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Numbers in Science Chapter 2

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Numbers in Science

Chapter 2

- 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

- 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

- 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

- Established in 1960 by an international agreement to standardize science units
- These units are in the metric system

- 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)

- About 3½ inches longer than a yard

- 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

- 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

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

- 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

- What is power of 10 Big?

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)

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

- 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…..

- 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

- 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

- We follow guidelines (i.e. rules) to determine what numbers are significant
- Nonzero integers are always significant
- 2753
- 89.659
- .281

- Nonzero integers are always significant
- Zeros
- Captive zeros are always significant (zero sandwich)
- 1001.4
- 55.0702
- 4780.012

- Captive zeros are always significant (zero sandwich)

- 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

- Leading zeros never count as 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

- 102,3400.0179692,017
- 1.0 x 1071,200.000.1192
- 1,908,021.00.0000028.01010 x 1014

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

- 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

- counting numbers

- 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

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.

- Round 0.00649710 to 3 sig figs.

- 5.18 x 0.0208
- 21 + 13.8 + 130.36
- 116.8 – 0.33

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

- 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 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

- counting numbers

Fundamental Unit 100

- 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

- How many 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

- 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

- 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

- 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.

- 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

- 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

- 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 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

- 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?

- 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 – 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.

- 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?