Water & Wastewater. Laboratory Math

Water & Wastewater Laboratory Math VWEA Good Lab Practices Meeting Presented by: Wayne Staples Virginia DEQ July 24, 2017

Required to: Math in the Laboratory Determine a test result Alkalinity calculation Assure data accuracy Relative Percent Difference Explain how a procedure or test works Beer s law Many other applications 7/24/17 Laboratory Math 2

Significant Figures Rules for significant figures of numbers Follow requirements of permits Otherwise: Non-zero integers are always significant Zeros Three classes Leading Captive Trailing Exact numbers 7/24/17 Laboratory Math 3

Leading Zeros Significant Figures Zeros that precede nonzero digits Not counted as significant Indicate the position of decimal Example: 0.0036 Two significant figures 7/24/17 Laboratory Math 4

Captive Zeros Significant Figures Zeros between nonzero digits Always count as significant figures Example: 2.009 Four significant figures 7/24/17 Laboratory Math 5

Trailing Zeros Significant Figures Zeros at the right end of numbers Significant only if there is a decimal Examples: 200 One significant figure 2.00 x 10 2 (AKA: 200) Three significant figures 7/24/17 Laboratory Math 6

Exact numbers Significant Figures Numbers determined by counting 11 BOD samples, 8 DO tests, etc. Infinite possible number of significant figures Defined numbers 1 in. is defined as exactly 2.54 cm. Neither limits the significant figures in a calculation 7/24/17 Laboratory Math 7

Your Turn!! How many significant figures? An extraction yields 0.0027 g of grease. 2 An analyst records 0.0705 g of solids. 3 The time required for a test is 9.060 minutes. 4 7/24/17 Laboratory Math 8

Significant Figures in Calculations Simple rules for determining significant figures in calculations Multiplication or Division Number of significant figures equals the number in the least precise value or measurement used in the calculation. 5.68 1.2 = 6.816 corrected 6.8 7/24/17 Laboratory Math 9

Significant Figures in Calculations Simple rules for determining significant figures in calculations. Addition or Subtraction Number of significant figures equals the number of decimal places in the least precise value or measurement used in the calculation. 13.12 + 12.2 + 2.045 = 27.365 corrected 27.4 7/24/17 Laboratory Math 10

Significant Figures Rules for Rounding Series of calculations carry extra digits through to end of calculation then round. If digit to be removed is less than 5 Round down 2.32 rounds to 2.3 If digit to be removed is five or higher Round up 2.37 rounds to 2.4 7/24/17 Laboratory Math 11

Significant Figures Rules for Rounding Look at first digit to the right of the digit to be rounded. Do not round in series Example: Round to two significant figures 3.448 3.4 Not 3.45, then 3.5 7/24/17 Laboratory Math 12

Universal Equation C 1 V 1 = C 2 V 2 Basis for three of four basic testing methodologies Titrimetric Colorimetric Electrometric 7/24/17 Laboratory Math 13

C 1 V 1 = C 2 V 2 Where: C 1 = Concentration of stock solution. V 1 = Volume of stock solution that will be added to standard, spike or sample. C 2 = Concentration of desired standard, spike or sample. V 2 = Volume of standard, spiked sample or dilution. 7/24/17 Laboratory Math 14

Target Concentrations Some test procedures cover a limited range of concentrations Samples with higher levels must be diluted before testing Requires some knowledge of the sample Results of past analyses Results for similar samples Initial testing always requires multiple dilutions to cover range of results 7/24/17 Laboratory Math 15

Procedure Determine concentration range of test procedure. Determine expected concentration range of sample based on available information. Determine volume of sample to be diluted to produce a concentration within the range of the test. Since concentrations can vary, it is best to select several dilution concentrations to ensure one or more will fall within the range of the test. 7/24/17 Laboratory Math 16

Calculation Highest Sample Volume Volume High Volume Test Concentration Concentration Lowest Sample Volume Volume Low Volume Test Concentration Test Sample High Concentration High Sample Low Test Low 7/24/17 Laboratory Math 17

