Brief review of important concepts for quantitative analysis in forensic chemistry

1
Brief review of important concepts for
quantitative analysis in forensic chemistry
CHM 380 Dr. SkrabalOct. 2005

• Significant figures
• Some important units of quantification
• Units for expressing concentrations in solids and
  liquids
• Concentration-dilution formula
• Precision and accuracy assessing accuracy

2
Significant Figures

• Definition Minimum of digits needed to
  express a number in scientific notation without a
  loss of accuracy
• Example Partial pressure of CO2 in atmosphere ?
  0.000356 atm. This number has 3 sig. figs., but
  leading zeros are only place-keepers and can
  cause some confusion. So express in scientific
  notation
• 3.56 x 10-4 atm
• This is much less ambiguous, as the 3 sig. figs.
  are clearly shown.

3
Avoiding ambiguity

• Consider the quantity 1000 g. A little
  ambiguoushow many sig. figs. are intended to be
  in this number?
• 1.000 x 103 g (4 sf)
• 1.00 x 103 g (3 sf)
• 1.0 x 103 g (2 sf)
• 1 x 103 g (1 sf)
• Using scientific notation takes away the
  ambiguity.

4
Rules for using sig. figs. in calculations

• Addition and Subtraction
• Answer goes to the same decimal place as the
  individual number containing the fewest number of
  sig. figs. to the right of decimal point.
• Example of addition Formula weight of PbS
• 207.2 32.066 239.266 ? round to 239.3
• Example of subtraction
• 4.5237 1.06 3.4637 ? round to 3.46

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Rules for using sig. figs. in calculations

• Adding and subtracting numbers in scientific
  notation
• First convert all numbers to same power, then
  apply rules for adding and subtracting.
• Example
• 1.032 x 104 ? 1.032 x 104
• 2.672 x 105 ? 26.72 x 104
• 3.191 x 106 ? 319.1 x 104
• ----------------
• 346.852 x 104 ? round to
• 346.9 x 104

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Rules for using sig. figs. in calculations

• Multiplication and division
• The number of sig. figs. in the answer should be
  equal to the number of sig. figs. found in the
  individual number which contains the fewest
  number of sig. figs., regardless of whether or
  not the numbers are expressed in scientific
  notation or to what power they are raised.
• Examples
• (A) (0.9987 g) (1.0032 mL g-1) 1.0018958 mL ?
  1.002 mL
• (B) (1.721) (1.8 x 10-4) 3.09780 x 10-4 ?
  3.1 x 10-4
• (C) 1.2215 x 10-3 / 0.831 1.4699158 x 10-3 ?
  1.47 x 10-3

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About rounding

• When rounding, look at all digits to the right of
  the last digit you want to keep. If more than
  halfway to the next digit, round up. If more
  than halfway down to next digit, round down.
• Examples
• (A) 4.9271 (round to 3 sf) ? 4.93
• (B) 39.0324 (round to 4 sf) ? 39.03
• (C) 5.43918 x 10-2 (round to 4 sf) ? 5.439 x
  10-2

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About rounding

• If exactly halfway, round to the nearest even
  digit. This avoids systematic round-off error.
• Examples
• (A) 4.25 x 10-2 (round to 2 sf) ? 4.2 x 10-2
• (B) 17.87500 (round to 4 sf) ? 17.88

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Fundamental SI units

• Remember the correct abbreviations!

10
Some other SI and non-SI units
11
Some common prefixes for exponential notation
Remember the correct abbreviations!
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Commonly used equalities

• 103 mg 1 g milli thousandth
• 1 mg 10-3 g
• 106 µg 1 g micro millionth
• 1 µg 10-6 g
• 109 ng 1 g nano billionth
• 1 ng 10-9 g
• 1012 pg 1 g pico trillionth
• 1 pg 10-12 g

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Concentration scales

• Molarity (M)
• Molality (m)

• Molarity is a temperature-dependent scale
  because volume (and density) change with
  temperature.
• Molality is a temperature-independent scale
  because the mass of a kilogram does not vary with
  temperature.

