ANALYTICAL CHEMISTRY CHEM 3811

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					INSTRUMENTAL ANALYSIS
      CHEM 4811

           CHAPTER 1



  DR. AUGUSTINE OFORI AGYEMAN
    Assistant professor of chemistry
    Department of natural sciences
        Clayton state university
     CHAPTER 1

FUNDAMENTAL CONCEPTS
     WHAT IS ANALYTICAL CHEMISTRY

    - The qualitative and quantitative characterization of matter

 - The scope is very wide and it is critical to our understanding of
                  almost all scientific disciplines


                         Characterization
 - The identification of chemical compounds or elements present
                      in a sample (qualitative)

- The determination of the amount of compound or element present
                     in a sample (quantitative)
             CHATACTERIZATION

                      Qualitative Analysis
 - The identification of one or more chemical species present
                           in a sample


                   Quantitative Analysis
- The determination of the exact amount of a chemical species
                     present in a sample


                    Chemical Species
- Could be an element, ion or compound (organic or inorgnic)
                CHATACTERIZATION

                         Bulk Analysis
             - Characterization of the entire sample
Example: determination of the elemental composition of a mixture
                             (alloys)

                        Surface Analysis
         - Characterization of the surface of a sample
   Example: finding the thickness of a thin layer on the surface
                       of a solid material

   - Characterization may also include Structural Analysis and
         measurement of physical properties of materials
           WET CHEMICAL ANALYSIS

                      Volumetric Analysis
                      - Analysis by volume

                     Gravimetric Analysis
                      - Analysis by mass

- Wet analysis is time consuming and demands attention to detail

                            Examples
Acid-base titrations, redox titrations, complexometric titrations,
                      precipitation reactions
    WET CHEMICAL ANALYSIS

            Nondestructive Analysis
  - Useful when evidence needs to be preserved

- Used to analyze samples without destroying them


                    Examples
                Forensic analysis
                    Paintings
          INSTRUMENTAL ANALYSIS

 - Use of automated instruments in place of volumetric methods

   - Carried out by specially designed instruments which are
                    controlled by computers

- Samples are characterized by the interaction of electromagnetic
                      radiation and matter

   - All the analytical steps (from sample preparation through
                  data processing) are automated
      INSTRUMENTAL ANALYSIS

                 This course covers

- The fundamentals of common analytical instruments

        - Measurements with these instruments

- Interpretation of data obtained from the measurements

    - Communication of the meaning of the results
    THE ANALYTICAL APPROACH

   - Problems continuously occur around the world in
               - Manufacturing industries
                   - The environment
             - The health sector (medicine)
                           etc.

- The analytical chemist is the solution to these problems

       -The analytical chemist must understand the
                     analytical approach
uses, capabilities, and limitations of analytical techniques
  THE ANALYTICAL APPROACH

                       Analyte
   - A substance to be measured in a given sample


                      Matrix
           - Everything else in the sample


                   Interferences
- Other compounds in the sample matrix that interfere
         with the measurement of the analyte
      THE ANALYTICAL APPROACH
                    Homogeneous Sample
           - Same chemical composition throughout
 (steel, sugar water, juice with no pulp, alcoholic beverages)


                  Heterogeneous Sample
- Composition varies from region to region within the sample
     (pudding with raisins, granola bars with peanuts)

 - Differences in composition may be visible or invisible to
       the human eye (most real samples are invisible)

  - Variation of composition may be random or segregated
    THE ANALYTICAL APPROACH

                  Analyze/Analysis
          - Applied to the sample under study

               Determine/Determination
- Applied to the measurement of the analyte in the sample

                    Multiple Samples
       - Identically prepared from another source

                    Replicate Samples
        - Splits of sample from the same source
        THE ANALYTICAL APPROACH
              General Steps in Chemical Analysis

      1. Formulating the question or defining the problem
       - To be answered through chemical measurements

    2. Designing the analytical method (selecting techniques)
            - Find appropriate analytical procedures

                3. Sampling and sample storage
         - Select representative material to be analyzed

                      4. Sample preparation
- Convert representative material into a suitable form for analysis
THE ANALYTICAL APPROACH
     General Steps in Chemical Analysis

   5. Analysis (performing the measurement)
- Measure the concentration of analyte in several
                identical portions

              6. Assessing the data

              7. Method validation

               8. Documentation
           DEFINING THE PROBLEM

- Find out the information that needs to be known about a sample
                (or what procedure is being studied)

