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A revised algorithm for calculating sample concentrations from spectrophotometric absorbances
Wat. Res. Vol. 29, No. 6, pp. 1589-1590, 1995
Pergamon
0043-1354(94)00281-9
Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0043-1354/95 $9.50 + 0.00
TECHNICAL NOTE A REVISED ALGORITHM FOR CALCULATING SAMPLE CONCENTRATIONS FROM SPECTROPHOTOMETRIC ABSORBANCES J A M E S R. M c E W A N , t* and A L B E R T J. G A B R I C 2 ~Department of Chemical Engineering, The University of Queensland, Brisbane, Qld 4072 and 2Faculty of Environmental Sciences, Griffith University, Nathan, Qld 4111, Australia
(First received April 1994; accepted in revised form October 1994) Abstract--Standard colorimetric techniques require conversion of measured absorbances to concentrations. When blank absorbances are significant, use of the conventional algorithm under-estimates the true sample concentration. A revised algorithm is presented which allows a more accurate determination of sample concentration. Key words--spectrophotometry, a n a l y t i c a l t e c h n i q u e s , w a t e r q u a l i t y a s s e s s m e n t
INTRODUCTION
Field studies of water quality in marine and freshwater environments often require accurate determination of extremely low ( < 1 # M ) ambient nutrient levels. Difficulties in obtaining sufficiently pure water necessary for the requisite laboratory analysis techniques has led to a re-appraisal of the conventional method of calculating sample concentrations based on spectrophotometrically measured absorbances. Spectrophotometric techniques for measurement of concentrations of the target analyte in a sample are based on the Beer-Lambert law, which relates absorbance at a particular wavelength linearly to sample concentration, after addition of suitable reagents to develop colour in the sample (e.g. Parsons et al., 1984; A P H A , 1985). Standards of known concentration are measured simultaneously to establish a calibration curve, which allows the sample concentration to be calculated from the sample absorbance after subtraction of the reagent blank, by a simple algorithm. When very low levels of analyte are being measured, the water used to make up standards, blanks and reagents can often contain background concentrations of the analyte which are of the same order of magnitude as those in the sample. Addition of reagent to the blank does not change the concentration in the blank regardless of the level of background contamination, since both are from the same matrix. When " c o n t a m i n a t e d " reagent is added to samples and standards, however, their respective concentrations will change by different amounts depending on their initial concentrations and volumes relative to that of the reagent. *Author to whom all correspondence should be addressed.
The conventional algorithm does not take this into account and its use can result in under-estimates of the true concentration of the analyte in the sample and even negative values where blank absorbances are significant. In the following we outline a more correct procedure, which is to subtract a blank absorbance weighted by the relative volumes of reagent and sample before conversion of the sample absorbance to concentration. DERIVATION OF REVISED A L G O R I T H M
Sample concentration is usually calculated by,
1
Cs = ~ ( A s - Ab)
(l)
where As = sample absorbance blank absorbance C~ = sample concentration k = calibration factor (slope of calibration curve i.e. absorbance per unit concentration).
Ab=
Consider a situation where the water used to make up standards and blanks contains a detectable concentration of the analyte, the reagents themselves have no intrinsic absorbance at the wavelength used, and two standards and one blank are used to construct the calibration curve. Let Cb = analyte concentration in make-up water Cs = true sample concentration C~,,2a = nominal concentration in standards 1 and 2 (obtained by serial dilution) C~b,2b = true concentration in standards 1 and 2 C~¢.2~= concentration in standards 1 and 2 after addition of reagents
1590
Technical Note V~= volume of reagent added to samples and standards V~= volume of sample and standard used A = measured absorbance.
From the Beer-Lambert law, absorbance is proportional to concentration i.e.
A = kC.
therefore
A I - A 2 fV~+ V,~ C from which Cb
(2)
The true concentration in the standards is given by CIb = CIa -~- C b
(3a)
C2b = C2, + Cb-
(3b)
If reagents of total volume V~ are added to the standards, both of volume V~, the concentrations are
Vr+V~
b t A l _ _ A s , ].
(12)
From equations (9) and (12), the true concentration in the sample is given by
C
Cla -- C2a[- --( Vr ] A, [As iV'+ VJ bJ
(13)
which can be written as
now
I Cl~ -
V r C b -}-
Vs
(C,. + Cb)
(4a)
and, similarly
C2c ~ _
_
V~+Vs
cb + ~ Vs (c2. + CO.
