Chapter 3: Fundamental Measurements

1
Chapter 3 Fundamental Measurements
2
Measurement

• Components of Measurement
• Numerical quantity
• Unit
• Name of substance
• For example,
• 325.0 mL water

Numerical quantity
unit
Name of substance
3
Metric System

• Also called the International or SI system.
• Based on Units of 10
• SI base Units
• Other units derived from base units
• Prefixes indicating power of ten

4
SI Units
Quantity Name Symbol Length meter m Mass kilog
ram kg Temperature Kelvin K Time second s Amount
mole mol
5
Common Metric Prefixes
Prefix Symbol Decimal Exponential Mega M 1,000,00
0 106 Kilo k 1000 103 Centi c 0.01 10-2 M
illi m 0.001 10-3 Micro ? 0.000001 10-6
6
Metric Units of Length

• Base Unit is the meter (m), a little longer than
  a yard.
• Other common Units
• Centimeter, cm 1 cm 0.01m, 0.4 inch
• Kilometer, km 1 km 1000 m, 5/8 mile
• Millimeter, mm 1 mm 0.001 m, thickness of a
  dime.

7
Units of Length
Insert Fig. 3.4
8
Dimensional Analysis

• (Also called the factor-label method)
• To convert a measurement from one unit to
  another, multiply the known quantity and unit(s)
  by a conversion factor to equal the desired
  quantity and unit(s)

Known quantity and unit(s)
Quantity with desired unit(s)
Conversion Factor(s)

X
9
Converting Units Using Conversion Factors

• Equivalencies yield two conversion factors
• For example
• 1 ft 12 in
• 1ft/12 in 1
• or 12 in/1 ft 1
• How many inches in 5 ft?
• Number of inches 5ft x

12 in 1 ft
60 in.
10
Considerations in Choosing Conversion Factors

• The chosen conversion factor cancels all units
  except those required for the answer.
• Set up the calculation so that the unit you are
  converting from (beginning unit) is on the
  opposite part of the conversion factor (numerator
  or denominator) from the unit you are converting
  to (final unit).

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Formulas
final unit
beginning unit x
final unit
beginning unit
ft
Mi x
ft
mi
12
Choosing the correct conversion factor

• Choose the conversion factor which will yield the
  correct units in the denominator and numerator
  after the units are cancelled.
• Problem How many cm are in 50 m?
• 1. Unit equivalency 1 cm 0.01 m
• 2. Conversion factors 1cm/0.01m or 0.01m/1cm
• 3. In the answer, you want cm in the numerator so
  choose 1 cm/0.01m.

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Finish the Problem

• 4. Multiply given by chosen conversion factor
• Answer Given x Conversion factor
• Cm 50 m x

1 cm
5000 cm
0.01 m
5. Cancel units to get correct answer
14
Conversion Factors May Be Multiplied Together in
Series

• Example Convert 179,800 s to days.
• Solution Plan s min hours
  days
• Conversion equivalents 1 min 60 s,
• l hr 60 min, 1 day 24 hr
• Conversion factors 1 min/60 s, 1 hr/60min,
• 1 day/ 24 hr

179800 s x
1min/60s x
1hr/60 min x
1 day/24hr
2.08 days
15
Ratios with New Final Unit in Denominator

Final unit1
Beginning unit
Final unit1
X

Beginning unit
Final unit2
Final unit2
mi
mi
h
x

min
h
min
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Problem Which Has New Final Unit in Denominator

• Convert 55.0 mi/h to mi/min
• 1 h 60 min
• Conversion factor 1 h/60 min
• mi/min 55.0

mi
1 h
0.917
x

mi
h
60 min
min
17
Volume Conversions

• The Volume of a rectangular solid
• Volume length x width x height
• or V (l)(w)(h)
• For a cube V (l)(l)(l) (l)3
• The SI unit of length is the meter.
• The SI unit of volume is the cubic meter, (a box
  1 m on each side) m3
• m3 is a derived unit.

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Atomic Number    Atomic Symbol    Atomic Mass
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Volume Equivalencies

• 1 m3 (100 cm)3 (100)3 cm3
• 1,000,000 cm3
• 1 cm3 1 cc 1 mL (milliliter)
• 1mL 0.001 L, 1000mL 1 L (liter)
• 1 L 1 dm3 .001 m3
• 1mL 1000 mL (microliter, a millionth of a liter)

20
Metric Volume and Length Relationships
Insert figure 3.5
21
Volume Conversions

• Convert 324 mL to Liters
• 1 mL 0.001 L
• Conversion factor 0.001L/mL

324 mL x
0.001 L/mL
0.324 L
22
Mass Conversions

• Important Mass Equivalencies
• 1 kg 1000g 1 g 0.001 kg
• 1 mg 0.001 g 1 g 1000 mg
• 1 µg 0.001 mg 0.000001 g
• 1 g 1,000,000 µg