Example The phosphorus test requires: A 50 ml sample volume A concentration in the range 2 5 mg/l Previous tests indicate the sample has a concentration of 12.5 18 mg/l 7/24/17 Laboratory Math 18

Step 1 Calculate the sample volume required to fall within the upper concentration limit. Volume High Volume 50 ml 5 mg/l 18 mg/l 13.9 ml Test Concentration Concentration Sample High 13.9 ml of sample would be diluted to 50 ml Test High 7/24/17 Laboratory Math 19

Step 2 Calculate the sample volume required to fall within the lower concentration limit. Volume Low Volume 50 ml 2 mg/l 8 ml Test Concentration 12.5 mg/l Concentration 8 ml of sample would be diluted to 50 ml Sample Low Test Low 7/24/17 Laboratory Math 20

Chemical Dilutions Not all labs do this Useful when: Solutions are not available at the required concentration Shelf life is extremely short at the required concentration 7/24/17 Laboratory Math 21

Dilution Formula Works with any concentrations if: Stock and diluted concentrations are expressed in the same units (i.e. %, mg/l, Normality, Molarity, etc.) Stock and diluted volumes are expressed in the same units (ml, L, gallons, etc.) Formula can also be used when preparing dilutions of process chemicals 7/24/17 Laboratory Math 22

Dilution Formula Volume 1 Stock 2 1 Volume 2 Concentration Concentration Solution ( solution used to prepare Standard Solution ( solution being prepared) 1 2 dilution) 7/24/17 Laboratory Math 23

Example Prepare 1,000 ml of 0.025N sodium thiosulfate solution. The concentration of the stock sodium thiosulfate solution is 1.0N. What is the volume of stock solution required? Volume 1.0N 1,000 ml = 25 ml 0.025N 1.0 N 0.025N 7/24/17 Laboratory Math 24

Calculate a Final Result Titration test method Alkalinity, Hardness, Ammonia, etc. Gravimetric test method TS, TSS, TVS, Oil & Grease, etc. Colorimetric test method TP, NO2-, NO3-, TKN, Ammonia, etc. Electrometric test method Ammonia 7/24/17 Laboratory Math 25

Titration Determine concentration of one chemical using the amount of another chemical required to just react in a known volume of sample. Examples: Winkler D.O., Alkalinity, Ammonia, TRC Oddly, BOD (?) 7/24/17 Laboratory Math 26

Titration Calculation Calculate alkalinity as mg/l CaCO 3. Alkalinity, mg/l CaCO3 (B A) ml of A Start Buret Reading B Final Buret Reading N Normality of Standard Acid N 50,000 Sample 7/24/17 Laboratory Math 27

Given Titration Example Start Buret Reading = 0.0 ml Final Buret Reading = 22.5 ml Sample Volume = 100 ml Standard Acid = 0.02 N Alkalinity, mg/l CaCO3 (22.5 0.0) 225 mg/l 100 0.02 50,000 7/24/17 Laboratory Math 28

Gravimetric Measurements Determining the amount of material (usually a solid) by a difference in weight. Examples: TS TSS TVSS Oil and Grease 7/24/17 Laboratory Math 29

Gravimetric Calculation After completing preparation steps of the procedure and determining the tare and sample weights: TSS, A B mg/l Weight Weight (A (grams) (grams) of of Sample dry filter solids, Volume, & dish filter B) 1,000 mg/gram 1,000 ml/l ml & dish 7/24/17 Laboratory Math 30

Example Given the following data, what is the concentration of suspended solids? Weight of dish, filter 1.8996 grams Weight of dish, filter & dry solids 2.0113 grams Sample volume 500 ml TSS, mg/l (2.0113 g 1.8996) 1,000 mg/g 1,000 ml/l = 223 mg/l 500 ml 7/24/17 Laboratory Math 31