      - How accurate and precise the information must be

- Whether qualitative or quantitative analysis or both is required

           - How much sample is available for study

      - Whether nondestructive analysis must be employed
       DEFINING THE PROBLEM

 - Bulk analysis or analysis of certain parts is required

            - Sample is organic or inorganic

        - Sample a pure substance or a mixture

       - Homogeneous or heterogeneous sample

- Chemical information or elemental information needed
        DEFINING THE PROBLEM

                  Qualitative Analysis

- Provides information about what is present in the sample


- If quantitative analysis is required, qualitative analysis
                    is usually done first


    - Capabilities and limitations of analysis must be
                     well understood
       DEFINING THE PROBLEM

                Qualitative Analysis

            Qualitative Elemental Analysis
    - Used to identify elements present in a material
- Can provide empirical formula of organic compounds
              (X-Ray Fluorescence, AAS)

            Qualitative Molecular Analysis
   - Used to identify molecules present in a material
       - Can be used to obtain molecular formula
     - Can be used to distinguish between isomers
                    (NMR, IR, MS)
         DEFINING THE PROBLEM
                   Qualitative Analysis

                     Empirical Formula
- The simplest whole number ratios of atoms of each element
                   present in a molecule

                     Molecular Formula
 - Contains the total number of atoms of each element in a
              single molecule of the compound

                           Isomers
   - Different structures with the same molecular formula
                   (n-butane and iso-butane)
         DEFINING THE PROBLEM
                   Qualitative Analysis

                         Enantiomers
        - Nonsuperimposable mirror-image isomers
                      - Said to be chiral
             - Have the same IR, NMR, and MS
              - Mostly same physical properties
       (boiling-point, melting point, refractive index)

- Chiral Chromatography can be used to distinguish between
             such optically active compounds

                (erythrose, glyceraldehyde)
     DEFINING THE PROBLEM

                Qualitative Analysis

          Mixtures of Organic Compounds
- Mixtures are usually separated before the individual
              components are identified

           - Separation techniques include
                          GC
                          LC
                        HPLC
                          CE
          DEFINING THE PROBLEM
                   Quantitative Analysis

- The determination of the amount of analyte in a given sample

         - Often expressed in terms of concentrations

                        Concentration
- The quantity of analyte in a given volume or mass of sample

        Molarity = moles/liters, ppm = µg/g sample
            ppb = ng/g sample, ppt = pg/g sample
   Percent by mass [%(m/m)], Percent by volume [%(v/v)]
          DEFINING THE PROBLEM
                    Quantitative Analysis

     - Early methods include volumetric, gravimetric, and
                     combustion analysis

   - Automated and extremely sensitive methods are being
          used today (GC, IR, HPLC, CE, XRD)


         - Require micron amounts and a few minutes

Hyphenated techniques are used for qualitative and quantitative
 measurements of the components mixtures (GC-MS, LC-MS)
DESIGNING THE ANALYTICAL METHOD

 - Analytical procedure is designed after the problem
                   has been defined

                 Analyst must consider
               - Accuracy and precision

            - Amount of sample to be used

                    - Cost analysis

                  - Turnaround time
(time between receipt of sample and delivery of results)
DESIGNING THE ANALYTICAL METHOD

 Green chemistry processes preferred for modern
             analytical procedures

   - The goal is to minimize waste and pollution

   - Use of less toxic or biodegradable solvents

      - Use of chemicals that can be recycled

    - Standard methods are available in literature
 (reproducible with known accuracy and precision)
   DESIGNING THE ANALYTICAL METHOD

- Do not waste time developing a method that already exists

       - Method of choice must be reliable and robust

             - Interferences must be evaluated


                        Interference
- Element or compound that respond directly to measurement
                 to give false analyte signal
          - Signal may be enhanced or suppressed
     DESIGNING THE ANALYTICAL METHOD
               Fundamental Features of Method

                    - A blank must be analyzed

- The blank is usually the pure solvent used for sample preparation

  - Used to identify and correct for interferences in the analysis

                - Analyst uses blank to set baseline

Reagent blank: contains all the reagents used to prepare the sample
  Matrix blank: similar in chemical composition to the sample
                     but without the analyte
   DESIGNING THE ANALYTICAL METHOD

              Fundamental Features of Method

 - Methods require calibration standards (except coulometry)


