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Quantitative Analysis

This analysis focuses on interpreting intensities in mixtures to determine concentrations accurately. By utilizing measured intensities and key equations, concentrations can be derived. It explores the complexities of mass attenuation coefficients in mixtures and solving for unknowns. Special cases like polymorph mixtures and reflective intensities comparisons are also covered to enhance understanding. The emphasis is on practical applications and methodologies, such as the intensity ratio and internal standard methods, in analyzing unknown mixtures, including steps to calibrate and measure reflections.

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Quantitative Analysis

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  1. Quantitative Analysis Intuitive - in a mixture, higher intensities mean higher concentrations

  2. Quantitative Analysis Intuitive - in a mixture, higher intensities mean higher concentrations ….. but …..

  3. Quantitative Analysis So, use measured intensities ….. but not straightforward

  4. Quantitative Analysis KiaXa Iia= a  So, use measured intensities ….. but not straightforward Main equation for quantitative analysis

  5. Quantitative Analysis Iia = intensity of reflection i of phase a KiaXa Iia= a  So, use measured intensities ….. but not straightforward Main equation for quantitative analysis

  6. Quantitative Analysis KiaXa Iia= a  ais density of phase a So, use measured intensities ….. but not straightforward Main equation for quantitative analysis

  7. Quantitative Analysis Kia contains structure factor, multiplicity, Lorentz-polarization factor, temperature factor + scale factor for reflection i of phase a KiaXa Iia= a  So, use measured intensities ….. but not straightforward Main equation for quantitative analysis

  8. Quantitative Analysis Xa = wt fraction of phase a (want this!!) KiaXa Iia= a  So, use measured intensities ….. but not straightforward Main equation for quantitative analysis

  9. Quantitative Analysis Xa = wt fraction of phase a (want this!!) KiaXa Iia= a  So, use measured intensities ….. but not straightforward Main equation for quantitative analysis measure this!! But what do we do with ?

  10. Quantitative Analysis Xa = wt fraction of phase a (Want this!!) KiaXa Iia= a  So, use measured intensities ….. but not straightforward Main equation for quantitative analysis Measure this!! But what do we do with ?  is mass attenuation coefficient for mixture…. unknown, unless we know

  11. Quantitative Analysis If sample contains >one phase, then  unknown if wt fractions unknown KiaXa Iia= a   for each phase may be calculated from the chemical composition of each phase compd = X11 + X22 + X33 + ……… mix = X1phase1 + X2phase 2 + X3phase 3 + ………

  12. Quantitative Analysis KiaXa Iia= a  So, use measured intensities ….. but not straightforward Main equation for quantitative analysis also be nice not to calc Kia or get a

  13. 1. Special case - mixture of 2 polymorphs Since polymorphs have same composition mix = a = b

  14. 1. Special case - mixture of 2 polymorphs Suppose we measure Iia of pure a can eliminate mix (& other stuff) KiaXa Iia= a Kia Iia= aa Since polymorphs have same composition mix = a = b = Xa

  15. 1. Special case - mixture of 2 polymorphs Iia = Xa Iia Measure reflection intensities for pure mat’l & same phase in mixture

  16. 1. Special case - rutile/anatase mixture Pure rutile Reflection Intensity Background Net 1 102309 1508 100801 2 47564 1453 46111 Rutile-anatase mixture Reflection Intensity Background Net 1 36987 1486 35501 2 17408 1439 15969 1: 35501/100801=0.352 2: 15969/46111=0.346 wt. fract. anatase: 1-wt. fract. rutile=1-0.349=0.651

  17. 2. Quantitative Analysis KiaXa Iia= a  Main equation for quantitative analysis Determine reflection intensities for unknown & compare to those of standard Mix known amount of standard with unknown Measure reflection intensities for: 1) pure standard 2) pure analyte 3) analyte in unknown 4) standard in unknown

  18. 2. Quantitative Analysis Is KiaXa Xs s KsXs KsXs Ks Ks = Is = Is = Iia= Is= Is=  s s ss ss Is a  Ka KaXa Ia= Ia= aa a

  19. 2. Quantitative Analysis Ia KiaXa Xa a KsXs KaXa Ka Ks = Is = Ia = Iia= Ia= Is=  s a aa ss Ia a  Ka KaXa Ia= Ia= aa a

  20. 2. Quantitative Analysis Xs s Xs s Is Is Xa a Xa a Ia Ia = = = =     Is Is Ia Ia Xs s Is Ia = Is Ia Xa a

  21. Robs t = 1.0 sec Rtrue = 1 – t * Robs 2. Quantitative Analysis - intensity ratio method Now for the problem…. Determine amount of quartz in a rock Add known amount of KCl to rock sample Correct for deadtime and subtract background