(4b)
Thus, by equation (2), the measured absorbances of standards 1 and 2 will be given by
As=
V,
v~+v~
v~ - Cb + ~ V~+Vs
A, = k
A
Vs(Cla + Cb)
V~+ Vs C, + ~
V~
(C.
/<[v, [Vr+VsCb+~(C2,
+ Cb)]
+ cu)].
(5a)
(5b)
Solving equations (5a) and (5b) for k, gives
k
Aj - A2 (Vr q- Vs]. ~Cs~\ V~ ,]
(6)
Similarly, if reagents are added to a sample the actual concentration in the sample becomes C -
vsC~+ v~cb V~ +
(7)
where K is the slope of the calibration curve (absorbance per unit concentration) obtained from any desired number of standards, using the nominal standard concentrations. Comparison of equations (1) and (14) implies that the difference in computed sample concentration between the two formulae will depend on both the magnitude of the blank absorbance relative to the sample absorbance, and the relative volumes of reagent and sample. Use of equation (I) with low reagent/sample volume ratios will cause greater under-estimates of true concentrations. As an example, an analysis of reactive silicate in seawater by a standard technique (Parsons et al., 1984) produced the following results: Vs = Vr= K= As = Ab =
25 ml 10ml 100 absorbance units/pM 300 absorbance units 80 absorbance units.
From equation (1), sample concentration = 2.20 p M. From equation (14), sample concentration = 2.77 #M. Using equation (1) results in a 20% underestimate of the true concentration in the sample.
Vr
CONCLUSION
The measured sample absorbance is then
c,. cs.k--VS--~)L V-----~s~Vr J" (8) Solving for C~ gives As ] - ~ Cb.
(9)
Since the blank contains only the background concentration
By allowing for the dilution effect of the added reagents, the revised algorithm allows a more accurate determination of sample concentrations by colorimetry when water used for analysis contains trace levels of the target analyte. REFERENCES
APHA (1985) Standard Methods"for the Examination of Water and Wastewater, 16th edn. Am. Publ. Hlth Assoc., Washington, D.C. Parsons T. R., Maita Y. and Lalli C. M. (1984) A Manual
of Chemical and Biological Methodsfor Seawater Analysis, A b = kC b
(10)
1st edn. Pergamon Press, Oxford.
Pergamon
0043-1354(94)00281-9
Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0043-1354/95 $9.50 + 0.00
TECHNICAL NOTE A REVISED ALGORITHM FOR CALCULATING SAMPLE CONCENTRATIONS FROM SPECTROPHOTOMETRIC ABSORBANCES J A M E S R. M c E W A N , t* and A L B E R T J. G A B R I C 2 ~Department of Chemical Engineering, The University of Queensland, Brisbane, Qld 4072 and 2Faculty of Environmental Sciences, Griffith University, Nathan, Qld 4111, Australia
(First received April 1994; accepted in revised form October 1994) Abstract--Standard colorimetric techniques require conversion of measured absorbances to concentrations. When blank absorbances are significant, use of the conventional algorithm under-estimates the true sample concentration. A revised algorithm is presented which allows a more accurate determination of sample concentration. Key words--spectrophotometry, a n a l y t i c a l t e c h n i q u e s , w a t e r q u a l i t y a s s e s s m e n t
INTRODUCTION
Field studies of water quality in marine and freshwater environments often require accurate determination of extremely low ( < 1 # M ) ambient nutrient levels. Difficulties in obtaining sufficiently pure water necessary for the requisite laboratory analysis techniques has led to a re-appraisal of the conventional method of calculating sample concentrations based on spectrophotometrically measured absorbances. Spectrophotometric techniques for measurement of concentrations of the target analyte in a sample are based on the Beer-Lambert law, which relates absorbance at a particular wavelength linearly to sample concentration, after addition of suitable reagents to develop colour in the sample (e.g. Parsons et al., 1984; A P H A , 1985). Standards of known concentration are measured simultaneously to establish a calibration curve, which allows the sample concentration to be calculated from the sample absorbance after subtraction of the reagent blank, by a simple algorithm. When very low levels of analyte are being measured, the water used to make up standards, blanks and reagents can often contain background concentrations of the analyte which are of the same order of magnitude as those in the sample. Addition of reagent to the blank does not change the concentration in the blank regardless of the level of background contamination, since both are from the same matrix. When " c o n t a m i n a t e d " reagent is added to samples and standards, however, their respective concentrations will change by different amounts depending on their initial concentrations and volumes relative to that of the reagent. *Author to whom all correspondence should be addressed.