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Concentration scales (cont.)

• Weight / weight (w/w) basis
• (w/w)
• ppt (w/w)
• ppm (w/w)
• ppb (w/w)
• ppt (w/w)

? percent
? ppt parts per thousand
? ppt parts per million
? ppt parts per billion
? ppt parts per trillion
This scale is useful for solids or solutions.
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Concentration scales (cont.)

• Weight / volume (w/v) basis
• (w/v)
• ppt (w/v)
• ppm (w/v)
• ppb (w/v)
• ppt (w/v)

? percent
? ppt parts per thousand
? ppt parts per million
? ppt parts per billion
? ppt parts per trillion
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Concentration scales (cont.)

• Volume / volume (v/v) basis
• (v/v)
• ppt (v/v)
• ppm (v/v)
• ppb (v/v)
• ppt (v/v)

? percent
? ppt parts per thousand
? ppt parts per million
? ppt parts per billion
? ppt parts per trillion
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Concentration examples

• Concentrated HCl
• Alcoholic beverage
• Color indicator for titrations

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Concentration scales (cont.)

• Parts per million, billion, trillion are very
  often used to denote concentrations of aqueous
  solutions (whose densities are approximately 1 g
  per mL)

Note ppt parts per trillion
19
Concentration scales (cont.)

• It will be very useful to memorize
• 1 part per million (ppm) 1 mg / L
• 1 part per billion (ppb) 1 µg / L
• 1 part per trillion (ppt) 1 ng / L

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Concentration examples

• Conversion of molarity to ppm
• Solution of 0.02500 M K2SO4

21
Concentration examples

• What is concentration (in ppm) of K in this
  solution?
• Solution of 0.02500 M K2SO4

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Concentration-dilution formulaA very versatile
formula that you absolutely must know how to use

• C1 V1 C2 V2
• where C conc. V volume
• M1 V1 M2 V2
• where M molarity
• Cconc Vconc Cdil Vdil
• where conc refers to the more concentrated
  solution and dil refers to the more dilute
  solution
• Keep in mind
• concentration times volume equals moles or mass
• Examples (A) (mol/L x L) mol (B) (mg/L x L)
  mg

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Concentration-dilution formula example

• Problem You have available 12.0 M HCl (conc.
  HCl) and wish to prepare 0.500 L of 0.750 M HCl
  for use in an experiment. How do you prepare
  such a solution?
• Cconc Vconc Cdil Vdil
• Write down what you know and what you dont know

24
Concentration-dilution formula example

• Problem You have available 12.0 M HCl (conc.
  HCl) and wish to prepare 0.500 L of 0.750 M HCl
  for use in an experiment. How do you prepare
  such a solution?
• Cconc Vconc Cdil Vdil
• Cconc 12.0 mol L-1 Cdil 0.750 mol L-1
• Vconc ? Vdil 0.500 L
• Vconc (Cdil)(Vdil) / Cconc
• Vconc (0.750 mol L-1) (0.500 L) / 12.0 mol L-1
• Vconc 3.12 x 10-2 L 31.2 mL

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Concentration-dilution formula example

• Great! So how do you prepare this solution of
  0.750 M HCl?
• Use a pipet or graduated cylinder to measure
  exactly 31.2 mL of 12.0 M
• Transfer the 31.2 mL of 12.0 M HCl to a 500.0 mL
  volumetric flask
• Gradually add deionized water to the volumetric
  flask and swirl to mix the solution
• As the solution gets close to the 500.0 mL
  graduation on the flask, use a dropper or squeeze
  bottle to add water to the mark
• Put the stopper on the flask and invert 20 times
  to mix

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Types of experimental error

• Random or indeterminate error
• Arise from inherent limitations in ability to
  make measurements. Assumed to cancel out over
  time. Can minimize with experimental setup, but
  can never eliminate completely.
• Examples electrical noise differences in
  visual determination.
• Systematic or determinate error
• Can be determined and corrected for.
• Example improperly calibrated instrument

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Precision vs. accuracy
CHM 235---Dr. Skrabal

