Langmuir 1988,4, 1193-1197
I
0I Figure 9. Schematic diagram of proposed chemisorption bond between Lewis acid poly(dimethylsi1oxane)polymer and Lewis acid silanolate group on a silica substrate.
constituent because of its small size, slight polarizability, and its empty d-orbitals, which can accept electrons from a base. The Lewis acid character of poly(dimethylsi1oxane)explains its solubility in the basic synthetic ester lubricant trimethylolpropane heptanoate compared to its insolubility in a hydrocarbon lubricant as well as its solubility in benzene and toluene, both Lewis bases. It also explains the widely used process by which silica is made hydrophobic by interaction with poly(dimethylsiloxane),which is an important step in the manufacture of agents to combat foam in black liquor in paper making and for other sources of troublesome foam in various industries. Silica normally has a hydrophilic surface, which is to say it is perfectly wetted by water, and separates readily into its individual particles (which are aggregates of primary particles) without clumping, clotting, or agglomerating when shaken with water. After treatment with poly(dimethylsiloxane) the surface of each particle is drastically altered. The particle is no longer wetted by water but floats on the surface, just as would an oiled needle, in spite of its density being greater than that of water. This behavior points to an angle of contact larger than 90°. A prerequisite for producing such large angles of contact is that the silica be first treated with a solution of caustic soda, which has the effect of creating silanolate groups on
1193
the surface: -Si-O-. The higher the pH at which the caustic treatment is performed, the more effective is the foam-inhibiting composition made from the silica so treated.l0 Patterson has shown a three-way correlation with the pH of the caustic treatment.ll As pH is increased above a value of 7, the amount of bound polymer on the silica surface so treated increases, the contact angle of black liquor on the surface of the silica increases, and the more effective a foam-inhibiting agent is the final manufactured composition containing the hydrophobic silica. These results support an interpretation that the interaction between the silica surface and poly(dimethy1siloxane) is one between a Lewis acid and a Lewis base. The caustic treatment creating silanolate groups on the silica surface introduces anions, which are clearly electron donors and capable of interacting with the electron-accepting silicon atoms of the poly(dimethylsi1oxane). The interaction is therefore an example of chemisorption. The polymer becomes attached by multiple bonds to the silica surface (Figure 9). These bonds are separately weak enough to form and break in a dynamic equilibrium with the bulk phase, but since not all of them break simultaneously, the polymer remains tenaciously attached to the surface and resists elutriation by solvents. The nature of the bonding of poly(dimethylsi1oxane)to a silica surface has long been uncertain.12 The present finding of the Lewis acid character of poly(dimethy1siloxane) provides a definitive answer. Registry No. TMP, 78-16-0; N-phenyl-1-naphthylamine, 90-30-2. (10) Miller, J. R.; Pierce, R. H.; Linton, R. W.; Wills, J. H. U.S.Patent 3 714068 to Philadelphia Quartz Company, Jan 30, 1973. (11) Patterson, R. E. TAPPI Non-Woven Conference, Nashville, TN, April 5-8, 1988. Preprinted paper "Influence of Silica Properties on Performance of Antifoams in Pulp and Paper Applications". (12) Ross, S.; Nishioka, G. M. In Emulsions, Latices, and Dispersions; Becher, P.; Yudenfreund, M. N., Eds.; Marcel Dekker: New York, 1978; p 254.
1291Mossbauer Studies of Iodine Sorbed on Charcoal? M. L. Hyder,* R. L. Postles, and D. G. Karraker E. I. du Pont de Nemours & Company, Savannah River Laboratory, Aiken, South Carolina 29808 Received June 16, 1987. I n Final Form: May 25, 1988 lZ9I2sorbed on several different samples of commercial charcoal was investigated by 1291Mossbauer spectroscopy. The spectra obtained were similar for all types of charcoal studied. They show two quadrupole split patterns: for the first (A), 6 = 1.42 mm/s, e2qQ12,/h= -2470 MHz, and 7) = 0.09; for the second (B), 6 = 0.17 mm/s, e2qQ12,/h= -1230 MHz, and 7 = 0.20. The data are interpreted in terms of the sorption of I through one iodine atom at approximately a right angle to the carbon surface by comparison with the iizsIMossbauer spectrum and crystal structure of the 12-hexamethylenetetraminecharge-transfer complex. In this interpretation, the spectrum A corresponds to the iodine atom nearest to the carbon surface and spectrum B to the remote iodine atom. The A atom is electronically positive, the B atom is negative, and the charcoal surface contributes 0.22e to the iodine molecule.
Introduction Speciality charcoals and some other sorbents are commonly used to remove radioactive iodine from the ventilation systems of nuclear reactors and other nuclear fa'This paper was prepared in connection with work done under Contract No. DE-AC09-76SR00001 with the U.S.Department of Energy.
0743-7463/8S/2404-1193$01.50/0
cilities. At the Savannah River Plant (SRP), the exhaust air from the nuclear reactor buildings is passed through beds of granular charcoal before being released to the atmosphere. The purpose of these beds is to minimize the release of fission product radioiodine to the environment in the event of fuel damage or a reactor accident. Iodine uptake by iodine has been found to be very fast and efficient, and the iodine is very strongly held by the 0 1988 American Chemical Society
Hyder et al.
