Math and Measurement

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					Math and Measurement

        Unit 2
Numbers…which ones are important?
What is 13/7?

Is it 1.8571428?

Or…is it 1.86? Or 1.9? Or 2?

Where do we round?
Significant Digits
  Rules for Counting Significant
         Figures - Details
• Exact numbers have an infinite
 number of significant figures.

   1 inch = 2.54 cm, exactly
   Rules for Counting Significant
          Figures - Details
• Nonzero integers always count as
  significant figures.

             3456 has
             4 sig figs.
    Rules for Counting Significant
           Figures - Details
•   Zeros
    - Leading zeros do not count as

      significant figures.

            • 0.0486 has
              3 sig figs.
    Rules for Counting Significant
           Figures - Details
•   Zeros
     - Captive zeros always count as
       significant figures.

            • 16.07 has
             4 sig figs.
    Rules for Counting Significant
           Figures - Details
•   Zeros
    Trailing zeros are significant only if
    the number contains a decimal
    point.

               9.300 has
               4 sig figs.
             Sig Fig Practice #1
How many significant figures in each of the following?

          1.0070 m         5 sig figs
           17.10 kg        4 sig figs

         100,890 L         5 sig figs

      3.29 x 103 s         3 sig figs
        0.0054 cm          2 sig figs
        3,200,000          2 sig figs
       Rules for Significant Figures in
         Mathematical Operations

•    Multiplication and Division: # sig figs in
    the result equals the number in the least
    precise measurement used in the
    calculation.

               6.38 x 2.0 =
          12.76  13 (2 sig figs)
                Sig Fig Practice #2
   Calculation        Calculator says:      Answer
3.24 m x 7.0 m         22.68 m2              23 m2
100.0 g ÷ 23.7 cm3     4.219409283 g/cm3   4.22 g/cm3
0.02 cm x 2.371 cm     0.04742 cm2          0.05 cm2
710 m ÷ 3.0 s          236.6666667 m/s      240 m/s
1818.2 lb x 3.23 ft    5872.786 lb·ft      5870 lb·ft
1.030 g ÷ 2.87 mL      2.9561 g/mL         2.96 g/mL
  Rules for Significant Figures in
    Mathematical Operations
• Addition and Subtraction: The number of
  decimal places in the result equals the
  number of decimal places in the least
  precise measurement.

           6.8 + 11.934 =
      18.734  18.7 (3 sig figs)
               Sig Fig Practice #3
   Calculation        Calculator says:   Answer
3.24 m + 7.0 m           10.24 m         10.2 m
100.0 g - 23.73 g         76.27 g        76.3 g
0.02 cm + 2.371 cm        2.391 cm       2.39 cm
713.1 L - 3.872 L         709.228 L      709.2 L
1818.2 lb + 3.37 lb       1821.57 lb     1821.6 lb
2.030 mL - 1.870 mL       0.16 mL        0.160 mL
            Practice Question
Questions 1-2 refer to the following sets of
  numbers.
A.1.023 g
B.0.0030 mL
C.40,500 m

1.Is a number containing three significant
  figures
2.Is a measure of mass
Scientific Notation

In science, we deal with some very LARGE
numbers:
1 mole = 602000000000000000000000

In science, we deal with some very SMALL
numbers:
Mass of an electron =
0.000000000000000000000000000000091 kg
 Imagine the difficulty of calculating the
 mass of 1 mole of electrons!


0.000000000000000000000000000000091 kg
      x 602000000000000000000000
   ???????????????????????????????????
Scientific Notation:
A method of representing very large or very small
numbers in the form:
        M x 10n

     M is a number between 1 and 10
     n is an integer
          2 500 000 000.
              9 8 7 6 5 4 3 2 1
Step #1: Insert an understood decimal point
Step #2: Decide where the decimal must end
      up so that one number is to its left
Step #3: Count how many places you bounce
      the decimal point
Step #4: Re-write in the form M x 10n
2.5 x   10 9


The exponent is the
number of places we
moved the decimal.
           0.0000579
                 1 2 3 4 5


Step #2: Decide where the decimal must end
      up so that one number is to its left
Step #3: Count how many places you bounce
      the decimal point
Step #4: Re-write in the form M x 10n
       5.79 x           10 -5


       The exponent is negative
       because the number we
       started with was less than 1.

When you move the decimal to the left, the exponent is
                     positive.
When you move the decimal to the right, the exponent is
                     negative
                     You Try!
Convert the following
   numbers to scientific     Answers:
   notation
1. 21.9                      1. 2.19 X 101
2. 6022                      2. 6.022 X 103
3. 0.12011                   3. 1.2011 X 10-1
Convert the following into
   expanded form             4. 1800
4. 1.8 X 103                 5. .000081
5. 8.1 X 10-5                6. 720
6. 7.2 X 102
       Multiplying and Dividing in
          Scientific Notation
1. Multiply or divide the      Example #1
   “M” values
2. If multiplying, add the     (1.35 x 104) x (2.35 x 105)
   exponents
3. If dividing, subtract the
   exponents.
4. If necessary, move
   decimal and adjust          Example #2
   exponent to get
   numbers back into           (2.6 x 104) / (4.6 x 103)
   scientific notation.
                            You try
1. (6 X 103) x (4 X 10-3)       Answers:

                                1. 2.4 x 101
2. (3 x 104) x (4.5 x 105)      2. 1.35 x 1010
                                3. 5 x 10-3
3. ( 4.5 x 10-5) / (9 x 10-3)
        Nature of Measurement
Measurement – quantitative observation
consisting of 2 parts
            Part 1 – number
              Part 2 – unit
                Examples:
                20 grams
        6.63 x 10-34 Joule seconds
      Accurate or Precise?
Accurate
measurements are
close to the actual or
accepted value.


