Numbers in Science
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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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Numbers in science chapter 2

Numbers in Science

Chapter 2


Measurement

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

The Unit


The measurement system units

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

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


Numbers in science chapter 2

Fundamental Unit 100


Fundamental si units

Fundamental SI Units

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

  • These units are in the metric system


Length

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)


Figure 2 1 comparison of english and metric units for length on a ruler

Figure 2.1: Comparison of English and metric units for length on a ruler.


Volume

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


Numbers in science chapter 2

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

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


Thermometers based on the three temperature scales in a ice water and b boiling water

Thermometers based on the three temperature scales in (a) ice water and (b) boiling water.


The number

The number


Scientific notation

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

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

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

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

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


Are you sure about that number

Are you sure about that number?


Uncertainty in measured numbers

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

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

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

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


  • Have a little fun remembering sig figs

    Have a little fun remembering sig figs

    • http://www.youtube.com/watch?v=ZuVPkBb-z2I


    Exact numbers

    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

    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

    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

    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

    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


    Moving unit to unit conversion

    Moving unit to unit: Conversion


    Exact numbers1

    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

    The Metric System

    Fundamental Unit 100


    Movement in the metric system

    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

    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


    Numbers in science chapter 2

    Converting Between Metric and non-Metric (English) units


    Converting non metric 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 units1

    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

    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

    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

    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


    Numbers in science chapter 2

    Measurements

    and

    Calculations


    Density

    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


    Using density in calculations

    Using Density in Calculations


    Density example problems

    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?


    How could you find your density

    How could you find your density?


    Volume by displacement

    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?


    Numbers in science chapter 2

    • 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

    • 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

    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?


    Numbers in science chapter 2

    • 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?


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