Colorimetric Measurements In colorimetric procedures chemicals are added to samples/standards. Reactions occur that produce a characteristic color. In many cases, the intensity of color produced is directly related to the concentration of the chemical of interest. 7/24/17 Laboratory Math 32

Colorimetric Principle When light of an appropriate wavelength is passed through a colored solution part of the light is absorbed If absorbance of different concentrations is linear it demonstrates that the procedure adheres to the Beer-Lambert law (commonly called Beer s Law). 7/24/17 Laboratory Math 33

Beer s Law Reactions that adhere to Beer s Law can form the basis of a colorimetric test procedure. Where : A Absorbance concentration of through solution (in cm) absorbing parameter molar absorptivity/molar extiction coefficient (in L/mol cm) l length of lightpath c A l c (in mol/l) 7/24/17 Laboratory Math 34

Beer s Law For a particular parameter and wavelength, the product of ϵ and l can be determined. Where : A l 7/24/17 Laboratory Math 35 A c Absorbance measured using a spectrophotometer/colorimeter c Concentration of a known solution

Beer s Law Assuming the product of ϵ and l are known for a particular parameter and wavelength, by determining the absorbance (A) of a sample using the test procedure, the concentration of the parameter can be calculated. c A 7/24/17 Laboratory Math 36 l

Calibration Curve Key to any colorimetric determination Multiple prepared standards Treated exactly the same as samples Determine Correlation Coefficient ( r ) Linear regression Evaluate for linearity ( ±0.995) Move on or repeat! 7/24/17 Laboratory Math 37

Correlation Coefficient, r r = [ ( n x x 2 [ ( n x xy ) - ( x ) 2 ) - ( ] x [ x x y ) ( n x y 2 ] ) - ( y ) 2 ) ] x 2 x = Absorbance or xy = product of = square of Absorbance or y 2 r = correlation coefficent n = number of test values = square of y = Concentration = the sum of individual values ( % Transmittance paired x and y values % Transmittance values concentration values i.e. x, y, xy,x and y 7/24/17 Laboratory Math 38 2 2 )

Calibration Curve Use to determine concentration or mass Multiple prepared standards Treated exactly the same as samples Determine line-of-best-fit y = mx + b Where: y = concentration of parameter x = absorbance value m = Slope of the line b = point where the line crosses the y-axis when x = 0 7/24/17 Laboratory Math 39

Calibration Curve Example Total Phosphorus Test Data Calibration Curve Date: June 19, 2017 Std Concentration (mg) Abs. 0.125 0.08 0.25 0.17 0.375 0.26 0.5 0.36 0.75 0.52 1.25 0.82 7/24/17 Laboratory Math 40

Linear Regression The linear regression of the concentration and absorbance data for the standards produces the three key elements in calculating the Total Phosphorus results (correlation coefficient, slope of the line-of-best-fit & y axis intercept). Total Phosphorus Regression Statistics Calibration Calibration Curve Curve Date: Date: June June 19, 19, 19, 2017 2017 Regression Output: Constant (b (b = = y y intercept) -0.01694192 Std Err Err of of Y Y Est Est 0.024348846 R Squared 0.997154242 Correlation Coefficient (r): (r): 0.99857611 No. No. of of Observations 6 6 Degrees of of Freedom 4 4 X Coefficient(s) (m (m = = slope slope of of line) line) 1.516584398 7/24/17 Laboratory Math 41

Calculating Results Calibration Curve Date: Total Phosphorus Final Results Because the standards were prepared as mg P, the final result must be calculated using the calculated P and the sample volume. mg P / L June 19, 2017 Sample Site Sample Date Calculated P, mg Abs. Sample Vol., ml a1 6/14/2011 0.15 0.11 50 a2 6/14/2011 0.17 0.12 50 a3 6/14/2011 0.18 0.13 50 a4 6/14/2011 0.13 0.10 50 a5 6/14/2011 0.17 0.12 50 mg P ( in approximately ml final volume) 1,000 ml original sample P Concentration, mg/l 7/24/17 Laboratory Math 42 3.00 3.30 3.60 2.69 3.30

