at the variable molar ratio 10 : 1 up to 300 : 1 and pH 1 up to 9. The free concentration c_{1} of H^{+} (i.e. pH) and the analytical concentration C_{2} of maltol and C_{3} of UO_{2}^{2+} are known in each of the 148 solutions. The dependence of the absorbance at 3 wavelengths (390, 400 and 410 nm using 2 different optical lengths) on the free concentration c_{1} and the analytical concentrations C_{2} and C_{3} was measured¶. The origin of individual parameters is displayed as an “extension” of the matrix q (the chemical model is taken from Havel J.^{13}).

H^{+} maltol UO_{2}^{2+}
1  0  0  H^{+}  148 × c_{1}
0  1  0  maltol  148 × C_{2}
0  0  1  UO_{2}^{2+}  148 × C_{3}
−1  1  0  H−1maltol−  β_{−110}
0  1  1  UO_{2}maltol^{2+}  3 × a_{2}, β_{011}
−1  1  1  UO_{2}H−1maltol^{+}  3 × a_{3}, β_{−111}
−1  2  1  UO_{2}H−1maltol_{2}^{+}  3 × a_{4}, β_{−111}
−2  2  1  UO_{2}H−2maltol_{2}  3 × a_{5}, β_{−221}
−2  3  1  UO_{2}H−2maltol_{3}  3 × a_{6}, β_{−231}
−3  3  1  UO_{2}H−3maltol_{3}−  3 × a_{7}, β_{−331}
−3  1  1  UO_{2}H−3maltol−  3 × a_{8}, β_{−311}
−6  4  2  (UO_{2})_{2}H−6maltol_{4} 2−  3 × a_{9}, β_{−642}
−6  3  2  (UO_{2})_{2}H−6maltol_{3} 2−  3 × a_{10}, β_{−632}
−6  1  2  (UO_{2})_{2}H−6maltol2−  3 × a_{11}, β_{−612}

In spite of the enormous number of solutions, the absorbance data do not contain enough information for computation of estimates of all the unknown parameters (a, β) together (and no “overall” calculation has ever been performed). The 10 unknown complexes were determined by dividing the studied pH range into several parts and treating them separately.

We did not follow this procedure, but searched for the best “overall” model. For the sake of simplicity, we chose the 10 already “determined” complexes and obtained ∑_{i=1}^{10} \binom{10}{i} = 1023 combinations of order 1 up to 10.

There was no problem to test \underline{all the combinations}†. 740 solutions were found, 535 of them are “positive” (i.e. point estimates of all the unknown stability constants are positive), but most of them are highly correlated. That is why only positive solutions of orders 1 up to 5 are listed (421 solutions).

1. {5} o.k. 1.5468e-2 -4517.5667 -9.9441 -10.0559 1.9543e-4 "uo2malt +10 9 5 3 1"
2. {5} o.k. 1.6792e-2 -4481.1064 -9.8620 -9.9737 2.1216e-4 "uo2malt +10 7 5 3 1"
3. {5} o.k. 1.7515e-2 -4462.3811 -9.8198 -9.9316 2.2130e-4 "uo2malt +8 7 5 3 1"

¶ 3 calibration functions of type 8 are used † the task took several minutes on R10000 175MHz, which is about 20 × faster than i486DX 33MHz

4. {5} o.k. 1.7573e-2 -4460.9123 -9.8165 -9.9283 2.2203e-4 "uo2malt +10 8 5 3 1"
5. {5} o.k. 2.0239e-2 -4398.2185 -9.6753 -9.7870 2.5570e-4 "uo2malt +9 7 5 3 1"
6. {5} o.k. 2.1360e-2 -4374.2718 -9.6214 -9.7331 2.6987e-4 "uo2malt +9 8 5 3 1"
7. {5} o.k. 2.6417e-2 -4279.9353 -9.4089 -9.5206 3.3376e-4 "uo2malt +10 9 7 5 3"
8. {4} o.k. 2.7340e-2 -4272.6845 -9.4389 -9.5283 2.8653e-4 "uo2malt +[2 4 6 8]"
9. {5} o.k. 3.3617e-2 -4172.9169 -9.1679 -9.2796 4.2473e-4 "uo2malt +10 9 8 3 1"
10. {5} o.k. 3.4558e-2 -4160.6572 -9.1403 -9.2520 4.3662e-4 "uo2malt +10 9 8 5 3"
11. {4} o.k. 3.5413e-2 -4157.8077 -9.1801 -9.2695 3.7114e-4 "uo2malt +[2 4 6 7]"

. .

421. {1} o.k. 12.6952 -1570.2417 -3.4906 -3.5130 5.3995e-2 "uo2malt +[1]"

The best models (of orders 5 and 4) are presented in Table 6.

1.  2.  3.  5.  8.  11.
log β_{−111}  −0.40(2)  −0.40(2)  −0.39(2)  −0.39(2)  −0.38(2)  −0.40(3)
log β_{−221}  −2.33(3)  −2.31(4)  −2.30(4)  −2.28(4)  −2.25(5)  −2.33(5)
log β_{−331}  −7.57(4)  −7.59(4)  −7.49(4)  −7.50(5)  −7.42(5)  −7.48(6)
log β_{−311}  −12.20(6)  -  -  -  -  −11.92(6)
log β_{−642}  −15.49(8) −15.1(1)  -  −15.45(9) −15.6(1)  -
log β_{−632}  -  −17.4(1) −17.17(9)  -  -  -
log β_{−612}  -  -  −22.7(1)  −23.8(1)  -  -
10^{4} · U  154.7  167.9  175.2  202.4  273.4  354.1
AIC  −4518  −4481  −4462  −4398  −4273  −4158

Table 6.

2.3 �Fast exchange� NMR

4,9-diaza-1,12-dodecanediamine (spermine, sprm)

The 4th example^{14,15} is the ^{1}H+^{13}C NMR “titration” of spermine. The free concentration c_{1} of H^{+} (i.e. pH) and the analytical concentration C_{2} of sprm are known in each of the 20 solutions. There are 10 different types of nuclei (5×H, 5×C) in the molecule and thus the dependence of 10 chemical shifts on the free concentration c_{1} was measured¶. The origin of individual parameters is displayed as an “extension” of the matrix q.

H^{+} sprm
1  0  H^{+}  20 × c_{1}
0  1  sprm  10 × a_{−2}, 20 × C_{2}
1  1  Hsprm^{+}  10 × a_{1}, β_{11}
2  1  H_{2}sprm^{2+}  10 × a_{2}, β_{21}
3  1  H_{3}sprm^{3+}  10 × a_{3}, β_{31}
4  1  H_{4}sprm^{4+}  10 × a_{4}, β_{41}

Principal component analysis gives the set of residual standard deviations

¶ 10 calibration functions of type −5 are used