# Numbers in Science Chapter 2 - PowerPoint PPT Presentation

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Numbers in Science Chapter 2. Measurement. What is measurement? Quantitative Observation Based on a comparison to an accepted scale. A measurement has 2 Parts – the Number and the Unit Number Tells Comparison Unit Tells Scale There are two common unit scales English Metric. The Unit.

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Numbers in Science Chapter 2

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#### Presentation Transcript

Numbers in Science

Chapter 2

### Measurement

• What is measurement?

• Quantitative Observation

• Based on a comparison to an accepted scale.

• A measurement has 2 Parts – the Number and the Unit

• Number Tells Comparison

• Unit Tells Scale

• There are two common unit scales

• English

• Metric

### The measurement System units

• English (US)

• Length – inches/feet

• Distance – mile

• Volume – gallon/quart

• Mass- pound

• Metric (rest of the world)

• Length – meter

• Distance – kilometer

• Volume – liter

• Mass - gram

### Related Units in the Metric System

• All units in the metric system are related to the fundamental unit by a power of 10

• The power of 10 is indicated by a prefix

• The prefixes are always the same, regardless of the fundamental unit

Fundamental Unit 100

### Fundamental SI Units

• Established in 1960 by an international agreement to standardize science units

• These units are in the metric system

### Length…..

• SI unit = meter (m)

• About 3½ inches longer than a yard

• 1 meter = distance between marks on standard metal rod in a Paris vault or distance covered by a certain number of wavelengths of a special color of light

• Commonly use centimeters (cm)

• 1 inch (English Units) = 2.54 cm (exactly)

### Volume

• Measure of the amount of three-dimensional space occupied by a substance

• SI unit = cubic meter (m3)

• Commonly measure solid volume in cubic centimeters (cm3)

• Commonly measure liquid or gas volume

in milliliters (mL)

• 1 L is slightly larger than 1 quart

• 1 mL = 1 cm3

### Mass

• Measure of the amount of matter present in an object

• SI unit = kilogram (kg)

• Commonly measure mass in grams (g) or milligrams (mg)

• 1 kg = 2.2046 pounds, 1 lbs.. = 453.59 g

### Temperature Scales

Any idea what the three most common temperature scales are?

• Fahrenheit Scale, °F

• Water’s freezing point = 32°F, boiling point = 212°F

• Celsius Scale, °C

• Temperature unit larger than the Fahrenheit

• Water’s freezing point = 0°C, boiling point = 100°C

• Kelvin Scale, K (SI unit)

• Temperature unit same size as Celsius

• Water’s freezing point = 273 K, boiling point = 373 K

### Scientific Notation

• Technique Used to Express Very Large or Very Small Numbers

• 135,000,000,000,000,000,000 meters

• 0.00000000000465 liters

• Based on Powers of 10

• What is power of 10 Big?

• 0,10, 100, 1000, 10,000

• 100, 101, 102, 103, 104

• What is the power of 10 Small?

• 0.1, 0.01, 0.001, 0.0001

• 10-1, 10-2, 10-3, 10-4

### Writing Numbers in Scientific Notation

1. Locate the Decimal Point : 1,438.

2. Move the decimal point to the right of the non-zero digit in the largest place

- The new number is now between 1 and 10

- 1.438

3. Now, multiply this number by a power of 10 (10n), where n is the number of places you moved the decimal point

- In our case, we moved 3 spaces, so n = 3 (103)

### The final step for the number……

4. 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

• We moved left, so 3 is positive

• 1.438 x 103

### 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

Try it for these numbers: 2.687 x 106 and 9.8 x 10-2

• We reverse the process and go from a number in scientific notation to standard form…..

