**4.1 Example**

312 Macro to Nano Spectroscopy

**(C**[M,S]**.**

The concentration matrix **C**[M,S] is expressed from relation (44). To this purpose, equation

**A**[,1] = **E<sup>T</sup>**[,]**.**

**-1.**

The particular case of standard addition method applied to a system with two components to be determined, is illustrated graphically in Figure 2. In this case, the procedure is reduced to determining the plane passing through a number of figurative points and to reading the

**E<sup>T</sup>**[,]**.**

**C T**[S,M]**)**

**E**[,M]**.**

[,]**.**

**-1**= **E**[,M] (49)

**C**[M,1] (50)

**E**[,M]**)-1**, the explicit form of

**A**[,1] = **C**[M,1] (51)

(1**/**d).

(1**/**d).

(1**/**d).

the concentration matrix results (51).

concentrations c1i and c2i (i = 1 , 2 , . . . , n)

**A**[,S]**.**

When the above relation is multipled on the left by (**E<sup>T</sup>**

**(E<sup>T</sup>** [,]**.**

(44) is multiplied on the left by the transpose of matrix **E**[,M].

**E<sup>T</sup>**[,]**.**

**C T**[S,M]**.**

**E**[,M]**)**

intersection points of this plane with the negative semi-axes of the concentrations.

Fig. 2. Graphic representation of absorbances Ai() in relation to the modifications of

of interest in the first solution, and finally their content in the primary sample.

At the graphic representation of absorbances Ai() vs. the increase of concentrations c1i and c2i (i = 1 , 2 , . . . , n), the figurative points are situated theoretically on a plane (denoted by in Figure 4-20). The axis of absorbances is intersected by plane in point P, corresponding to the absorbance A0(), measured in the case of the solution with i = 0. If at the selected wavelength () the absorbance of the ingredients can be left out, the points X and Y, situated at the intersection of plane with the negative parts of axes c1 and c2, have the coordinates –c10 respectively –c20 (in other words, the lengths of the segments OX and OY are proportional to the concentrations c10 and c20). From the values c10 and c20, and knowing the volumes Va, v and Vb, one may calculate the concentrations c1 and c2 of the components In order to illustrate the application of the standard addition method and of the subsequent data processing procedure, let consider the mixture of salicylic acid, caffeine and acetaminophen, discussed in a previous example. The aim is to determine the concentrations of the three chemical components. Table 2 includes the modifications of the component concentrations (5 modifications are performed) and the absorbances both for the original solution (where concentrations have not been modified) and for the five solutions in which the three chemical components have been modified. All absorbance values are read at the same set of 18 wavelengths ( = 18).

The elements of matrix **E** are calculated with relation (49) and are expressed in the tolerated unit of measure l/(mol. cm), employed in spectrophotometric practice, and the elements of matrix **C**, calculated with relation (51) are expressed in mol/l.


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$$\begin{array}{cc} \text{ } & \mathbf{C} = \begin{pmatrix} 17.819 \\ 10.290 \\ 5.307 \end{pmatrix} \text{ } \mu m \text{ol } / \text{lb} \end{array}$$