- Used to establish relationship between analytical signal being
           measured and the concentration of analyte


- This relationship (known as the calibration curve) is used to
  determine the concentration of unknown analyte in samples
      DESIGNING THE ANALYTICAL METHOD
                Fundamental Features of Method

             - Reference (check) standards are required

          - Standards of known composition with known
                      concentration of analyte

     - Run as a sample to confirm that the calibration is correct

    - Used to access the precision and accuracy of the analysis

Government and private sources of reference standards are available
     (National Institute of Standards and Technology, NIST)
                      SAMPLING

- The most important step is the collection of the sample of the
                   material to be analyzed

       - Sample should be representative of the material

    - Sample should be properly taken to provide reliable
               characterization of the material

       - Sufficient amount must be taken for all analysis

                    Representative Sample
  - Reflects the true value and distribution of analyte in the
                        original material
                        SAMPLING
                  Steps in Sampling Process
     - Gross representative sample is collected from the lot

- Portions of gross sample is taken from various parts of material

                   Sampling methods include
    - Long pile and alternate shovel (used for very large lots)

                       - Cone and quarter


                           Aliquot
   - Quantitative amount of a test portion of sample solution
                     SAMPLING

   - Care must be taken since collection tools and storage
            containers can contaminate samples

- Make room for multiple test portions of sample for replicate
      analysis or analysis by more than one technique


                   Samples may undergo
                        - grinding
                       - chopping
                         - milling
                         - cutting
                        SAMPLING
                         Gas Samples

             - Generally considered homogeneous

   - Samples are stirred before portions are taken for analysis

  - Gas samples may be filtered if solid materials are present

                         Grab samples
            - Samples taken at a single point in time

                      Composite Samples
- Samples taken over a period of time or from different locations
                         SAMPLING
                           Gas Samples

                            Scrubbing
             - Trapping an analyte out of the gas phase

                              Examples
  - Passing air through activated charcoal to adsorb organic vapors
   - Bubbling gas samples through a solution to absorb the analyte

                      Samples may be taken with
                           - Gas-tight syringes
  - Ballons (volatile organic compounds may contaminate samples)
- Plastic bags (volatile organic compounds may contaminate samples)
            - Glass containers (may adsorb gas components)
                      SAMPLING
                       Liquid Samples

  - May be collected as grab samples or composite samples

- Adequate stirring is necessary to obtain representative sample

    - Stirring may not be desired under certain conditions
                (analysis of oily layer on water)

     - Undesired solid materials are removed by filtration
                       or centrifugation

  - Layers of immiscible liquids may be separated with the
                     separatory funnel
                       SAMPLING

                         Solid Samples

   - The most difficult to sample since least homogeneous
               compared to gases and liquids

              - Large amounts are difficult to stir

- Must undergo size reduction (milling, drilling, crushing, etc.)
                    to homogenize sample

      - Adsorbed water is often removed by oven drying
                         SAMPLING
                          Sample Storage

      - Samples are stored if cannot be analyzed immediately

     - Sample composition can be changed by interaction with
                 container material, light, or air

   - Appropriate storage container and conditions must be chosen

   - Organic components must not be stored in plastic containers
                        due to leaching

- Glass containers may adsorb or release trace levels of ionic species
                      SAMPLING
                       Sample Storage

      - Appropriate cleaning of containers is necessary

  - Containers for organic samples are washed in solvent

     - Containers for metal samples are soaked in acid
                    and deionized water

- Containers must be first filled with inert gas to displace air

      - Biological samples are usually kept in freezers

   - Samples that interact with light are stored in the dark
                      SAMPLING
                       Sample Storage

            - Some samples require pH adjustment

       - Some samples require addition of preservatives
               (EDTA added to blood samples)

              - Appropriate labeling is necessary

- Computer based Laboratory Information Management Systems
          (LIMS) are used to label and track samples
            SAMPLE PREPARATION

- Make samples in the physical form required by the instrument


- Make concentrations in the range required by the instrument


          - Free analytes from interfering substances


             - Solvent is usually water or organic
SAMPLE PREPARATION

Type of sample preparation depends on
           - nature of sample
          - technique chosen
       - analyte to be measured
      - the problem to be solved

            Samples may be
- dissolved in water (or other solvents)
         - pressed into pellets
          - cast into thin films
                   - etc.
   SAMPLE PREPARATION METHODS
      - Specific methods are discussed in later chapters