23
Mass Conversions

• Example problem
• Convert 456.0 g to kg.
• Answer
• Equivalency 1000 g 1 kg
• Factor 1 kg/1000 g
• 456.0g x 1 kg/1000 g

0.4560 kg
24
Mass Equivalents
Insert figure 3.10
25
English/Metric Conversions

• Common English/Metric Conversion Equivalencies
  are shown in Table 3.5
• English/Metric conversions may also be done with
  conversion factors

Known quantity and unit(s)
Quantity with desired unit(s)
Conversion Factor(s)

X
26
Another Problem

• Convert 31.0 in to cm
• 1 in 2.54 cm
• 2.54cm/ in
• 31.0 in x

2.54 cm
78.7 cm
1 in
27
Conversion Factors May Be Multiplied Together in
Series

• Convert 55 mi/h to m/s
• Conversion equivalencies/factors
• 1 km 0.62 mi 1 km/0.62mi
• 1 km 1000 m 1000m/1 km
• 60 min 1 h 1h/60 min
• 60 s 1 min 1 min/60 s
• 55

mi
1km
1000m
1 h
x
x
1 min
x
x
h
0.62 mi
km

24.6 m/s
60 min
60 s
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Uncertainty in Measurement

• Differentiate between precision and accuracy.
• Precision- The closeness of a measurement to
  other measurements of the same phenomenon in a
  series of experiments
• Accuracy- The closeness of a measurement to the
  true value.

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A Comparison of Accuracy and Precision
30
Significant Figures

• The number of significant figures (digits) is a
  measure of the uncertainty of a measurement. The
  greater the number of significant digits, the
  less uncertain a number is.
• The number of significant figures equals the
  number of digits that are certain, plus one
  additional digit , which is an uncertain digit.

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Reading a Volumetric Device
Insert figure 3.13
Note the Meniscus
32
Determine the Number of Significant Digits

• All digits are significant except zeros that are
  not measured but are used only to position the
  decimal point.
• 1. The measured quantity should have a decimal
  point.
• 2. Start at the left of the number and move
  right until you reach the first non-zero digit.
• 3. Count that digit and every digit to its right
  as significant.

33
If the Number Has No Decimal Point

• For example 9500 m. It would be assumed that the
  zeros are not significant. Scientific notation
  is used to show which zeros are significant
• 9.5 x 103 has 2 significant digits.
• 9.50 x 103 has 3 significant digits.
• 9.500 x 103 has 4 significant digits.

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How many significant digits does each of the
following numbers have?

• Number significant digits
• 0.0050 m 2
• 0.0003056 L 4
• 56,980. Cm3 5
• 3.7890 x 104 s 5
• 1.2 x 10-8 mL 2

35
Exact Numbers

• There is no uncertainty associated with exact
  numbers. Numbers that are a result of counting
  are exact (if the counting is done exactly).
  Definitions contain exact numbers. For example
  60 s in one min, 1000 mg 1 g, 0.01m 1 cm
• Exact numbers do not limit the number of
  significant figures in an answer.
• Exact numbers have the number of significant
  digits the calculation requires.

36
Multiplying and Dividing

• When numbers are multiplied and divided, the
  answer has the same number of significant digits
  as measurement with the least number of
  significant digits. (Exact numbers have an
  infinite number of significant digits.)
• E. g., 0.00569 x 0.91 0.0052
• 3 sd 2 sd 2 sd

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Significant Figures in Division

• 9863./876.89 11.25
• 4 sd 5 sd 4 sd

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Significant Figures in Addition and Subtraction

• In addition and subtraction, the answer has the
  same number of decimal places as the measurement
  with the fewest number of decimal places
• E. g., 23.560 2.1 25.7
• 3 dp 1 dp 1 dp
• 0.0056 - 0.004067 0.0015
• 2 dp 4 dp 2 dp

39
Rounding

• The round a number to the proper number of
  significant figures or decimal points, start at
  the right of the number and remove all digits
  needed to have the correct number of digits.
• If the last digit removed is gt5, round up.
  (Increase the last remaining digit by 1.)
• If the last digit removed islt 5 leave the
  remaining last digit the same.

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Scientific Notation

• Exponential notation is also called Scientific
  Notation.
• A method of expressing very large and very small
  numbers.

41
Convert Between Notations

• Numbers in Scientific Notation are in the form
  X.YZ x 10n.
• X.YZ is the coefficient n is the exponent.
• To convert numbers gt 10 to Scientific Notation,
  move the decimal point to the left until the
  number has the proper form, then nthe number of
  places moved.
• E. g. 5286.2

5.2862 x 103
3 places
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Convert to Scientific Notation

• If the number is between 1 and 9.999, n0.
• E. g., 6.623 6.623 x 100
• If the number is lt 1, move the decimal point to
  the right until the number is in the proper form,
  n - the number of places moved.
• E.g., 0.0000652

6.52 x 10-5
5 places
43
Convert from Scientific to Normal Notation

• If n gt0, move decimal point n places to the
  right, adding 0s as necessary.
• E. g., 5.3 x 104 53000 53000
• If n 0, drop the exponential part.
• E. g., 2.5x100 2.5

