Proportional Relationships

1
Chapter 14

• Proportional Relationships

2
Ratios

• The word ratio is derived from the Latin word
  "ratio" which means "computation.
• A ratio can be defined as the relative size of
  two quantities expressed as the quotient of one
  divided by the other.
• Ex. The ratio of 7 to 4 is written 74 or 7/4.
• To solve this problem you would divide 4 from 7
• To simplify a compound or an equation you take
  the coefficients of each element or compound and
  reduce them.
• Ex. C6H12O6 --gt 6126 --gt 121

3
Ratios

• Here are some examples on how to determine the
  ratio from a chemical formula
• Ex. 2H2 O2 --gt 2H2O
• From this example you can see that for every two
  molecules of hydrogen gas there is one molecule
  of oxygen gas.
• The ratio is 21.
• If you have fifty molecules of oxygen gas then
  from this ratio you would have one hundred
  molecules of hydrogen gas.

4
Ratios

• Ex. 2Al(NO3)3 3H2SO4 --gt Al2(SO4)3 6HNO3
• In this chemical formula you can see that
  aluminum nitrate combined with hydrogen sulfate
  (sulfuric acid) yields aluminum sulfate and
  hydrogen nitrate.
• From this example you can see that for every two
  molecules of aluminum nitrate there are three
  molecules of hydrogen sulfate.
• The ratio is 23.
• If you have twenty molecules of aluminum nitrate
  then you would have thirty molecules of hydrogen
  sulfate

5
Proportion

• The word proportion is derived from the Latin
  word "proportio" which means "portion."
• A proportion can be described as an equation
  which states that two ratios are equal.
• Ex. 4 5     8   10
• To solve a proportion you cross-multiply.
• In this example you multiply the 4 and the 10
  together and the 5 and the 8 together

6
Proportion

• Ex. 6 x    5   10
• To solve for x in this proportion you cross
  multiply the 6 and 10 together, and the 5 and x
  together
• 5x 60.
• Then divide the 60 by 5 to get your answer
• 60/5 12.
• Ex. 4 8       x  16
• To solve for x in this proportion you cross
  multiply the 4 and 16 together, and the 8 and x
  together
• 8x 64.
• Then divide the 64 by 8 to get the value of x
• 64/8 8.

7
Proportion

• Ex. x1 6      x2    x4
• To solve for x in this proportion you must
  cross-multiply.
• In this example you multiply the x1 with the
  x4.
• When you multiply the two binomials together you
  will get one trinomial. (x1)(x4) x25x4.
• You can obtain this trinomial by using the
  F.O.I.L. method

8
Proportion

• F.O.I.L.
• First multiply the x's from each binomial
  together.
• Then multiply the Outer numbers and variables
  together (X x 4), and the Inner numbers and
  variables together (Last) (X x 1)
• Add the results
• 4x and 1x together.
• Finally you multiply the last numbers and
  variables in the two binomials together (4 x 1).
• Your final answer is x25x4.
• Next you cross-multiply the 6 with the (x2).
• Distribute the 6 through the binomial (x2).

9
Proportion

• F.O.I.L.
• You multiply the 6 with the X and then the 6 with
  the 2.
• You final answer is 6x12.
• Set the two equations equal to each other
• x25x46x12.
• Move all the numbers and variables together on
  one side,--gt x2-x-80.
• Break this equation into two binomials and you
  will have your answer.
• This is what you should get(x4) and (x-4).
• X is equal to 4.

10
Percents

• The word percent comes from the Latin per centum
  meaning "out of one hundred,"
• We can think of 22 as "22 out of 100.
• Thus a percent is a symbol representing a ratio
  of the part -- in this case 22 -- to the whole --
  100.
• A percent must be changed to a number (fraction
  or decimal) before we can compute with it. Hence
  22 is the ratio 22 100, which gives

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Percents

• Other examples

Percent notation Ratio notation Number notation
30 30 100 30 / 100 .3
8 8 100 8 / 100 .08
63.7 63.7 100 63.7 / 100 .637
100 100 100 100 / 100 1
212 212 100 212 / 100 2.12
5 3/4 5 3/4 100 or 5.75 100 5.75 / 100 .0575
We will most often use the decimal form (in bold
above) because it is easy to find on a calculator
by dividing by 100.
12
Percents

• Percents come up several times in chemistry.
• Remember that the key is to change a percent to a
  number before calculating with it
• If the answer is to be stated as a percent,
  convert the number to a percent before giving the
  answer.

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Percents

• When measuring a sample for its constituent
  parts, the amounts of each part are often stated
  as s
• Percent abundance of isotopes
• Percent composition of compounds.
• Part / whole.
• For example A sample of lead was tested in a
  mass spectrometer, and four isotopes were found
  along with their abundances 204 at 1.4, 206
  at 24.1, 207 at 22.1 and 208 at 52.4.
• How do we read these?
• 1.4 of the sample was isotope 204,
• 24.1 of the sample was 206, etc.
• Notice that the percents add up to 100. (All the
  parts together should total up to the whole!)