               Acid Dissolution and Digestion
  - Used for dissolving metals, alloys, ores, glass, ceramics

   - Used for dissolving trace elements in organic materials
                        (food, plastics)

    - Concentrated acid is added to sample and then heated

- Choice of acid depends on sample to be dissolved and analyte

        Acids commonly used: HCl, HNO3, H2SO4
     HF and HClO4 require special care and supervision
SAMPLE PREPARATION METHODS

            Fusion (Molten Salt Fusion)

- Heating a finely powdered solid sample with a finely
powdered salt at high temperatures until mixture melts

  - Useful for the determination of silica-containing
      minerals, glass, ceramics, bones, carbides


           Salts (Fluxes) Usually Used
   Sodium carbonate, sodium tetraborate (borax),
       sodium peroxide, lithium metaborate
SAMPLE PREPARATION METHODS

           Dry Ashing and Combustion

    - Burning an organic material in air or oxygen


- Organic components form CO2 and H2O vapor leaving
      inorganic components behind as solid oxides


      - Cannot be used for the determination of
           mercury, arsenic, and cadmium
SAMPLE PREPARATION METHODS

                    Extraction

      - Used for determining organic analytes

              - Makes use of solvents

 - Solvents are chosen based on polarity of analyte
                 (like dissolves like)

               Common Solvents
       Hexane, xylene, methylene chloride
 SAMPLE PREPARATION METHODS

                    Solvent Extraction

- Based on preferential solubility of analyte in one of two
                   immiscible phases

            For two immiscible solvents 1 and 2
- The ratio of concentration of analyte in the two phases is
                approximately constant (KD)


             KD    distribution coefficien t 
                                                A1
                                                A2
SAMPLE PREPARATION METHODS

                   Solvent Extraction

     - Large KD implies analyte is more soluble in
              solvent 1 than in solvent 2

   - Separatory funnel is used for solvent extraction

           Percent of analyte extracted (%E)
- V1 and V2 are volumes of solvents 1 and 2 respectively

 %E 
           A1 V1      x 100%        %E 
                                                100K D
      A1 V1  A2 V2                      K D  V2 /V1 
SAMPLE PREPARATION METHODS

                 Solvent Extraction

 - Multiple small extractions are more efficient than
                 one large extraction

      - Extraction instruments are also available


                        Examples
                      Extraction of
- pesticides, PCBs, petroluem hydrocarbons from water
                     - fat from milk
     SAMPLE PREPARATION METHODS

                 Other Extraction Approaches

                 Microwave Assisted Extraction
       - Heating with microwave energy during extraction

               Supercritical Fluid Extraction (SFE)
    - Use of supercritical CO2 to dissolve organic compounds
              - Low cost, less toxic, ease of disposal

                   Solid Phase Extraction (SPE)
                Solid Phase Microextraction (SPME)
              - The sample is a solid organic material
- Extracted by passing sample through a bed of sorbent (extractant)
                     STATISTICS

 - Statistics are needed in designing the correct experiment

                          Analyst must
             - select the required size of sample
                - select the number of samples
               - select the number of replicates
        - obtain the required accuracy and precision


Analyst must also express uncertainty in measured values to
         - understand any associated limitations
                - know significant figures
                       STATISTICS

                Rules For Reporting Results

                   Significant Figures =
      digits known with certainty + first uncertain digit

- The last sig. fig. reflects the precision of the measurement

- Report all sig. figs such that only the last figure is uncertain

                  - Round off appropriately
             (round down, round up, round even)
                    STATISTICS

              Rules For Reporting Results

   - Report least sig. figs for multiplication and division
of measurements (greatest number of absolute uncertainty)

 - Report least decimal places for addition and subtraction
of measurements (greatest number of absolute uncertainty)

    - The characteristic of logarithm has no uncertainty
         - Does not affect the number of sig. figs.

