.

22.1 cm3 x 4.3 g/cm3 = 95.03 g

3 sig figs 2 sig figs Answer must have2 significant figures

Round to 95 g (or 9.5 x 101 g)

3 decimal places 2 decimal places Answer must be rounded to two decimal places

Significant figures in calculationsThe results of a calculation based on measurements can not be more precise than the least precise measurement.

For addition and subtraction: The answer must be rounded to the same number of decimal places as the measurement with the lowest number of decimal places

5.375 inches + 96.11 inches = 101.485 inches

Round to 101.49 inches

3.1 miles/gallon x 84.5 gallons = 261.95 miles!!

30.25 gallons + 90.0 gallons = 120.25 gal!!

3.12 grams / 7.0 milliliters = 0.4457142 g/ml!

( 10.50 meters – 1.0 meters )= 3.46715 m/s

2.74 seconds

Report the answer to correct number of significant figures

Measurements and Units

A measurement of a physical property always consists of a numerical value together with a unit of that measurement!Examples:

12.5 meters 70.0 kilograms 89 °F

numericalvalue unit numerical

value unitnumerical

value unit

The Metric SystemA decimal system ( i.e., based on powers of 10) of units for measurements of mass, length, time, and other physical quantitiesExample: Length -- standard unit is the meter ( 1 m ≈ 3.28 ft )

megameter (Mm) 106 m

kilometer (km) 103 m

hectometer (hm) 102 m

dekameter (dam) 101 m

meter (m) 100 m ( = 1 m )

decimeter (dm) 10-1 m

centimeter (cm) 10-2 m

millimeter (mm) 10-3 m

micrometer (µm) 10-6 m

nanometer (nm) 10-9 m

Metric System Units SI Units

• Système International d’Unités!• A different base unit is used for each quantity!• Many other types of units derived from base units

LengthLength: Measure of distance or dimension

-- standard metric system unit (SI unit) is the meter (m)

-- 1 m of length is roughly equivalent to 3.28 ft !Metric system conversions

1 m = 0.001 km

1 m = 100 cm

1 m = 1000 mm

1 km 1000 m 1 km( ) 1000 mm

1 m( )

or 1 cm = 0.01 m

or 1 km = 1000 m

or 1 mm = 0.001 m

Example: How many millimeters are there in one kilometer?

= 106 mm or 1 mm = 10-6 m

Mass

1 kg 1000 g 1 kg( ) 1000 mg

1 g( )

or 1 mg = 0.001 g

or 1 g = 0.001 kg

or 1 mg = 10-6 kg= 106 mg

Example: How many milligrams are there in one kilogram?

Mass: The amount of matter that an object possesses-- standard metric system unit (SI unit) is the kilogram (kg)

-- 1 kg of mass is contained in an object that weighs 2.2 lbs (Note: mass and weight are not the same thing -- this will be explained shortly)Metric conversions

1 kg = 1000 g

1 g = 1000 mg

1 kg

2.2 lbs

1 kg

mass -- amount of matter contained by object

weight -- amount of force exerted on an object due to Earth’s gravitational pull

F = mg

F = (1.0 kg) (9.8 m/s2)F = 9.8 kg m/s2 = 9.8 Newtons

9.8 N = 2.2 lbs

g = 9.8 m / s2

Mass and Weight

25.5 lbs

The mass of an object is constantThe weight of an object can change

g = 1.6 m/s2

154 lbs

g = 9.8 m/s2

Earth’s surface Moon’s surface

mass = 70 kg mass = 70 kg

weight weight

Mass and Weight

TemperatureTemperature -- ability of a substance to transfer heat to another substance

-- temperature is commonly measured on three different scales:Fahrenheit, Celsius, and Kelvin (the SI unit of temperature is Kelvin)

T(K) = T(°C) + 273.15

T(°F) = [ 1.8 x T(°C) ] + 32

T(°C) = [ T(°F) – 32 ] / 1.8

0 °C = 273.15 K = 32 °F

100 °C = 373.15 K = 212 °F

VolumeVolume: The amount of space occupied by matter

-- standard metric system unit (SI unit) is the cubic meter (m3)

-- commonly used units include the liter (L) and milliliter (mL)

1 cubic centimeter (cm3 or cc) is equivalent to 1 mL, BUT...