4 places
44
Convert to Normal Notation

• If n lt 0, move decimal point n places to the
  left, adding 0s as necessary
• E. g. 7.23 x 10-4 0000723 0.000723

4 places
45
Multiply and Divide Numbers in Scientific Notation

• To multiply, algebraically add exponents and
  multiply coefficients. Adjust decimal point if
  necessary
• (5.3x104) x (6.5x10-7) 5.3 x 6.5 x 10(4-7)
  34.24 x 10-3 3.424 x 10-2 3.4 x 10-2
• If the decimal point is moved to the left, the
  exponent is increased by the number of places
  moved. If the decimal point is moved to the
  right the decimal point is decreased by the
  number of places moved.

Decimal point moved 1 place to the left
46
Dividing Numbers in Scientific Notation

• To divide, divide the numerator coefficient by
  the denominator coefficient and subtract the
  denominator exponent from the numerator
  exponent. Adjust decimal point as needed.
• E. g., x10(8-3) 1.8 x
  105

8.2x108
8.2

4.6x103
4.6
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Adding and Subtracting Numbers in Scientific
Notation

• Convert numbers to be added or subtracted to have
  the same exponent. Add or subtract the
  coefficients. Adjust decimal point and exponent
  to proper form.
• E. g., 4.24x102 3.6 x 103

Move decimal point 1 place to the left to
increase the exponent by 1

• 0.424 x 103 3.6 x 103 4.024 x 103 4.0 x 103

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Calculators

• Most calculators will do the exponential notation
  work for you.
• Learn how your calculator does exponential
  notation.

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Density and Specific Gravity

• What is density?
• Density mass/volume
• Specific Gravity
• S. G. (Density of Substance)/(Density of Water)
• Density of water 1.00 g/mL at 4o C
• Q A student took an unknown liquid and
  discovered that a volume of 9.02 mL had a mass of
  8.31g. Calculate the density and S. G.
• A d m/v 8.31g/9.02mL 0.92 g/mL
• S. G. (0.92 g/mL)/(1.00 g/mL) 0.92

(Note Specific gravity has no units.)
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Objective 5 (cont.)

• Q Calculate the mass of a 30 mL sample of a
  substance which has a density of 0.92 g/mL.
• A d m/v m d x v (0.92g/mL)x(30mL)

27.6 g
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Objective 5 (cont.)

• Q Calculate the volume of a 100g sample of a
  substance which has a density of 0.92 g/mL.
• A d m/v
• v m/d 100g/0.92g/mL

108.7 mL
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Density and Specific Gravity

• Immiscible liquids with a lesser density will
  float on top of liquids with a greater density.
• Two devices to measure specific gravity or
  density
• Hydrometer
• Pycnometer

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Hydrometers for Measuring Specific Gravity
Insert figure 3.18
54
Temperature Measurements

• Common Units of Temperature
• Fahrenheit (oF)
• Celsius (oC)
• Kelvin (K)
• Boiling Point of Water
• 212oF , 100oC, 373.15 K
• Freezing Point of Water
• 32oF, 0oC, 273.15 K
• Absolute zero is 0 K. You cant get any colder
  than this!

55
A Comparison of Temperature Scales
Insert Figure 3.21
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Conversion Among Temperature Units

• Because the temperature scales have different
  zero points, formulas must be used to carry out
  the conversions.
• K oC 273.15
• oC K- 273.15
• oC

(oF - 32)
5
(oF - 32)
or
9
1.8
9
(oC)
32
oF
5
or
1.8(oC) 32
57
Example Temperature Conversions

• Convert 350oF to oC and K.
• oC (350-32) (318)(5/9) 177oC
• K 177 273 450 K
• Convert -40oC to oF
• oF (9/5)(-40) 32 9x(-8) 32
• -7232 -40o F
• Convert 298 K to oC
• oC 298 - 273 25o C
  

5
9
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Heat Energy

• Energy (heat) can flow spontaneously from a high
  temperature source to a low temperature sink.
• The amount of heat flow depends on the nature of
  the substances and the temperature change (?T).
• Heat gained or lost (S. H.)(mass)(?T)
• S. H. is the specific heat, an intensive property
  of the substance.

59
Units of Heat

• Joules, J (Kg m2/s2)
• Kilojoules, kJ 1000 Joules
• calories, cal
• Kilocalories, kcal also called the large
  Calorie, Cal (capital C)
• 1 cal 4.184 J
• 1 kcal 4184 J 4.184 kJ

60
Sample Heat Problems

• Calculate the heat energy required to raise the
  temperature of 24.5 g of mercury from 5.0 oC to
  35.2oC.
• Answer ?T 35.2-5.0 30.2
• S. H. 0.139 J/g oC mass 24.5 g
• Heat gained or lost (S. H.)(mass)(?T)
• Heat gained or lost (0.139)(24.5)(30.2) 103 J