          - Discrete objects have no uncertainty
     - Considered to have infinite number of sig. figs.
        ACCURACY AND PRECISION
    - Accuracy is how close a measurement is to the true
                     (accepted) value

- True value is evaluated by analyzing known standard samples

   - Precision is how close replicate measurements on the
                 same sample are to each other

       - Precision is required for accuracy but does not
                       guarantee accuracy

          - Results should be accurate and precise
    (reproducible, reliable, truly representative of sample)
                       ERRORS
               - Two principal types of errors

   - Determinate (systematic) and indeterminate (random)

             Determinate (Systematic) Errors
        - Caused by faults in procedure or instrument
           - Fault can be found out and corrected
        - Results in good precision but poor accuracy

                            May be
- constant (incorrect calibration of pH meter or mass balance)
  - variable (change in volume due to temperature changes)
                  - additive or multiplicative
                         ERRORS

                 - Two principal types of errors

     - Determinate (systematic) and indeterminate (random)

         Examples of Determinate (Systematic) Errors
     - Uncalibrated or improperly calibrated mass balances
      - Improperly calibrated volumetric flasks and pipettes
           - Analyst error (misreading or inexperience)
                       - Incorrect technique
- Malfunctioning instrument (voltage fluctuations, alignment, etc)
        - Contaminated or impure or decomposed reagents
                          - Interferences
                         ERRORS

                 - Two principal types of errors

     - Determinate (systematic) and indeterminate (random)

         To Identify Determinate (Systematic) Errors
 - Use of standard methods with known accuracy and precision
                      to analyze samples

- Run several analysis of a reference analyte whose concentration
                      is known and accepted

          - Run Standard Operating Procedures (SOPs)
                    ERRORS

            - Two principal types of errors

- Determinate (systematic) and indeterminate (random)

         Indeterminate (Random) Errors
 - Sources cannot be identified, avoided, or corrected
               - Not constant (biased)

                       Examples
       - Limitations of reading mass balances
           - Electrical noise in instruments
                        ERRORS
 - Random errors are always associated with measurements

   - No conclusion can be drawn with complete certainty

- Scientists use statistics to accept conclusions that have high
probability of being correct and to reject conclusions that have
                low probability of being correct

 - Random errors follow random distribution and analyzed
                using laws of probability

          - Statistics deals with only random errors

   - Systematic errors should be detected and eliminated
     THE GAUSSIAN DISTRIBUTION
- Symmetric bell-shaped curve representing the distribution
                    of experimenal data

    - Results from a number of analysis from a single
          sample follows the bell-shaped curve

      - Characterized by mean and standard deviation

                                                 (x  ) 2
                                             
        The Gaussian function is f(x)  ae        2 2



                                1
                         a
                              σ 2π
      THE GAUSSIAN DISTRIBUTION
              - a is the height of the curve’s peak

     - µ is the position of the center of the peak (the mean)

 - σ is a measure of the width of the curve (standard deviation)

                 - T (or xt) is the accepted value

   - The larger the random error the broader the distribution

- There is a difference between the values obtained from a finite
     number of measurements (N) and those obtained from
                 infinite number of measurements
           THE GAUSSIAN DISTRIBUTION
           f(x) = frequency of occurrence of a particular results
       a                                           T (xt)
f(x)




                                                    Point of inflection




                -3σ   -2σ     -σ      μ      σ    2σ 3σ
                                                              x
                     SAMPLE MEAN ( x )
   - Arithmetic mean of a finite number of observations

               - Also known as the average

- Is the sum of the measured values divided by the number
                      of measurements
              N

         _    x     i
         x   i 1
                         
                             1
                               x1  x 2  x 3  ..... x N 
                N            N

       ∑xi = sum of all individual measurements xi
                  xi = a measured value
               N = number of observations
           POPULATION MEAN (µ)

- The limit as N approaches infinity of the sample mean

                                  N
                      lim             xi
                 μ 
                     N
                                 N
                                 i 1




          µ = T in the absence of systematic error
                            ERROR
Error (E)  the difference between T and either x i or x
                    E  x i  T or E  x  T
          Absolute error  Absolute value of E
                   Eabs  x i  T or E  x  T

   Total error = sum of all systematic and random errors

  Relative error = absolute error divided by the true value

                     Eabs                  Eabs
           E rel                  %Erel       x 100%
                      T                     T
           STANDARD DEVIATION

             Absolute deviation (di )  x i  x

Relative deviation (D) = absolute deviation divided by mean

                                   di
                            D      _
                                   x

               Percent Relative deviation [D(%)]

                           di
                  D(%)    _
                                x 100%  D x 100%
                           x
         STANDARD DEVIATION
            Sample Standard Deviation (s)
       - A measure of the width of the distribution