• Two important concepts regarding data are
  accuracy and precision
• Accuracy Closeness to a true value. Must
  eliminate systematic error to assure accuracy.
• Precision Reproducibility

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Not accurate, not precise
Accurate, not precise
Accurate and precise
Precise, not accurate
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Assessing accuracy

• Can never really know a true value, but there
  are practical methods for assessing accuracy
• Four important methods
• Blank analyses
• Use of certified reference materials or standard
  reference materials (SRM and CRM)
• Interlaboratory comparison (round-robin
  experiment)
• Multiple method comparison

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Blank analyses

• Analysis performed using some analytical
  procedure with everything but the analyte
• Purpose is to detect whether or not there is
  analyte present in reagents, water, laboratory
  implements, etc. used for an analytical procedure
• For example, for analysis of an analyte in an
  aqueous solution, use high-purity water in place
  of sample and measure as normal.
• Signal from blank is subtracted from each and
  every sample signal

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Example Blank analyses

• For a spectrophotometric analysis, a series of
  standards is prepared and tested. Then an
  unknown is analyzed.
• Note the blank value is subtracted from each
  standard and sample measurement.

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Certified (or standard) reference materials

• CRMs (or SRMs) are available for many different
  types of analytes in many different media.

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CRM for trace metals in estuarine water.
Available from National Research Council of
Canada.
34
Fish tissue CRM for various organic analytes.
Available from National Research Council of
Canada.
35
Interlaboratory comparison (round-robin
experiment)

• Sample containing analyte of interest is
  distributed to different laboratories for
  analysis (single blind or double blind)
• Results analyzed for similarity compared to
  certified value if available

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Lead in polyethylene intercomparison
From D.C. Harris (2003) Quantitative Chemical
Analysis, 6th Ed.
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Multiple method comparison

• Use different methods to analyze for the same
  analyte in the same sample

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CRM for trace metals in estuarine water.
Available from National Research Council of
Canada. Note that most metals were determined
using completely different analytical methods.
39
Replication confusion

• Typically, several measurements are made for an
  analyte on each sample. Sometimes called
  replicates, referring to number of measurements
  on each sample (n). Enables researcher to
  evaluate precision of analysis.
• True replicates are independently collected and
  measured. Enables researcher to evaluate
  variability in population or environment.

40
Replication confusion

• Three measurements of estrogen level in one
  sample of one individuals urine (n 3). Can
  evaluate precision of estrogen concentration in
  that urine sample.
• True replication Measurements of estrogen in
  urine from six randomly selected individuals.
  Can evaluate variability in estrogen levels among
  individuals.

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Basic Statistics

• Statistics Set of mathematical tools used to
  describe and make judgments about data
• Type of statistics we will talk about in this
  class has important assumption associated with
  it
• Experimental variation in the population from
  which samples are drawn has a normal (Gaussian,
  bell-shaped) distribution.
• - Parametric vs. non-parametric statistics

42
Normal distribution

• Estimate of mean value of population ?
• Estimate of mean value of samples
• Mean

43
Standard deviation and the normal distribution

• Standard deviation defines the shape of the
  normal distribution (particularly width)
• Larger std. dev. means more scatter about the
  mean, worse precision.
• Smaller std. dev. means less scatter about the
  mean, better precision.

44
Standard deviation and the normal distribution

• Degree of scatter (measure of central tendency)
  of population is quantified by calculating the
  standard deviation
• Std. dev. of population ?
• Std. dev. of sample s
• Characterize sample by calculating

45
Standard deviation and the normal distribution

• There is a well-defined relationship between the
  std. dev. of a population and the normal
  distribution of the population
• ? 1? encompasses 68.3 of measurements
• ? 2? encompasses 95.5 of measurements
• ? 3? encompasses 99.7 of measurements
• (May also consider these percentages of area
  under the curve)

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Example of mean and standard deviation calculation

• Consider measurements of lead in blood in a
  poisoning case
• 31.2, 32.4, 30.9, 31.3, 32.8 µg dL-1
• 31.72 µg dL-1
• s 0.8288 µg dL-1
• Real rule of sig figs says first uncertain
  digit is last sig fig. So round answer
  appropriately
• 31.72 0.82 µg dL-1 or 31.7 0.8 µg dL-1
• Learn how to use the statistical functions on
  your calculator. Do this example by longhand
  calculation once, and also by calculator to
  verify that youll get exactly the same answer.
  Then use your calculator for all future
  calculations.