The conventional algorithm does not take this into account and its use can result in under-estimates of the true concentration of the analyte in the sample and even negative values where blank absorbances are significant. In the following we outline a more correct procedure, which is to subtract a blank absorbance weighted by the relative volumes of reagent and sample before conversion of the sample absorbance to concentration. DERIVATION OF REVISED A L G O R I T H M
Sample concentration is usually calculated by,
1
Cs = ~ ( A s - Ab)
(l)
where As = sample absorbance blank absorbance C~ = sample concentration k = calibration factor (slope of calibration curve i.e. absorbance per unit concentration).
Ab=
Consider a situation where the water used to make up standards and blanks contains a detectable concentration of the analyte, the reagents themselves have no intrinsic absorbance at the wavelength used, and two standards and one blank are used to construct the calibration curve. Let Cb = analyte concentration in make-up water Cs = true sample concentration C~,,2a = nominal concentration in standards 1 and 2 (obtained by serial dilution) C~b,2b = true concentration in standards 1 and 2 C~¢.2~= concentration in standards 1 and 2 after addition of reagents
1590
Technical Note V~= volume of reagent added to samples and standards V~= volume of sample and standard used A = measured absorbance.
From the Beer-Lambert law, absorbance is proportional to concentration i.e.
A = kC.
therefore
A I - A 2 fV~+ V,~ C from which Cb
(2)
The true concentration in the standards is given by CIb = CIa -~- C b
(3a)
C2b = C2, + Cb-
(3b)
If reagents of total volume V~ are added to the standards, both of volume V~, the concentrations are
Vr+V~
b t A l _ _ A s , ].
(12)
From equations (9) and (12), the true concentration in the sample is given by
C
Cla -- C2a[- --( Vr ] A, [As iV'+ VJ bJ
(13)
which can be written as
now
I Cl~ -
V r C b -}-
Vs
(C,. + Cb)
(4a)
and, similarly
C2c ~ _
_
V~+Vs
cb + ~ Vs (c2. + CO.
(4b)
Thus, by equation (2), the measured absorbances of standards 1 and 2 will be given by
As=
V,
v~+v~
v~ - Cb + ~ V~+Vs
A, = k
A
Vs(Cla + Cb)
V~+ Vs C, + ~
V~
(C.
/<[v, [Vr+VsCb+~(C2,
+ Cb)]
+ cu)].
(5a)
(5b)
Solving equations (5a) and (5b) for k, gives
k
Aj - A2 (Vr q- Vs]. ~Cs~\ V~ ,]
(6)
Similarly, if reagents are added to a sample the actual concentration in the sample becomes C -
vsC~+ v~cb V~ +
(7)
where K is the slope of the calibration curve (absorbance per unit concentration) obtained from any desired number of standards, using the nominal standard concentrations. Comparison of equations (1) and (14) implies that the difference in computed sample concentration between the two formulae will depend on both the magnitude of the blank absorbance relative to the sample absorbance, and the relative volumes of reagent and sample. Use of equation (I) with low reagent/sample volume ratios will cause greater under-estimates of true concentrations. As an example, an analysis of reactive silicate in seawater by a standard technique (Parsons et al., 1984) produced the following results: Vs = Vr= K= As = Ab =
25 ml 10ml 100 absorbance units/pM 300 absorbance units 80 absorbance units.
From equation (1), sample concentration = 2.20 p M. From equation (14), sample concentration = 2.77 #M. Using equation (1) results in a 20% underestimate of the true concentration in the sample.
Vr
CONCLUSION
The measured sample absorbance is then
c,. cs.k--VS--~)L V-----~s~Vr J" (8) Solving for C~ gives As ] - ~ Cb.
(9)
Since the blank contains only the background concentration
By allowing for the dilution effect of the added reagents, the revised algorithm allows a more accurate determination of sample concentrations by colorimetry when water used for analysis contains trace levels of the target analyte. REFERENCES
APHA (1985) Standard Methods"for the Examination of Water and Wastewater, 16th edn. Am. Publ. Hlth Assoc., Washington, D.C. Parsons T. R., Maita Y. and Lalli C. M. (1984) A Manual
of Chemical and Biological Methodsfor Seawater Analysis, A b = kC b
(10)
1st edn. Pergamon Press, Oxford.