- Small standard deviation gives narrow distribution curve

         For a finite number of observations, N

                                     x             
                     N              N                2

                    d      2
                            i               i   x
               s    i 1
                                   i 1
                    N 1                   N 1


                xi = a measured value
             N = number of observations
              N-1 = degrees of freedom
   STANDARD DEVIATION
    Standard Deviation of the mean (sm)
- Standard deviation associated with the mean
        consisting of N measurements
                       s
                  sm 
                       N

      Population Standard Deviation (σ)
   - For an infinite number of measurements

                        N             2


                  lim    x   i    μ
            σ          i 1
                 N           N
          STANDARD DEVIATION
    Percent Relative Standard Deviation (%RSD)
                               s
                     %RSD      _
                                    x 100
                               x

                         Variance
         - Is the square of the standard deviation

                    - Variance = σ2 or s2

                 - Is a measure of precision
- Variance is additive but standard deviation is not additive
   - Total variance is the sum of independent variances
   QUANTIFYING RANDOM ERROR
                         Median
    - The middle number in a series of measurements
                arranged in increasing order
      - The average of the two middle numbers if the
              number of measurements is even


                          Mode
       - The value that occurs the most frequently


                         Range
- The difference between the highest and the lowest values
         QUANTIFYING RANDOM ERROR
- The Gaussian distribution and statistics are used to determine how
    close the average value of measurements is to the true value

- The Gaussian distribution assumes infinite number of measurements

                As N increases x  μ approacheszero

                         x μ     for N > 20

                         Random error  x  μ

    - The standard deviation coincides with the point of inflection
      of the curve (2 inflection points since curve is symmetrical)
           QUANTIFYING RANDOM ERROR
              Population mean (µ) = true value (T or xt)
       a                                      x=µ
f(x)




                                               Points of inflection




             -3σ   -2σ   -σ       μ     σ    2σ 3σ
                                                           x
      QUANTIFYING RANDOM ERROR
                       Probability
   - Range of measurements for ideal Gaussian distribution

- The percentage of measurements lying within the given range
(one, two, or three standard deviation on either side of the mean)


            Range           Gaussian Distribution (%)

            µ ± 1σ                     68.3
            µ ± 2σ                     95.5
            µ ± 3σ                     99.7
         QUANTIFYING RANDOM ERROR
- The average measurement is reported as: mean ± standard deviation

   - Mean and standard deviation should have the same number
                        of decimal places

          In the absence of determinate error and if N > 20
      - 68.3% of measurements of xi will fall within x = µ ± σ
     - (68.3% of the area under the curve lies in the range of x)

     - 95.5% of measurements of xi will fall within x = µ ± 2σ

     - 99.7% of measurements of xi will fall within x = µ ± 3σ
           QUANTIFYING RANDOM ERROR
                              x=µ±σ
       a
f(x)




                                                     68.3%
                                          known as the confidence level
                                                      (CL)




             -3σ   -2σ   -σ     μ     σ   2σ 3σ
                                                         x
           QUANTIFYING RANDOM ERROR
                              x = µ ± 2σ
       a
f(x)




                                                          95.5%
                                               known as the confidence level
                                                           (CL)




             -3σ   -2σ   -σ      μ         σ   2σ 3σ
                                                              x
           QUANTIFYING RANDOM ERROR
                              x = µ ± 3σ
       a
f(x)




                                                          99.7%
                                               known as the confidence level
                                                           (CL)




             -3σ   -2σ   -σ      μ         σ   2σ 3σ
                                                              x
    QUANTIFYING RANDOM ERROR

                    Short-term Precision
- Analysis run at the same time by the same analyst using the
             same instrument and same chemicals


                   Long-term Precision
  - Compiled results over several months on a regular basis


                        Repeatability
   - Short-term precision under same operating conditions
       QUANTIFYING RANDOM ERROR
                          Reproducibility
   - Ability of multiple laboratories to obtain same results on a
                            given sample


                           Ruggedness
  - Degree of reproducibility of results by one laboratory under
            different conditions (long-term precision)


                     Robustness (Reliability)
- Reliable accuracy and precision under small changes in condition
             CONFIDENCE LIMITS
- Refers to the extremes of the confidence interval (the range)


- Range of values within which there is a specified probability
           of finding the true mean (µ) at a given CL


   - CL is an indicator of how close the sample mean lies
                   to the population mean

                          µ = x ± zσ
             CONFIDENCE LIMITS
                          µ = x ± zσ

                          If z = 1
we are 68.3% confident that x lies within ±σ of the true value


                          If z = 2
we are 95.5% confident that x lies within ±2σ of the true value


                          If z = 3
we are 99.7% confident that x lies within ±3σ of the true value
                CONFIDENCE LIMITS

