A student in their lab dass measured the following spectroscopic data for some pure samples of three indicators thymol blue (TB), semithymol blue (STB) and methylthymol blue (MTB), below Figure.

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Rewrite the data in matrix form and by appropriate matrix manipulation; find the concentrations of TB, STB and MTB in the mixture. You should attempt the manual approach which we have used up until now and find the difficulties involved.

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Solution

This is the same fundamental problem as above, extended to a ternary mixture. It is assumed that the concentrations of the dyes are low and their absorbances at a range of wavelengths are additive, i.e. they are each dominant in at a given wavelength and they don’t interact chemically with one another. First let us try the manual matrix approach.

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Writing this in matrix form, entering the student’s lab results (without units for clarity) and then rearranging the matrix equation into the standard simultaneous equations form we have

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but trying to solve this manually gives us problems, here is the determinant

 

|A|= +4800(11200 x 4450- 11800 x 13900)- 11100(7350 x 4450- 11800 x 36400) +18900(7350 x 13900- 11200 x 36400)- 7350(11100 x 4450- 18900 x 13900) +11200(4800 x 4450- 18900 x 36400)- 11800(4800 x 13900- 11100 x 36400) +36400(11100 x 11800- 18900 x 11200, 13900(4800 x 4450- 18900 x 36400)+4450(4800 x 11200- 11100 x  7350)

The above determinant is awful! The manual method of matrix algebra for this type of numerical matrix equation has been useful for initial learning purposes also for small 2×2 matrices and matrices with “sparse” numbers. The manual method is also useful for examination questions where computers are not allowed! However, the manual method is a problem when finding the determinant and the matrix of cofactors for 3×3 or larger matrices and for those with numerically “rich” matrices as in the one above. It is slow, clumsy and prone to errors. Lees see how easy it is to use a spreadsheet and the inversion of matrix A. and then directly use x = A-1 b which is shown in below figure.

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The wavelengths are not used in the calculation, they are just helpful reminders. The matrix A is in the array B3:D5 and the matrix b is in array E3:E5. We select the three empty cell F3:F5 and type in the formula window the following array formula

=MMULT(MINVERSE(B3D5),E3:E5)

Do not press enter or return as it is an array function rather than a simple Maths function. Instead on a PC press Shift+Control+Return or on Mac Command+Return and the solution appear in the array F3:F5. The spreadsheet commands are MMULT for matrix multiplication the two matrices. The first matrix is the inverse of A (Le. A-1) MINVERSE and the second is the absorbances of the mixture, b.

cTB = 3.52 x 10-6 mo1 L-1

cSTB = 6.51 x 10-6 mo1 L-1

cMTB = 2.04 x 10-5 mo1 L-1