  22. 2. Quantitative Analysis - intensity ratio method corr corr Phase  I pure Ipure Imix Imix Quartz 35.0 48360 2648 KCl 124.0 19072 6160

  23. 2. Quantitative Analysis - intensity ratio method corr corr Phase  I pure Ipure Imix Imix Quartz 35.0 48360 50578 2648 2415 KCl 124.0 19072 19203 6160 5958

  24. 2. Quantitative Analysis - intensity ratio method

  25. 2. Quantitative Analysis - intensity ratio method

  26. 2. Quantitative Analysis - intensity ratio method

  27. 2. Quantitative Analysis - intensity ratio method

  28. 2. Quantitative Analysis - intensity ratio method

  29. KiaXa Iia= a  3. Quantitative Analysis - internal standard method Mix known amount of standard with unknown Make calibration sample containing known amounts of standard & analyte Measure reflections for standard & analyte in: 1) calibration sample 2) unknown mixture

  30. Ks Xs Ka Xa Is= Ia= s a Ka Xa Ia= Ia a Ka Xas = a Ks Xs Ks Xs Is Is= s Ia Kas Xa Xa = = C Xs Ksa Xs Is 3. Quantitative Analysis - internal standard method Measure reflections for standard & analyte in: 1) calibration sample 2) unknown mixture

  31. Ia Xs C = Is Xa Ia Kas Xa Xa = = C Xs Ksa Xs Is 3. Quantitative Analysis - internal standard method Measure reflections for standard & analyte in: 1) calibration sample 2) unknown mixture Get C from calibration sample (Xs & Xa known)

  32. Ia Xs C = Is Xa Iq XKCl (8041) (0.5) = 0.7064 = C = IKCl Xq (11383) (0.5) 3. Quantitative Analysis - internal standard method From calibration sample: Phase Robs Rtrue Inet Quartz 8105 8171 8041 KCl 11382 11513 11383

  33. Iq Xq = C IKCl XKCl Iq XKCl (2415) (0.1667) Xq = = IKCl C (5958) (0.7064) = 0.096 1200 Xqrock = 0.096 = 0.115 1000 3. Quantitative Analysis - internal standard method Now everything known to get wt. fract. of quartz in rock

  34. KiaXa Iia= a  4. Quantitative Analysis - reference intensity ratios Problem, as always, is what to do with 

  35. For corundum For analyte KaXa KcXc Ia= Ic= c a KaXa Ia KaXac Xa Ia= = = K a Ic KcXca Xc KcXc Ic= c For 50-50 mixture of analyte with corundum Ia = K …..denoted (I/Ic)a Ic 4. Quantitative Analysis - reference intensity ratios

  36. 4. Quantitative Analysis - reference intensity ratios For 50-50 mixture of analyte with corundum Then, for any a + corundum mixture (I/Ic)a Ia Ia = = K …..denoted (I/Ic)a Ic Ic Xa Xc

  37. For any a + b mixture Ia KaXac Ia = = K = Ic KcXca Ib Ka c Xa Xa Kc a (I/Ic)a Xb Xc = (I/Ic)b Kb c Ka Kc b a Kb = b 4. Quantitative Analysis - reference intensity ratios

  38. 4. Quantitative Analysis - reference intensity ratios Problem: rutile/anatase

  39. 4. Quantitative Analysis - reference intensity ratios Problem: rutile/anatase

  40. 4. Quantitative Analysis - reference intensity ratios Problem: quartz in rock

  41. 4. Quantitative Analysis - reference intensity ratios Problem: quartz in rock

  42. 4. Quantitative Analysis - reference intensity ratios Problem: quartz in rock

  43. 4. Quantitative Analysis - reference intensity ratios Problem: quartz in rock Fast….no calibration req’d I/Ics not accurate……instrument conditions change

  44. Quantitative Analysis: RIR - procedure • Use intensities from workshop 1. Will use multiple reflections with different relative intensities. Must normalize: divide intensities for Al2O3 and ZnO by relative intensities given in PDF • Calculate average Imeas/IPDF for Al2O3 and ZnO • Divide average for ZnO by average for Al2O3 to determine I/Ic for ZnO • Do same for CaF2 & Al2O3

  45. Quantitative Analysis: RIR Procedure • Follow similar procedure for unknown #1 to calculate average Imeas/IPDFs for ZnO and CaF2 • Get IZnO/ICaF2 • Using IZnO/ICaF2 & K (= I/Ic)s previously determined, solve for XCaF2 • Get XZnO from XZnO = 1 – XCaF2 • Repeat for unknowns #2 and #3

  46. RIR 5C5C

  47. RIR

  48. RIR

  49. RIR 5C5Z

  50. RIR

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