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Standard error

• Tells us that standard deviation of set of
  samples should decrease if we take more
  measurements
• Standard error
• Take twice as many measurements, s decreases by
  
• Take 4x as many measurements, s decreases by

48
Relative standard deviation (rsd) or coefficient
of variation (CV)

• rsd or CV
• From previous example,
• rsd (0.82 µg dL-1/31.72 µg dL-1 ) 100 2.5 or
  3

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Some useful statistical tests

• To characterize or make judgments about data
• Tests that use the Students t distribution
• Confidence intervals
• Comparing a measured result with a known value
• Comparing replicate measurements (comparison of
  means of two sets of data)

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From D.C. Harris (2003) Quantitative Chemical
Analysis, 6th Ed.
51
Confidence intervals

• Quantifies how far the true mean (?) lies from
  the measured mean, . Uses the mean and standard
  deviation of the sample.
• where t is from the t-table and n number of
  measurements.
• Degrees of freedom (df) n - 1 for the CI.

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Example of calculating a confidence interval

• Consider measurement of benzene in drinking
  water
• Data 0.28, 0.32, 0.35, 0.27, 0.33, 0.31, 0.29
  ppb
• DF n 1 7 1 6
• 0.307 nM or 0.31 ppb
• s 0.028 or 0.03 ppb
• 95 confidence interval
• t(df6,95) 2.447
• CI95 0.307 0.026 ppb
• 50 confidence interval
• t(df6,50) 0.718
• CI50 0.307 0.0076 ppb

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Interpreting the confidence interval

• For a 95 CI, there is a 95 probability that
  the true mean (?) lies between the range 0.307
  0.026 ppb, or between 0.281 and 0.333 ppb
• For a 50 CI, there is a 50 probability that the
  true mean lies between the range 0.307 0.0076
  ppb, or between 0.2994 and 0.3146 ppb
• Note that CI will decrease as n is increased
• Useful for characterizing data that are regularly
  obtained e.g., quality assurance, quality control

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Comparing a measured resultwith a known value

• Known value would typically be a certified
  value from a standard reference material (SRM)
• Will compare tcalc to tabulated value of t at
  appropriate df and CL.
• df n - 1 for this test

55
Comparing a measured resultwith a known
value--example

• Blood lead analysis verified using Certificated
  Reference Materials of B.C.R. (European Community
  Reference Office) CRM
• Certified value 126 µg L-1
• Experimental results 118 9 µg L-1 (n 10)
• (Keep 3 decimal places for comparison to table.)
• Compare to ttable df 10 - 1 9, 95 CL
• ttable(df9,95 CL) 2.262
• If tcalc lt ttable, results are not
  significantly different at the 95 CL.
• If tcalc ? ttable, results are significantly
  different at the 95 CL.
• For this example, tcalc gt ttest, so experimental
  results are significantly different at the 95 CL

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Comparing replicate measurements or comparing
means of two sets of data

• Another application of the t statistic
• Example Given the same sample analyzed by two
  different methods, do the two methods give the
  same result?
• Will compare tcalc to tabulated value of t at
  appropriate df and CL.
• df n1 n2 2 for this test

57
Comparing replicate measurements or comparing
means of two sets of dataexample
Determination of ethanol in liquid using two
different methods

• Method 1 Alcohol oxidase method
• Data 8.82, 8.77, 9.09, 8.99 (v/v)
• 8.91 mg/g
• 0.14 mg/g
• 4

• Method 2 Headspace GC method
• Data 8.46, 8.67, 8.83, 8.50 (v/v)
• 8.61 mg/g
• 0.16 mg/g
• 4