Colorimetric Results In some cases the calibration curve is prepared using mg/l values vs absorbance or transmittance. Record the result directly from calculated value May need to correct for dilution factor 7/24/17 Laboratory Math 43

Electrometric Measurements Similar to colorimetric procedures Uses electrical devices (electrodes) to measure current produced by chemical reactions Internal circuitry allows meter to: Read current generated across electrodes Be adjusted to calibrate or match meter output to specific concentrations 7/24/17 Laboratory Math 44

Electrometric Examples ph (meter method) Dissolved Oxygen (probe method) Specific Ion Electrodes Ammonia & TKN (probe method) Nitrate (probe method) Total residual chlorine (probe method) Other parameters 7/24/17 Laboratory Math 45

Direct Measurements ph (standard units) Dissolved oxygen (mg/l) Requires calibration under test conditions No calculations for test result QA/QC may require math 7/24/17 Laboratory Math 46

Specific Ion Electrodes AKA: Ion Selective Electrodes Similar to colorimetric procedures Requires: Series of standards for calibration curve Calculation of correlation coefficient, r Calculation of final result from curve May require mathematical manipulation 7/24/17 Laboratory Math 47

Calibration Curve Example Ammonia Nitrogen Test Data Calibration Curve Date: 6/19/2017 Slope Check mv Std Concentration Log Conc mv 1 ml 1000 mg/l ammonia 114.4 0.2-0.69897 147.7 10 ml 1000 mg/l ammonia 57.2 0.5-0.30103 130.4 1 0.00000 114.4 Difference (Slope) -57.2 10 1.00000 57.2 100 2.00000 0.8 7/24/17 Laboratory Math 48

Linear Regression The linear regression of the concentration and absorbance data for the standards produces the three key elements in calculating the Total Phosphorus results (correlation coefficient, slope of the line-of-best-fit & y axis intercept). Ammonia Nitrogen Regression Statistics Calibration Curve Date: 6/19/2017 Regression Output: Constant (b (b = = y y intercept) 2.0297412 Std Err Err of of Y Y Est Est 0.04419666 R R Squared 0.99877384 Correlation Coefficient (r): (r): -0.99938673 No. No. of of Observations 5 5 Degrees of of Freedom 3 3 X X Coefficient(s) - (m - (m = = Slope) -0.01808814 7/24/17 Laboratory Math 49

Calculating Results Ammonia Nitrogen Final Results Calibration Curve Date: 6/19/2017 Sample Site Sample Date Calculated NH 3, mg/l Log Conc. mv Sample Vol., ml Effluent 6/15/2017 0.05-1.27134392 182.5 100 Filter Influent 6/15/2017 0.06-1.2297412 180.2 100 Influent 6/15/2017 50.81 1.70596354 17.9 100 Strange Inf. 6/15/2017 10.61 1.02584956 55.5 20 Because the standards were prepared as mg NH3-N/L, the final result only needs to be adjusted for sample dilution volume. mg NH 3 mg NH N 100 N / L 3 ml original sample NH 3 Concentration, mg/l 7/24/17 Laboratory Math 50 0.05 0.06 50.81 53.07

QA/QC CALCULATIONS 7/24/17 Laboratory Math 51

Quality Assurance/Quality Control (QA/QC) Quality Control: Individual things you do to check that data is acceptable. Examples: blanks, duplicate samples, and known additions. Quality Assurance: Combination of all of the QC things combined that ensure the quality of data. 7/24/17 Laboratory Math 52

Standard Methods - Part 1000 Part 1000 applies to ALL standard methods. 7/24/17 Laboratory Math 53

QA/QC Part 1020 22 nd Ed. Initial Demonstration of Capability (B.1) [Certification of operator competence] Operational Range (B.2) Externally Supplied Standards (B.3) [Ongoing Demonstration of Capability] Method Detection Level (B.4) Analysis of Reagent Blanks (B.5) Laboratory Fortified Blank/Control Standard (B.6) 7/24/17 Laboratory Math 54

