Uncertainty in Measurement

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• Uncertainty in Measurement
• Exact numbers numbers having a defined value or
  are counted they have no uncertainty
• The 100 multiplier in converting a ratio to
  percent
• The 12 items in a dozen
• The decimal multipliers associated with naming SI
  units
• Inexact numbers numbers having uncertainty
  associated with them
• Measured values always have some uncertainty
• Precision is the reproducibility of measurements
  made in the same way.
• Accuracy is the closeness a measurement or series
  of measurements is to the true or accepted
  value.
• The Significant figure convention involves
  writing a number representing a measurement so
  that all the digits that are certain are given
  plus one uncertain digit.
• The number of digits thus written is the number
  of significant figures in the number

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Rules for Counting Significant Figures All
non-zero digits are significant Leading zeros are
not significant 0123 m .0123 m 0.0123
m Embedded zeros are significant 1023 102.3 10.
23 Trailing Zeros to the right of non-zero
digits Digits to the left of an implied decimal
point are not significant 1230
12300 123000 Digits to the right of a decimal
point are significant 123.0 123.00 12.30 1230.
0
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Scientific Notation Write a number with a decimal
point between the first and second digit (if
there is more than one significant figure) and
multiply by an appropriate power of 10 123.0
1.230 x 103 1200 1.200 x 103 1200 1.2 x
103 sig. fig. not sig. fig.
0.0123 1.23 x 10-2 0.01230 1.230 x
10-2 Sometimes it may be convenient to place the
decimal at a different location, such as when
adding numbers with a different power of 10
multiplier 12345 1.2345 x 104 12.345 x 103
123.45 x 102 1234.5 x 101 1.2345 x 104
1.234 x 103 1.3579 x 104 easy with a
calculator! 12.345 x 103 1.234 x
103 ______________ 13.579 x 103 1.3579 x
104 sometimes this method will be used.
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Sig. Figs. In Calculated Results The number of
Sig. Figs. in a derived result depends on the
kind of arithmetic performed Multiplication and
division The derived result has no more sig.
figs. than the least number of sig. figs. of the
factors used in the calculation. 4.0 x
1023 mL 4092.0 g 4100 g 4.1 x 103 g
2 sig. figs. x 4 sig. figs. 2 sig.
figs Addition and subtraction The number of sig.
figs. in the result is determined by the number
with the least number of significant decimal
places. 4.0
1.067 x 104 1.30 x 102 102
106.7 x 102 106
1.30 x 102

108.00 x 102 1.080 x 104


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Rounding Retain all certain digits. Round the
uncertain digit - the least significant
digit If next number beyond the uncertain
digit is gt 5, increase the uncertain digits
value by 1 1.368 to 3 sig. figs. 1.37 If next
number beyond the uncertain digit is lt 5, drop
the later digits 1.363 to 3 sig. figs. 1.36 If
next number beyond the uncertain digit is 5,
round to the nearest even digit 1.365 to 3 sig.
figs. 1.36 1.375 to 3 sig. figs. 1.38
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Complicated Calculations Round after carrying out
a complex calculation Dimensional
Analysis Since all measured values have units and
most exact numbers have units, the units should
be included in calculations. Rationalizing the
units in calculations must give the correct units
associated with the answer. Incorrect units in
a result suggests an incorrect solution to a
problem. Density - An Intensive Physical
Property Definition usual
units mass in g and volume in mL or cm3
units of density g/mL or g/cm3
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• Atomic Theory
• Daltons Atomic Theory
• Each element is composed of extremely small
  particles called atoms.
• All atoms of a given element are the same, that
  is, they have the same properties such as mass
• Compounds are formed when atoms of more than one
  element combine a given compound always has
  the same atoms in a fixed ratio.
• All atoms of an element are not changed into
  different types of atoms during chemical
  change atoms are indestructible and thus are
  neither created nor destroyed during chemical
  change
• Daltons theory explains some known laws
• Law of Constant composition in a pure compound,
  the ratio of the masses of the elements is
  constant
• If atoms of a given element have have the same
  mass and compounds are formed from the atoms of
  elements in a fixed ratio, this law must result
• Law of Conservation of Mass During chemical
  change, the mass of products equals the mass of
  the reactants. Matter cannot be created or
  destroyed, but can be converted from one form to
  another.

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• Atomic Theory
• Daltons theory predicts new laws
• Law of Multiple proportions if two elements form
  two different compounds, the mass ratio of the
  elements in one compound is a whole number
  multiple of the mass ratio of the elements in the
  second
• Example Ammonia, NH3, and hyrazine, N2H4