                - For N measurements CL for µ is

                             μ  x  zs m

- s is not a good estimate of σ since insufficient replicates are made

            - The student’s t-test is used to express CL

         - The t-test is also used to compare results from
                        different experiments

                             t
                                  x  μ 
                                     s
             CONFIDENCE LIMITS
                              _
                                     ts
                      μ  x
                                      N

         - That is, the range of confidence interval is
     – ts/√n below the mean and + ts/√n above the mean


- For better precision reduce confidence interval by increasing
                     number of measurements


       - Refer to table 1.9 on page 37 for t-test values
            CONFIDENCE LIMITS

              To test for comparison of Means

      - Calculate the pooled standard deviation (spooled)

                         - Calculate t

 - Compare the calculated t to the value of t from the table

- The two results are significantly different if the calculated t
    is greater than the tabulated t at 95% confidence level
                 (that is tcal > ttab at 95% CL)
CONFIDENCE LIMITS
      For two sets of data with
     - N1 and N2 measurements
         averages of x1 and x 2
 - standard deviations of s1 and s2

             s1 N1  1  s2 N 2  1
              2
s pooled                   2
                   N1  N 2  2


       x1  x2           N1N 2
   t 
       s pooled         N1  N 2


Degrees of freedom = N1 + N2 - 2
               CONFIDENCE LIMITS
          Using the t-test to Test for Systematic Error

                              
                         t  x μ      N
                                         s
- A known valid method is used to determine µ for a known sample

- The new method is used to determine mean and standard deviation

               - t value is calculated for a given CL

          - Systematic error exists in the new method if
                    tcal > ttab for the given CL
                          F-TEST
   - Used to compare two methods (method 1 and method 2)

        - Determines if the two methods are statistically
                 different in terms of precision

         - The two variances (σ12 and σ22) are compared

F-function = the ratio of the variances of the two sets of numbers

                             σ1
                              2
                           F 2
                             σ2
                           F-TEST
          - Ratio should be greater than 1 (i. e. σ12 > σ22)

- F values are found in tables (make use of two degrees of freedom)

               - Table 1.10 on page 39 of text book

    Fcal > Ftab implies there is a significant difference between
                           the two methods

                      Fcal = calculated F value
                      Ftab = tabulated F value
            REJECTION OF RESULTS
                                Outlier
              - A replicate result that is out of the line
               - A result that is far from other results
 - Is either the highest value or the lowest value in a set of data

   - There should be a justification for discarding the outlier

      - The outlier is rejected if it is > ±4σ from the mean

    - The outlier is not included in calculating the mean and
                         standard deviation

- A new σ should be calculated that includes outlier if it is < ±4σ
     REJECTION OF RESULTS
                    Q – Test

            - Used for small data sets

           - 90% CL is typically used

        - Arrange data in increasing order
 - Calculate range = highest value – lowest value
- Calculate gap = |suspected value – nearest value|
          - Calculate Q ratio = gap/range
            - Reject outlier if Qcal > Qtab

             - Q tables are available
    REJECTION OF RESULTS
                  Grubbs Test

- Used to determine whether an outlier should be
               rejected or retained

- Calculate mean, standard deviation, and then G
                         outlier  x
                   G
                              s

           - Reject outlier if Gcal > Gtab

             - G tables are available
     PERFORMING THE EXPERIMENT

                            Detector
 - Records the signal (change in the system that is related to the
     magnitude of the physical parameter being measured)

     - Can measure physical, chemical or electrical changes


                      Transducer (Sensor)
- Detector that converts nonelectrical signals to electrical signals
                          and vice versa
      PERFORMING THE EXPERIMENT
                       Signals and Noise

     - A detector makes measurements and detector response
                is converted to an electrical signal

   - The electrical signal is related to the chemical or physical
property being measured, which is related to the amount of analyte

      - There should be no signal when no analyte is present

    - Signals should be smooth but are practically not smooth
                          due to noise
PERFORMING THE EXPERIMENT
         Signals and Noise

      Noise can originate from
       - Power fluctuations

           - Radio stations

          - Electrical motors

        - Building vibrations

      - Other instruments nearby
   PERFORMING THE EXPERIMENT
                     Signals and Noise