1 cubic meter (m3) is NOT equivalent to 1 L

1 L = 1000 ml or 1 ml = 0.001 L

1 m3 = 1000 L

1 m3 is equivalent to the volume of a cube with sides 1 m in length

1 cm3 is equivalent to the volume of a cube with sides 1 cm in length

1 cm3 = 1 ml

1 m

1 m

1 m

1 cm1 cm

1 cm

1 m3 = 1000 L1000 ml

1 L( ) = 106 ml

The volume of a regularly shaped solid can be calculated

wh

l

For a rectangular solid:volume = length x width x height

rFor a spherical solid:volume = ( 4 / 3 ) π r 3

For a right circular cylinder:volume = ( π r 2 ) hh

r

How to determine the volume of an irregularly shaped solid

A submerged object displaces a volume of fluid equal to its own volume

Archimedes of Syracuse (287 - 212 BCE)

-- we will take a closer look at this in the lab

Volume of liquidsLIQUIDS

definite volume -- incompressibleindefinite shape -- takes the shape of the containerThe volume of a given amount of liquid is constant, regardless of the shape of the container

Example: 100 g of water occupied a volume of 100 ml

(note that 100 g of a different fluid will not necessarily occupy 100 ml of volume)

10050

150 ml

Density

density -- the ratio of the mass of a substance to the volume occupied by that mass

-- i.e., the mass per unit volume of a substance

2.0 cm

2.0 cm

2.0 cm

mass = 16.0 g

volume = 8.0 cm3

density (ρ) = mass / volume

= 16.0 g / 8.0 cm3

= 2.0 g / cm3

Density and weight are not the same

1 ton

1 ton

A: Neither -- they both weigh the same ( 1 ton )

Q: What weighs more, a ton of feathers or a ton of lead?


1 m3

1 m3


For a given volume, the lead will weigh more since it has a greater density


1 lb

1 ton

So you can say that lead has a greater density than feathers(density is a characteristic property of a substance)

…but you can’t say that lead weighs more than feathers(weight depends on how much of the substance is present)

Note: Density varies with temperatureThe volume of a substance (especially liquids and gasses) varies with temperature

-- i.e., volume increases as temperature increases


mass = 100 g volume = 150 ml

density = 0.67 g / mlmass = 100 g volume = 100 ml density = 1 g / ml

Non-miscible fluids with different densitiesNote: If two fluids are not miscible (i.e., they form two separate phases), the fluid with lower density will float on top of the fluid with higher density.

Ethanol Density = 0.79 g/ml

Soluble in water

Mixtures with water ( density of water = 1.00 g / ml )

Cyclohexane Density = 0.78 g/mlInsoluble in water

Methylene chloride Density = 1.32 g/mlInsoluble in water

Homogeneous mixture of water

and ethanol

cyclohexane

water

water

methylene chloride

Intensive and extensive properties of matter

All substances have intensive and extensive properties !An intensive property is independent of the amount of the substance present

Examples: temperature, pressure, density, melting point, boiling pointIntensive properties can often be used to identify a substance

!An extensive property varies with the amount of the substance present

Examples: mass, volume, length, energy

Density calculations

density =mass

volume

For problems involving density:• you will typically be given the values for two of these variables (or

the information to calculate their values)• you then have to solve for the value of the third variable

Sample density problems

Calculate the density of mercury if 1.00 x 102 g of mercury occupies a volume of 7.36 cm3

density =mass

volume=

1.00 x 102 g7.36 cm3

= 13.6 g/cm3

What is the mass in grams of a cube of gold if the length of the cube is 2.00 cm? The density of gold is 19.32 g/cm3.

