The complication of matching the matrix of the standards to that of the sample can be avoided by conducting the standardization in the sample. This is known as the method of standard additions.The simplest version of a standard addi-
C S
A samp
= – .ppb = =
.
. – .
. .
–
0 003 0 296
0 397 0 003
0 296 1 33
1 ppb ppb
–1
Reported
Calibration curve obtained in standard’s matrix
Calibration curve obtained in sample’s matrix
Actual Amount of analyte
Signal
Figure 5.4
Effect of the sample’s matrix on a normal calibration curve.
matrix matching
Adjusting the matrix of an external standard so that it is the same as the matrix of the samples to be analyzed.
method of standard additions A standardization in which aliquots of a standard solution are added to the sample.
Colorplate 1 shows an example of a set of external standards and their corresponding normal calibration curve.
Figure 5.5
Illustration showing the method of standard additions in which separate aliquots of sample are diluted to the same final volume.
One aliquot of sample is spiked with a known volume of a standard solution of analyte before diluting to the final volume.
tion is shown in Figure 5.5. A volume, Vo, of sample is diluted to a final volume, Vf, and the signal, Ssamp is measured. A second identical aliquotof sample is spiked with a volume, Vs, of a standard solution for which the analyte’s concen- tration, CS, is known. The spiked sample is diluted to the same final volume and its signal, Sspike, is recorded. The following two equations relate Ssampand Sspiketo the concentration of analyte, CA, in the original sample
5.5
5.6
where the ratios Vo/Vfand Vs/Vfaccount for the dilution. As long as Vsis small rela- tive to Vo, the effect of adding the standard to the sample’s matrix is insignificant, and the matrices of the sample and the spiked sample may be considered identical.
Under these conditions the value of kis the same in equations 5.5 and 5.6. Solving both equations for kand equating gives
5.7 Equation 5.7 can be solved for the concentration of analyte in the original sample.
S C V V
S
C V V C V V
samp
A o f
spike
A o f S s f
( / ) = ( / ) ( / )
+
S k C V
V C V
spike A o V
f
S s f
= +
S kC V
samp AVo
f
=
Chapter 5 Calibrations, Standardizations, and Blank Corrections
111
Add Vo of CA
Dilute to Vf
Total concentration of analyte
CA Vo Vf
Add Vo of CA
Dilute to Vf
Total concentration of analyte
Add VS of CS
CA Vo
Vf + CS VS Vf
aliquot
A portion of a solution.
Figure 5.6
Illustration showing an alternative form of the method of standard additions. In this case a sample containing the analyte is spiked with a known volume of a standard solution of analyte without further diluting either the sample or the spiked sample.
112
Modern Analytical ChemistryEXAMPLE
5.4
A third spectrophotometric method for the quantitative determination of the concentration of Pb2+in blood yields an Ssampof 0.193 for a 1.00-mL sample of blood that has been diluted to 5.00 mL. A second 1.00-mL sample is spiked with 1.00 µL of a 1560-ppb Pb2+standard and diluted to 5.00 mL, yielding an Sspikeof 0.419. Determine the concentration of Pb2+in the original sample of blood.
SOLUTION
The concentration of Pb2+in the original sample of blood can be determined by making appropriate substitutions into equation 5.7 and solving for CA. Note that all volumes must be in the same units, thus Vsis converted from 1.00 µL to 1.00×10–3mL.
Thus, the concentration of Pb2+in the original sample of blood is 1.33 ppb.
It also is possible to make a standard addition directly to the sample after mea- suring Ssamp(Figure 5.6). In this case, the final volume after the standard addition is Vo+Vsand equations 5.5–5.7 become
Ssamp=kCA
S k C V 5.8
V V C V
V V
spike A o
o s
S s
o s
= + +
+
0 193 1 00 5 00
0 419 1 00
5 00 1560 1 00 10
0 193 0 200
0 419 0 200 0 312
0 0386 0 0602 0 0838
0 0452 0 0602
3
. . .
. .
.
.
. .
.
. .
. . .
. .
–
C C
C C
C C
C C
A A
A A
A A
A
mL mL
mL
mL ppb
ppb ppb
ppb
mL 5.00 mL
=
+ ×
= +
+ =
=
A
A =1 33. ppb
Total concentration of analyte
CA
Total concentration of analyte
Add VS of CS
Vo Vo
CA Vo
Vo + VS + CS VS Vo + VS
Colorplate 2 shows an example of a set of standard additions and their corresponding standard additions calibration curve.
Chapter 5 Calibrations, Standardizations, and Blank Corrections
113
5.9
EXAMPLE
5.5
A fourth spectrophotometric method for the quantitative determination of the concentration of Pb2+in blood yields an Ssampof 0.712 for a 5.00-mL sample of blood. After spiking the blood sample with 5.00 µL of a 1560-ppb Pb2+
standard, an Sspikeof 1.546 is measured. Determine the concentration of Pb2+in the original sample of blood.