 - Signal-to-noise ratio (S/N) is a useful tool for comparing
                   methods or instruments

      - Noise is random and can be treated statistically

- Signal can be defined as the average value of measurements

      - Noise can be defined as the standard deviation
                   S  x      mean
                      
                   N s standarddeviation
PERFORMING THE EXPERIMENT
                   Types of Noise

                   1. White Noise
                    - Two types

                   Thermal Noise
- Due to random motions of charge carriers (electrons)
          which result in voltage fluctuations

                    Shot Noise
    - When charge carriers cross a junction in an
                 electrical circuit
    PERFORMING THE EXPERIMENT
                        Types of Noise

    2. Drift (Flicker) Noise (origin is not well understood)

           3. Noise due to surroundings (vibrations)

- Signal is enhanced or noise is reduced or both to increase S/N

      - Hardware and software approaches are available

 - Another approach is the use of Fourier Transform (FT) or
     Fast Fourier Transform (FFT) which discriminates
        signals from noise (FT-IR, FT-NMR, FT-MS)
           CALIBRATION CURVES
                        Calibration
 - The process of establishing the relationship between the
   measured signals and known concentrations of analyte

 - Calibration standards: known concentrations of analyte

   - Calibration standards at different concentrations are
                   prepared and measured

  - Magnitude of signals are plotted against concentration

- Equation relating signal and concentration is obtained and
  can be used to determine the concentration of unknown
              analyte after measuring its signal
           CALIBRATION CURVES
   - Many calibration curves have a linear range with the
         relation equation in the form y = mx + b

- The method of least squares or the spreadsheet may be used

   - m is the slope and b is the vertical (signal) intercept

- The slope is usually the sensitivity of the analytical method

    - R = correlation coefficient (R2 is between 0 and 1)

   - Perfect fit of data (direct relation) if R2 is closer to 1
       BEST STRAIGHT LINE
   (METHOD OF LEAST SQUARES)
            The equation of a straight line


                      y = mx + b


                m is the slope (y/x)


b is the y-intercept (where the line crosses the y-axis)
        BEST STRAIGHT LINE
    (METHOD OF LEAST SQUARES)
              The method of least squares
               - finds the best straight line

   - adjusts the line to minimize the vertical deviations


     Only vertical deviations are adjusted because
  - experimental uncertainties in y values > in x values

- calculations for minimizing vertical deviations are easier
          BEST STRAIGHT LINE
      (METHOD OF LEAST SQUARES)
                   N  x i y i       x y
            m                                i          i

                                    D


             b 
                    x  y
                       2
                       i       i     x i y i  x i
                                   D

                D  N  x i    x i 
                           2                   2




              - N is the number of data points

Knowing m and b, the equation of the best straight line can
be determined and the best straight line can be constructed
         BEST STRAIGHT LINE
     (METHOD OF LEAST SQUARES)


xi      yi      xiyi        xi2




∑xi =   ∑yi =   ∑(xiyi) =   ∑xi2 =
             ASSESSING THE DATA

             A good analytical method should be
                 - both accurate and precise
                     - reliable and robust

  - It is not a good practice to extrapolate above the highest
              standard or below the lowest standard

        - These regions may not be in the linear range

    - Dilute higher concentrations and concentrate lower
concentrations of analyte to bring them into the working range
             ASSESSING THE DATA

                Limit of Detection (LOD)

- The lowest concentration of an analyte that can be detected

   - Increasing concentration of analyte decreases signal
                        due to noise

- Signal can no longer be distinguished from noise at a point

  - LOD does not necessarily mean concentration can be
                measured and quantified
                ASSESSING THE DATA
                    Limit of Detection (LOD)

 - Can be considered to be the concentration of analyte that gives
         a signal that is equal to 2 or 3 times the standard
                        deviation of the blank

- Concentration at which S/N = 2 at 95% CL or S/N = 3 at 99% CL

             LOD  x blank  2σblank or LOD  x blank  3σblank


- 3σ is more common and used by regulatory methods (e.g. EPA)
               ASSESSING THE DATA
               Limit of Quantification (LOQ)

- The lowest concentration of an analyte in a sample that can be
 determined quantitatively with a given accuracy and precision

               - Precision is poor at or near LOD

      - LOQ is higher than LOD and has better precision

      - LOQ is the concentration equivalent to S/N = 10/1

              - LOQ is also defined as 10 x σblank

				
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