density =mass

volumex volumevolume x

density= volume xmass

volume = l x w x h = (2.00 cm) (2.00 cm) (2.00 cm) = 8.00 cm3

= ( 8.00 cm3) 19.32g

cm3= 155 g

Sample density problems

Calculate the volume of 65.0 g of liquid methanol if its density is 0.791 g/ml

density =mass

volumex volumevolume x

volume x density = mass

density density

volume =mass

density65.0 g

0.791 g/ml= = 82.2 ml

Unit conversionsTo convert a measurement from one type of unit to another type of unit you must use a conversion factor

1. To obtain a conversion factor, start with a known equality between the units you are converting from and the units you are converting to

1 in = 2.54 cm

2. Set up a fraction ( i.e., the conversion factor) based on the equality-- put the units that your are converting to in the numerator-- put the units you are converting from in the denominator

2.54 cm1 in

3. Multiply your measurement (with the original units) by this conversion factor to get an equivalent measurement with the desired units

5.30 in 2.54 cm1 in

= 13.5 cmCheck to see that your units cancel out correctly

Check sig figs in your answer

Example: How may centimeters are there in 5.30 inches?

1 gal = 3.785 L

3.785 L1 gal

10.0 gal 3.785 L1 gal

= 37.85 L= 37.9 L

To convert a measurement from one type of unit to another type of unit you must use a conversion factor

Example: How may liters are there in 10.0 gallons?

Unit conversions

1. To obtain a conversion factor, start with a known equality between the units you are converting from and the units you are converting to

2. Set up a fraction (i.e., the conversion factor) based on the equality-- put the units that your are converting to in the numerator-- put the units you are converting from in the denominator

3. Multiply your measurement (with the original units) by this conversion factor to get an equivalent measurement with the desired units

Check to see that your units cancel out correctly

Check sig figs in your answer

Dimensional analysisdimensional analysis -- using units as a guide to solving problems

-- this is particularly useful when carrying out complex unit conversions

Example: How many centimeters are there in 1.76 yards? ( 1 m = 1.094 yd )

Known information:

Based on this information alone, you cannot covert directly from yds to cm

-- you need two steps:

1 m = 1.094 ydyds

Starting point:

cm

Goal:

m

1.76 yd1 m

1.094 yd100 cm

1 m= 160.8775 cm = 161 cm

1 m = 100 cm

(1) yds to m (2) m to cm

Dimensional analysis

Example: How many seconds are there in a year?(note that there are exactly 365.25 days in a Julian year)

Known information:

1 yr = 365.25 dayyr

Starting point: Goal:

1 yr365.25 d

1 yr24 hr1 d

= 31,557,600 sec

1 day = 24 hrsecday hr min

1 hr = 60 min1 min = 60 sec

60 min1 hr

60 sec1 min

Units raised to a powerWhen constructing a conversion factor for units raised to a power, remember to raise both the number and the unit to the power

Example: What is the conversion factor you would use to convert a measurement of area in square inches to square centimeters?

1 in = 2.54 cmStart with a known equality:

( 1 in )2 = ( 2.54 cm )2Square both sides:

12 in2 = 2.542 cm2

1 in2 = 6.45 cm2

Express as a fraction to obtain conversion factor:

6.45 cm2

1 in2

Dimensional analysis

Example: The surface of a table has an area of 1.250 square meters. What is its area in square centimeters?

Known information:

1 m = 100 cm

Starting point: Goal:

1.250 m21 m2

= 12,500 cm2

this does NOT mean that 1m2 = 100 cm2

Be careful with unit conversions involving units raised to a power

m2 cm2

( 1 m ) = ( 100 cm )2 2

1 m2

10,000 cm2

= 1.250 x 104 cm2

= 10,000 cm2

unit 1 slides pt2 copy - laulima : gateway : welcome non-zero numbers are always signiﬁcant 2.38 g...

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