SOLUTION
The concentration of Pb2+in the original sample of blood can be determined by making appropriate substitutions into equation 5.9 and solving for CA.
Thus, the concentration of Pb2+in the original sample of blood is 1.33 ppb.
The single-point standard additions outlined in Examples 5.4 and 5.5 are easily adapted to a multiple-point standard addition by preparing a series of spiked sam- ples containing increasing amounts of the standard. A calibration curve is prepared by plotting Sspikeversus an appropriate measure of the amount of added standard.
Figure 5.7 shows two examples of a standard addition calibration curve based on equation 5.6. In Figure 5.7(a) Sspikeis plotted versus the volume of the standard so- lution spikes, Vs. When kis constant, the calibration curve is linear, and it is easy to show that the x-intercept’s absolute value is CAVo/CS.
EXAMPLE
5.6
Starting with equation 5.6, show that the equations for the slope, y-intercept, and x-intercept in Figure 5.7(a) are correct.
SOLUTION
We begin by rewriting equation 5.6 as
which is in the form of the linear equation
Y=y-intercept + slope×X
S kC V
V
kC
V V
spike A o
f
S f
= + × s
S C
S
C V V V C V V V
samp A
spike
A o o s S s o s
= [ /( + )]+ [ /( + )]
0 712 1 546
5 00
5 00 5 00 1560 5 00 10
10
0 712 1 546
0 9990 1 558
0 7113 1 109 1 546
1 33
3 3
. .
.
( . .
. (
. .
. .
. . .
.
– –
C C
C C
C C
C
A A
A A
A A
A
mL
mL ppb
ppb ppb
ppb
=
+ × + ×
×
= +
+ =
=
10 mL)
mL 5.00 mL + 5.00 mL)
–3
Figure 5.7
Examples of calibration curves for the method of standard additions. In (a) the signal is plotted versus the volume of the added standard, and in (b) the signal is plotted versus the concentration of the added standard after dilution.
114
Modern Analytical ChemistryVS Sspiked
x-intercept =
(a)
–CAVo
CS y-intercept =kCAVo Vf
Slope =kCS Vf
Sspiked
x-intercept =
(b)
–CAVo
Vf y-intercept =kCAVo Vf Slope = k
CS VS Vf
( )
where Yis Sspikeand Xis Vs. The slope of the line, therefore, is kCS/Vf, and the y-intercept is kCAVo/Vf. The x-intercept is the value of Xwhen Yis 0, or
Thus, the absolute value of the x-intercept is CAVo/CS.
Since both Voand CSare known, the x-intercept can be used to calculate the ana- lyte’s concentration.
EXAMPLE
5.7
A fifth spectrophotometric method for the quantitative determination of the concentration of Pb2+in blood uses a multiple-point standard addition based on equation 5.6. The original blood sample has a volume of 1.00 mL, and the standard used for spiking the sample has a concentration of 1560 ppb Pb2+. All samples were diluted to 5.00 mL before measuring the signal. A calibration curve of Sspikeversus Vsis described by
0 = + ×
= =
kC V V
kC
V x
x kC V V
kC V
C V C
A o f
S f A o f
S f
A o S
( -intercept)
-intercept –( / )
( / ) –
Sspike= 0.266 + 312 mL–1×Vs
Determine the concentration of Pb2+in the original sample of blood.
SOLUTION
To find the x-intercept we let Sspikeequal 0
0 = 0.266 + 312 mL–1×(x-intercept)
and solve for the x-intercept’s absolute value, giving a value of 8.526×10–4mL.
Thus
and the concentration of Pb2+in the blood sample, CA, is 1.33 ppb.
Figure 5.7(b) shows the relevant relationships when Sspikeis plotted versus the con- centrations of the spiked standards after dilution. Standard addition calibration curves based on equation 5.8 are also possible.
Since a standard additions calibration curve is constructed in the sample, it cannot be extended to the analysis of another sample. Each sample, therefore, re- quires its own standard additions calibration curve. This is a serious drawback to the routine application of the method of standard additions, particularly in labora- tories that must handle many samples or that require a quick turnaround time. For example, suppose you need to analyze ten samples using a three-point calibration curve. For a normal calibration curve using external standards, only 13 solutions need to be analyzed (3 standards and 10 samples). Using the method of standard additions, however, requires the analysis of 30 solutions, since each of the 10 sam- ples must be analyzed three times (once before spiking and two times after adding successive spikes).
The method of standard additions can be used to check the validity of an exter- nal standardization when matrix matching is not feasible. To do this, a normal cali- bration curve of Sstandversus CSis constructed, and the value of kis determined from its slope. A standard additions calibration curve is then constructed using equation 5.6, plotting the data as shown in Figure 5.7(b). The slope of this standard additions calibration curve gives an independent determination of k.If the two val- ues of kare identical, then any difference between the sample’s matrix and that of the external standards can be ignored. When the values of kare different, a propor- tional determinate error is introduced if the normal calibration curve is used.