58
Comparing replicate measurements or comparing
means of two sets of dataexample
Note Keep 3 decimal places to compare to
ttable. Compare to ttable at df 4 4 2 6
and 95 CL. ttable(df6,95 CL) 2.447 If
tcalc ? ttable, results are not significantly
different at the 95. CL. If tcalc ? ttable,
results are significantly different at the 95
CL. Since tcalc (2.823) ? ttable (2.447),
results from the two methods are significantly
different at the 95 CL.
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Evaluating questionable data points using the
Q-test

• Need a way to test questionable data points
  (outliers) in an unbiased way.
• Q-test is a common method to do this.
• Requires 4 or more data points to apply.
• Calculate Qcalc and compare to Qtable
• Qcalc gap/range
• Gap (difference between questionable data pt.
  and its nearest neighbor)
• Range (largest data point smallest data
  point)

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Evaluating questionable data points using the
Q-test--example

• Consider set of data Cu values in Uncle Georges
  fish stew
• 9.52, 10.7, 13.1, 9.71, 10.3, 9.99 mg/L
• Arrange data in increasing or decreasing order
• 9.52, 9.71, 9.99, 10.3, 10.7, 13.1
• The questionable data point (outlier) is 13.1
• Calculate
• Compare Qcalc to Qtable for n observations and
  desired CL (90 or 95 is typical). It is
  desirable to keep 2-3 decimal places in Qcalc so
  judgment from table can be made.
• Qtable (n6,90 CL) 0.56

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From G.D. Christian (1994) Analytical Chemistry,
5th Ed.
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Evaluating questionable data points using the
Q-test--example

• If Qcalc lt Qtable, do not reject questionable
  data point at stated CL.
• If Qcalc ? Qtable, reject questionable data point
  at stated CL.
• From previous example,
• Qcalc (0.670) gt Qtable (0.56), so reject data
  point at 90 CL.
• Subsequent calculations (e.g., mean and standard
  deviation) should then exclude the rejected
  point.
• Mean and std. dev. of remaining data 10.04 ?
  0.47 mg/L

63
Steps in a quantitative analysis

• Using the concentration-dilution formula to make
  standards
• Measuring blanks and standards using some
  analytical technique
• Making a calibration curve
• Quantifying the analyte in an unknown (sample)

64
The problemQuantifying the cyanide (CN-)
concentration in a water sample

• Will quantify using a spectrophotometric
  technique
• You have available a 1000 ppm ( 1000 mg L-1) CN-
  standard, pipets, 100.0 mL volumetric flasks,
  deionized water, etc.

65
Prepare CN- standards

• Need to make standards of concentrations 0, 10.0,
  25.0, 50.0, and 100.0 mg L-1
• 100.0 mL of each standard will be made
• You need to determine how much of the 1000 mg L-1
  standard to use to prepare each standard
• The blank contains no added standard

66
Preparing standards

• The first standard is 10.0 mg/L
• Cconc Vconc Cdil Vdil
• Cconc 1000 mg L-1 Cdil 10.0 mg L-1
• Vconc ? Vdil 0.1000 L
• Vconc (Cdil)(Vdil) / Cconc
• Vconc (10.0 mg L-1) (0.1000 L) / 1000 mg L-1
• Vconc 1.00 x 10-3 L 1.00 mL

67
Preparing standards

• Similarly, you should be able to calculate that
  2.50 mL of 1000 mg L-1 concentrated standard will
  be required for the 25.0 mg L-1 standard 5.00 mL
  for the 50 mg L-1 standard and 10.00 mL for the
  100 mg L-1 standard.
• Each volume of conc. standard gets added to a
  100.0 mL flask, reagents are added, deionized
  water is added exactly to the mark, the flasks
  are capped and inverted 20 times to mix

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Make measurements
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Prepare calibration curve
70
Measure unknown sample and use calibration curve
to quantify concentration
71
Measure unknown sample and use calibration curve
to quantify concentration

• Suppose you prepare an unknown and measure its
  absorbance
• Abs 0.455
• Blank-corrected abs. 0.455 0.001 0.454

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Quantifying the unknown
Now run the rest of the samples!