Solution 2.11 - Chemometrics: Data Analysis for the Laboratory and Chemical Plant
Education Article
- Published: Jan 1, 2000
- Channels: Chemometrics & Informatics
1. A mixture diagram is as follows. The new triangle is indicated. Note how there are binary mixtures of components 1 and 2 (points A and B on the triangle), since the sum of the lower and upper limits for each of these components equal 1, but ternary mixtures when component 3 is at its upper limit. A triangle is possible because the sum of lower limits for any two components and upper limit for the third component is always 1.
2. The true design matrix is as follows.
x_{1} |
x_{2} |
x_{3} |
x_{1}x_{2} |
x_{1}x_{3} |
x_{2}x_{3} |
x_{1}x_{2}x_{3} |
0.9 |
0.1 |
0 |
0.09 |
0 |
0 |
0 |
0.2 |
0.8 |
0 |
0.16 |
0 |
0 |
0 |
0.2 |
0.1 |
0.7 |
0.02 |
0.14 |
0.07 |
0.014 |
0.55 |
0.45 |
0 |
0.2475 |
0 |
0 |
0 |
0.55 |
0.1 |
0.35 |
0.055 |
0.1925 |
0.035 |
0.01925 |
0.2 |
0.45 |
0.35 |
0.09 |
0.07 |
0.1575 |
0.0315 |
0.4333 |
0.3333 |
0.2333 |
0.1444 |
0.1011 |
0.0778 |
0.0337 |
3. The model is
y = 2.713 x_{1}-1.040 x_{2}-1.072 x_{3}+18.245 x_{1}x_{2}+15.775 x_{1}x_{3}
+16.041 x_{2}x_{3}-83.878 x_{1}x_{2}x_{3}
Note that the quite large terms for the higher order interactions are not necessarily very significant. The range of variation for x_{1} is 0.7 (maximum-minimum) whereas for x_{1}x_{2}x_{3} it is 0.0337 or almost 20 times less.
4. The predicted conductivity when the salts are in proportions 0.209, 0.146 and 0.645 is 2.27 W ^{-1}cm^{-1}.