### Let’s Practice…..

• Change these numbers to Scientific Notation:

• 1,340,000,000,000

• 697, 000

• 0.00000000000912

• Change these numbers to Standard Form:

• 3.76 x 10-5

• 8.2 x 108

• 1.0 x 101

1.34 x 1012

6.97 x 105

9.12 x 10-12

0.0000376

820,000,000

10

### Uncertainty in Measured Numbers

• A measurement always has some amount of uncertainty, you always seem to be guessing what the smallest division is…

• To indicate the uncertainty of a single measurement scientists use a system called significant figures

• The last digit written in a measurement is the number that is considered to be uncertain

cm

### Rules, Rules, Rules….

• We follow guidelines (i.e. rules) to determine what numbers are significant

• Nonzero integers are always significant

• 2753

• 89.659

• .281

• Zeros

• Captive zeros are always significant (zero sandwich)

• 1001.4

• 55.0702

• 4780.012

### Significant Figures – Tricky Zeros

• Zeros

• Leading zeros never count as significant figures

• 0.00048

• 0.0037009

• 0.0000000802

• Trailing zeros are significant if the number has a decimal point

• 22,000

• 63,850.

• 0.00630100

• 2.70900

• 100,000

### Significant Figures

Scientific Notation

• All numbers before the “x” are significant. Don’t worry about any other rules.

• 7.0 x 10-4 g has 2 significant figures

• 2.010 x 108 m has 4 significant figures

• How many significant figures are in these numbers?

• 102,3400.0179692,017

• 1.0 x 1071,200.000.1192

• 1,908,021.00.0000028.01010 x 1014

### Exact Numbers

• Exact Numbers are numbers known with certainty

• Unlimited number of significant figures

• They are either

• counting numbers

• number of sides on a square

• or defined

• 100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm

• 1 kg = 1000 g, 1 LB = 16 oz

• 1000 mL = 1 L; 1 gal = 4 qts.

• 1 minute = 60 seconds

### Calculations with Significant Figures

• Exact numbers do not affect the number of significant figures in an answer

• Answers to calculations must be rounded to the proper number of significant figures

• round at the end of the calculation

• For addition and subtraction, the last digit to the right is the uncertain digit.

• Use the least number of decimal places

• For multiplication, count the number of sig figs in each number in the calculation, then go with the smallest number of sig figs

• Use the least number of significant figures

### Rules for Rounding Off

If the digit to be removed

• is less than 5, the preceding digit stays the same

• Round 87.482 to 4 sig figs.

• is equal to or greater than 5, the preceding digit is increased by 1

• Round 0.00649710 to 3 sig figs.

In a series of calculations, carry the extra digits to the final result and then round off

Don’t forget to add place-holding zeros if necessary to keep value the same!! Round 80,150,000 to 3 sig figs.

### Examples of Sig Figs in Math

• 5.18 x 0.0208

• 21 + 13.8 + 130.36

• 116.8 – 0.33

Answers must be in the proper number of significant digits!!!

### Solutions:

• 0.107744 round to proper # sig fig

• 5.18 has 3 sig figs, 0.0208 has 3 sig figs so answer is 0.108

• 165.47

• Limiting number of sig figs in addition is the smallest number of decimal places = 12 (no decimals) answer is 165

• 116.47

• Same rule as above so answer is 116.5

### Exact Numbers

• Exact Numbers are numbers known with certainty

• They are either

• counting numbers

• number of sides on a square

• or defined

• 100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm

• 1 kg = 1000 g, 1 LB = 16 oz

• 1000 mL = 1 L; 1 gal = 4 qts.

• 1 minute = 60 seconds

### The Metric System

Fundamental Unit 100

### Movement in the Metric system

• In the metric system, it is easy it is to convert numbers to different units.

• Let’s convert 113 cm to meters

• Figure out what you have to begin with and where you need to go..

• How many cm in 1 meter?

• 100 cm in 1 meter

• Set up the math sentence, and check that the units cancel properly.

• 113 cm [1 m/100 cm] = 1.13 m

### Let’s Practice converting metric units

• 250 mL to Liters

• 0.250 mL

• 1.75 kg to grams

• 1,750 grams

• 88 µL to mL

• 0.088 mL

• 475 cg to kg

• 47,500,000 or

• 4.75 x 107

• 328 mm to dm

• 3.28 dm

• 0.00075 nL to µL

• 0.75 µL

Converting Between Metric and non-Metric (English) units

### Converting non-Metric Units

• Many problems involve using equivalence statements to convert one unit of measurement to another

• Conversion factors are relationships between two units

• Conversion factors are generated from equivalence statements

• e.g. 1 inch = 2.54 cm can giveor

### Converting non-Metric Units

• Arrange conversion factor so starting unit is on the bottom of the conversion factor

• Convert kilometers to miles

• You may string conversion factors together for problems that involve more than one conversion factor.