14
Percents

• When conducting an experiment to synthesize a
  chemical compound, you'll compare the amount you
  should get (according to the theory of how
  chemicals bind together) to the actual amount you
  did get from your experiment -- percent yield
  experiment / theory.
• For example Suppose we know that if we take
  formic acid and geraniol, we can make a synthetic
  rose perfume.
• If we start with 1000.0 g of geraniol added to
  formic acid, the theory of chemical reactions
  (stoichiometry!) can be used to calculate a
  theoretical yield of 1182.2 g of the rose
  essence.
• Our experiment actually produces 871.2 g. What is
  the percent yield?
• Experiment / Theory 871.2 / 1182.2 .736931145
  73.69 to four sig figs.

15
Percents

• When working with measurements there is often
  some associated error -- usually measured as
  percent error
• For example, suppose we test a new thermometer
  for accuracy by using it to find the boiling
  point of pure water.
• The boiling point of pure water is 212F but the
  thermometer measures 212.9F.
• What is the error of the reading?

16
Significant Figures

• In scientific work, most numbers are measured
  quantities and thus are not exact.
• All measured quantities are limited in
  significant figures (SF) by the precision of the
  instrument used to make the measurement.
• The measurement must be recorded in such a way as
  to show the degree of precision to which it was
  made-- no more, no less.
• Calculations based on the measured quantities can
  have no more (or no less) precision than the
  measurements themselves.
• The answers to the calculations must be recorded
  to the proper number of significant figures.
• To do otherwise is misleading and improper. 

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Significant Figures

• Determining Which Figures are Significant
• Non-zero integers are always significant.
  example 23.4g and 234g both have 3 SF
• Captive zeroes, those bounded on both sides by
  non-zero integers, are always significant.
  example 20.05 has 4 SF 407 has 3 SF
• Leading zeros, those not bounded on the left by
  non-zero integers are never significant.
• Such zeros just set the decimal point they
  always disappear if the number is converted to
  powers-of-10 notation. example 0.04g has 1 SF
  0.00035 has 2 SF. They can be written as 4x10-3
  and 3.5x10-4 respectively.
• Trailing zeros, those bounded only on the left by
  non-zero integers may or may not be significant.
  example 45.0L has 3 SF 450L has only 2 SF
  450.L has 3 SF.
• Note To clarify whether a trailing zero is
  significant, it is preferable to use scientific
  notation to express the final answer. example
  450.L can be expressed as 4.50 x 102 or 4.50E02
  whereas 450 L would be expressed as 4.5x102.
• Exact numbers are those not obtained by
  measurement but by definition or by counting
  numbers of objects. They are assumed to have an
  unlimited number of significant figures.

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Significant Figures

• Multiplication and Division Involving Significant
  Figures
• Calculations involving only multiplication and/or
  division of measured quantities shall have the
  same number of significant figures as the fewest
  possessed by any measured quantity in the
  calculation.
• example 14.0 x 3 40, not 42, because one of
  the multipliers has only one SF.
• example 14.0 x 3.0 42, because one of the
  multipliers has only two SF.
• example 14.0 / 3 5, not 4.6, because the
  denominator has only one SF.

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Significant Figures

• Addition and Subtraction Involving Significant
  Figures
• Calculations where measured quantities are added
  or subtracted shall correspond to the position of
  the last significant figure in any of the
  measured quantities.
• That is, the final answer is only as precise as
  the decimal position of the least precise value.
• The number of significant figures can change
  during these calculations.
• example 14.16 3.2 17.36 (this is not the
  final answer!)17.4 is the correct answer
• example 46.6 5.72 52.32 (this is not the
  final answer!) 52.3 is the correct final answer.

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Significant Figures

• Combined Calculations
• In calculations involving addition/subtraction
  and multiplication/division, significant figure
  guidelines must be applied before and after each
  calculation involving addition or subtraction.
• example (3.2 x 4 x 0.035 / 7) (12 x 0.5)
  0.06 6 6 

21
Significant Figures

• Tips for Rounding Off Numbers
• A number is rounded off to the desired number of
  significant figures by dropping one or more
  digits to the right.
• If you are rounding to the tens, hundreds place
  or higher, you must put zeroes in the lesser
  places (the ones place, for example) to indicate
  to what place you have rounded.
• The following guidelines should be observed when
  rounding off numbers.
• When the first digit dropped is less than 5, the
  last digit remains unchanged.
• When the first digit dropped is more than or
  equal to five, the last digit retained is
  increased by 1.
• example round 2,457 to the tens place - 2,460
  not 246 to the hundreds place - 2,500 not 25.

22
Significant Figures

• Conclusion
• The precision of the instruments used in
  collecting data determines the degree to which
  your results are accurate.
• Significant figures provide an easy way of
  indicating that accuracy.
• Using these guidelines assures that the data
  resulted from your procedures is not only
  reproducible, but also allows an observer to
  understand the degree to which your data is
  accurate.