Chemistry 155 Introduction to Instrumental Analytical Chemistry

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Chem 155 Unit 1 Page 32 of 3131.9.1 Linear Least Squares CalibrationThe most common method for determining theconcentration of an unknown analyte is the simplecalibration curve.In the calibration curve method, one measures theinstrument signal for a range of analyteconcentrations (called standards) and develops anapproximate relationship (mathematically orgraphically) between some signal ‘S’ and analyteconcentration ‘C’.If the signal-concentration relationship is linear, then:S = S B + mC y = a + bxBut, one can not just draw the line between any twopoints because all the points have some error. So,one mathematically attempts to minimize theresiduals.χ 2 = Σ(y i – (a + bx i )) 21.51.5Instrument Signal (S)S iS iF i10.5F i10000 2 4 6 8 10Page 32 of 3130 C iAnalyte Concentration (C)
Chem 155 Unit 1 Page 33 of 313The values of a and b for which χ 2 is a minimum arethe following:The errors in the coefficients, σ a and σ b can be found,similarly, using:These quantities are best found using a computerprogram. Modern versions of Microsoft Excel willcalculate a and b for you (use ‘display equation’option on the ‘trend line’), and σ a and σ b if you havethe data analysis ‘toolpack’ option installed. Excel isalso fairly well suited to doing the sums and formulas.Page 33 of 313
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Chemistry 155 Introduction to Instrumental Analytical Chemistry

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