QA/QC Part 1020 22 nd Ed. Laboratory Fortified Matrix/Spike (B.7) Analysis of Duplicates (B.8) Internal Standards (B.9) Surrogates and Tracers (B.10) Calibration (B.11) Instrument calibration (B.11.a.) Initial calibration (B.11.b) Calibration verification (B.11.c) Performance evaluation samples (C.1) 7/24/17 Laboratory Math 55

% Recovery Most common calculation used for QA/QC procedures! Test Result % Recovery 100 Known Conc. 7/24/17 Laboratory Math 56

% Recovery Example 23 mg/l % Recovery 100 20 mg/l Typically acceptance criteria may vary with application: 70 to 130% 80 to 120% 90 to 110% 115% 7/24/17 Laboratory Math 57

DUPLICATES 2 identical samples collected at the same time and site. Must be prepared and analyzed as separate samples. 5% of samples Relative Percent Difference (RPD) for low level samples [conc. near lowest calibration standard] ±25% RPD for higher level samples [conc. near middle calibration standard] ±10 % 7/24/17 Laboratory Math 58

RPD Example: RPD Calculation ( Sample Concentration ( Sample Concentration Duplicate Concentration) Duplicate Concentration) Sample Concentration = 3.5 mg/l Duplicate Concentration = 3.0 mg/l RPD 3.5 mg/l 3.5 mg/l 2 3.0 mg/l 3.0 mg/l 2 100% = 15.4% 0.5 mg/l 3.25 mg/l 100% 100% 7/24/17 Laboratory Math 59

Matrix Spikes SPIKES (Known Additions) Addition of known amount of parameter Spiking Conc. - 1-10X usual sample value. If not analyzing dups, spike 10% of samples. 1 out of 10 If analyzing dups and spikes, combined total must be at least 10% of samples. 80-120% recovery. 7/24/17 Laboratory Math 60

Matrix Spike Concentration What spike concentration is required? Sample conc. = 0.3 mg/l Standard conc. = 10 mg/l Spike at 10 mg/l? Dilutes sample too much. Spike at 1.5 mg/l? Awkward! Think KISS Spike at 1.0 mg/l? Good choice 7/24/17 Laboratory Math 61

Volume of Matrix Spike Need to spike a sample sample that usually is 0.3 mg/l. What do you do? 1. Determine Conc. of final spike. Don t include 0.3 ppm already in sample. 2. Vol. of spiked sample. 3. Conc. of spiking solution. 4. Vol. of spiking solution needed. 7/24/17 Laboratory Math 62

Volume of Matrix Spike Determine volume of spiking solution needed, if: 1. Final spike conc. = 1.0 mg/l 2. Final test volume = 50 ml 3. Spiking solution conc. = 10 mg/l Spike volume, ml 1.0 mg/l 10 mg/l 50 ml 5.0 ml 7/24/17 Laboratory Math 63

Spike Recovery Analyze sample without spike added (sample). Analyze sample with spike added (matrix spike or MS). Measured Spike Conc. Measured Sample Conc. % Spike Recovery 100 True Value of Spike Conc. 7/24/17 Laboratory Math 64

Example Spike Recovery Spiked Sample Result = 6.7 mg/l Dup. Spiked Sample Result = 7.4 mg/l Matrix Spike = 18.3 mg/l Concentration of Spike =10 mg/l % Spike Recovery % Spike Recovery 18.3 mg/l 18.3 mg/l 10 mg/l 10 mg/l 6.7 mg/l 7.4 mg/l 100 116% 100 109% 7/24/17 Laboratory Math 65

Spike Recovery When a duplicate is run on a sample that is spiked, calculate recovery using both sample value and dup value. Don t average. If one of the two fails, flag data on benchsheet and DMR 7/24/17 Laboratory Math 66

Questions? 7/24/17 Laboratory Math 67