• Convert kilometers to inches

• Find the relationship(s) between the starting and final units.

• Write an equivalence statement and a conversion factor for each relationship.

• Arrange the conversion factor(s) to cancel starting unit and result in goal unit.

### Practice

• Convert 1.89 km to miles

• Find equivalence statement 1mile = 1.609 km

• 1.89 km (1 mile/1.609 km)

• 1.17 miles

• Convert 5.6 lbs to grams

• Find equivalence statement 454 grams = 1 lb

• 5.6 lbs(454 grams/1 lb)

• 2500 grams

• Convert 2.3 L to pints

• Find equivalence statements: 1L = 1.06 qts, 1 qt = 2 pints

• 2.3 L(1.06 qts/1L)(2 pints/1 qt)

• 4.9 pints

### Temperature Conversions

• To find Celsius from Fahrenheit

• oC = (oF -32)/1.8

• To find Fahrenheit from Celsius

• oF = 1.8(oC) +32

• Celsius to Kelvin

• K = oC + 273

• Kelvin to Celsius

• oC = K – 273

### Temperature Conversion Examples

• 180°C to Kelvin

• To convert Celsius to Kelvin add 273

• 180+ 273 = 453 K

• 23°C to Fahrenheit

• Use the conversion factor: F = (1.80)C + 32

• F = (1.80)23 + 32

• F=73.4 or 73°F

• 87°F to Celsius

• Use the conversion factor C=5/9(F-32)

• C = 5/9(87-32)

• C = 30.5555555… or 31°C

• 694 K to Celsius

• To convert K to C, subtract 273

• 694-273= 421°C

Measurements

and

Calculations

### Density

• Density is a physical property of matter representing the mass per unit volume

• For equal volumes, denser object has larger mass

• For equal masses, denser object has small volume

• Solids = g/cm3

• Liquids = g/mL

• Gases = g/L

• Volume of a solid can be determined by water displacement

• Density : solids > liquids >>> gases

• In a heterogeneous mixture, denser object sinks

### Density Example Problems

• What is the density of a metal with a mass of 11.76 g whose volume occupies 6.30 cm3?

• What volume of ethanol (density = 0.785 g/mL) has a mass of 2.04 lbs?

• What is the mass (in mg) of a gas that has a density of 0.0125 g/L in a 500. mL container?

### Volume by displacement

• To determine the volume to insert into the density equation, you must find out the difference between the initial volume and the final volume.

• A student attempting to find the density of copper records a mass of 75.2 g. When the copper is inserted into a graduated cylinder, the volume of the cylinder increases from 50.0 mL to 58.5 mL. What is the density of the copper in g/mL?

• A student masses a piece of unusually shaped metal and determines the mass to be 187.7 grams. After placing the metal in a graduated cylinder, the water level rose from 50.0 mL to 60.2 mL. What is the density of the metal?

• A piece of lead (density = 11.34 g/cm3) has a mass of 162.4 g. If a student places the piece of lead in a graduated cylinder, what is the final volume of the graduated cylinder if the initial volume is 10.0 mL?

### Percent Error

• Percent error – absolute value of the error divided by the accepted value, multiplied by 100%.

% error = measured value – accepted value x 100%

accepted value

• Accepted value – correct value based on reliable sources.

• Experimental (measured) value – value physically measured in the lab.

### Percent Error Example

• In the lab, you determined the density of ethanol to be 1.04 g/mL. The accepted density of ethanol is 0.785 g/mL. What is the percent error?

• The accepted value for the density of lead is 11.34 g/cm3. When you experimentally determined the density of a sample of lead, you found that a 85.2 gram sample of lead displaced 7.35 mL of water. What is the percent error in this experiment?

• Joe measured the boiling point of hexane to be 66.9 °C. If the actual boiling point of hexane is 69 °C , what is the percent error?

• A student calculated the volume of a cube to be 68.98 cm3. If the true volume is 71.08 cm3, what is the student’s percent error?

• Tom used the density of copper and the volume of water displaced to measure the mass of a copper pipe to be 145.67 g. When he actually weighed the sample, he found a mass of 146.82 g. What was his percent error?