Precise
measurements are
close to one another.

More than one
measurement must be
taken to determine if
the measurements are
precise.
The Fundamental SI Units
(le Système International, SI)
                    SI Prefixes
 Prefix     Abbr.         Meaning       Exponent

 Giga        G        1,000,000,000       109
 Mega        M          1,000,000         106
   Kilo      K             1,000          103
 Hecto       h              100           102
  Deca       D               10           101
Base unit                     1
  deci       d             1/10           10-1
  Centi      c            1/100
                                          10-2
  Milli      m            1/1000
                                          10-3
 Micro                1/1,000,000
                                          10-6
  Nano        n      1/ 1,000,000,000
                                          10-9
     Converting Among SI Units
Convert 2.6 grams to   Convert 5.25 decigrams to
  milligrams             micrograms
                      You Try!
• Convert the following        Answer:

1. 3.4 liters to milliliters   1. 3,400 mL
2. 7,899 milligrams to         2. 7.899 g
   grams                       3. .0277 cg
3. 277 kilograms to            4. 2,000,000 ug
   centigrams
4. 2 meters to
   micrometers
                  Derived SI Units
 • Produced by multiplying or dividing standard
   units.
                                  5m
 For Example:
Area = (Length)(Width)              Width


                   2.5 m   Length



          Area =     (2.5 m)(5 m)   = 12.5 m2
                  Density
The ratio of mass to volume, or mass divided by
                     volume.
                 mass                  m
   Density =                     D=
                volume                 V
                    Density
• A measure of how closely matter is packed
  into a volume.
• Unique for each compound.
  – Density of water is 1.00 g/mL at 25˚C.
  – Increasing temperature decreases the density, so
    densities are given with temperatures.
• An intensive property.
• Substances that are less dense float in
  substances more dense.
                Density Problems
 A sample of aluminum metal has a mass of 8.4 g. The
 volume of the sample is 3.1 cm3. Calculate the density
                    of aluminum.

Given:   m = 8.4 g            8.4 g             Don’t
         V = 3.1 cm3   D=                       forget
                            3.1 cm3             units!
Unknown: D = ?
Equation:                                           Box
                       D = 2.7   g/cm3
            m                                     answer!
   D=       v
                        Don’t forget units!!!
               You try!
An unknown liquid is discovered at a crime
scene. A volume of 2.3 mL has a mass of 4.1
     grams, what the liquid’s density?
            Density Problems
Diamond has a density of 3.26 g/cm3. What is the mass
    of a diamond that has a volume of 0.350 cm3?
          Density Problems
A sample of metal is found to have a mass of
4.56 g and a density of 1.98 g/mL. What is the
            volume of this metal?
  Density Problem (No calculator)
The typical battery in a car is filled with a
  solution of sulfuric acid, which is
  approximately 39.9% sulfuric acid. If the
  density of this solution is 1.3 g/mL, determine
  the number of grams of acid present in 500.0
  mL of battery solution.
1. What is the
   volume of 5
   grams of this
   substance?

2. What is the
   approximate
   density of the
   substance?
Converting Temperatures
             C = 5/9 (F-32)

             F=

             K = C + 273

             C=
                     You Try!
Convert the following      Answers
   temperatures
1. 293 K to Celsius        1. 20 K
2. Room temperature to     2. About 24 C
   Celsius                 3. About 309 K
3. Internal body
   temperature to Kelvin
   Steps to complete these problems

Step 1: Read the problem CAREFULLY.
Step 2: Determine the unit for the answer
Step 3: Write down all values given in the problem
  and retrieve any needed conversion factors
Step 4: Set up the problem (watch carefully as teacher
  does this step)
Step 5: Calculate—Multiply by numbers on the top
  and divide by those on the bottom
                 Practice #1
The record long jump is 349.5 inches. Convert
  this to meters. There are 2.54 cm in an inch.
                Practice #2
A car is traveling 55.0 miles per hour. Convert
  this to meters per second. One mile is equal to
  1.61 km.
               Practice #3
How many mg are there is a 5.00 grain aspirin
  tablet?
1 grain = 0.00229 oz.
There are 454 grams/lb. There are 16 oz./lb
                Practice #4
Convert 24 km/h to m/s (write out all steps
  before using calculator).
                 Practice #5
In 1980, the US produced 18.4 billion (18.4 X
  109) pounds of phosphoric acid to be used in
  the manufacture of fertilizer. The average cost
  of the acid is $318/ton. (1 ton = 2000 lbs).
  What was the total value of the phosphoric
  acid produced?

				
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