1194 Langmuir, Vol. 4, No. 5, 1988
carbon. E x p e r i m e n t s have shown that up to 99.998% of elemental iodine is retained on carbon during 4 h in flowing a i r at 180 "C,a t e m p e r a t u r e near the boiling point of T h i s suggests that chemisorption is involved. A t temperatures in this range, the active groups on the carbon surface are beginning to decompose, so i t was not considered fruitful to carry desorption measurements to higher temperatures as a means of studying the interaction. However, i t was noted that the rates of both thermal and radiolytic evolution of iodine from carbon varied with t h e carbon history. In both cases, the evolution rate increased with t i m e of exposure to flowing air.l This observation raised the possibility that the sorption process might differ among carbons of different t y p e and history and inspired t h i s investigation. Mossbauer spectroscopy can, in favorable cases, identify iodine species b y t h e i r characteristic spectra, and i t is applicable to iodine sorbed on solid surfaces. T h i s s t u d y reports the use of lZgIMossbauer spectroscopy to investigate the iodine species sorbed on charcoal and o t h z r sorbents used in air-cleaning technology.
3,P-
iodine.'
3-
5 2.
7 2-
1
-
Experimental Section Sorbents. Carbon samples of varying types were investigated (1) One was coconut shell carbon type GX-176, North American Carbon Co., impregnated with 1% triethylenediamine (TEDA)
and 2% KI; the surface area of this carbon is approximately lo00 m2/g. (2) Several samples of this carbon that had been used for various times in the plant. These were generaly free of TEDA as a result of its loss by evaporation or chemical destruction. (These samples are identified by their serial numbers in Tables 11-IV). (3) A coal-based carbon (Type 208C), surface area about 800 m2/g, prepared by Sutcliffe-Speakman Ltd., impregnated with 5% KI:, was used. (4) A laboratory-grade carbon, Witco 950, prepared by Witco Chemical Co., 8-30 mesh size, was the last sample. Sample Preparation. Iodine-129 was purchased from the Isotopes Division of ORNL as a solution of NalBI in 0.1 M NaaO,. Elemental iodine was released from the solution by acidifying with H2S04and oxidizing with H202. The elemental iodine was swept by a nitrogen gas stream through a small bed (-100 mg) of each sorbent. The sorbent samples contained about 10 mg of lBI, were packaged for Mossbauer measurements in hard plastic holders, and were contained by a flexible polyethylene disk. Mossbauer spectra were measured with an Austin Science Associates constant-acceleration instrument that was operated with a triangular wave form. The source was Mg3'29mTe06(ref 2) prepared by irradiation of Mg3'28Te06 in the HIFR reactor at ORNL. Spectra were measured with both source and sample in a Janis Veri-Temp cryostat in vertical transmission geometry. Spectra were normally measured at temperatures in the range 5-20 K to minimize liquid helium use. The Na(T1)I detector was shielded by an 80 mg/cm2 indium foil to minimize the background from the 27.2-keV T e K X-rays3s4 The velocity was calibrated with a 57Co-Rh source replacing the Mg3129Te06source and an a-iron absorber replacing the sample. The correction to zero velocity for the 57Co-Rh source was taken as -0.11 m m / ~ . ~lZgIMossbauer isomer shifts are reported as relative to Zn129Te = 0. The correction from the Mg3129Te0, source was taken as -3.49 m m / ~ . ~ With * ~ these corrections, the isomer shift of C u ' q was measured as -0.44mm/s, (1) Evans, A. G. "Confinement of Airborne Radioactivity: Progress Report"; DP-1497, January, 1979. ( 2 ) Pasternak, M.; Van der Heyden, M.; Langouche, G. Nucl. Instrum. Methods Phys. Res. 1984, 34, 152. (3) Sanders, R.; de Waard, H. Phys. Reu. 1966, 146, 907. (4) Hafemeister, E. W.; de Pasquall, G.; de Waard, H. Phys. Rev. 1964, 135L3, 1089. (5) Stevens, J. G.; Gettys, W. L. In Mossbauer Isomer Shifts; Shenoy, G. K. Wagner, F. E., Eds.; North-Holland Amsterdam, 1978; p 903. (6) de Waard, H. In Chemical Mossbauer Spectroscopy; Herbert, R. L., Ed.; Plenum: New York, 1984; p 295. (7) de Waard, H. In Mossbauer Effect Data Index; Stevens, J. G.; Stevens, V. E., Eds.; IFI/Plenum: New York, 1973; p 447.
5
7
2
Absorber
Source
-v
1
0
+v
Figure 1. Nuclear energy levels for l'?I'e source and "1 absorber (eq assumed positive) and line positions in a lZsI Miissbauer spectrum. in reasonable agreement with the recommended value of 0.41 mm/~.~ lBMossbauer Spectroscopy?*7 Two iodine isotopes, natural 1271 and the 1.7 X lo7yr lz9I,have y transitions from an excited state to the ground state of the nucleus that are suitable for Mossbauer spectroscopy. The resolution and sensitivity of the lBI Mossbauer effect, so most iodine Mossbauer studies are done with lBI. The disadvantage with lZsIMossbauer experiments is that absorbers must be prepared with the radioactive lBI isotope. Normally, 5-15 mg of 1291 in a sample is sufficient for Mossbauer measurements. The source isotope for the 27.7-keV y transition is 33-day 12%Te,produced by neutron irradiation of isotopically separated ' T e . The excited state has a half-life of 16.8 ns and a 512 nuclear spin; the 1291 ground state has a spin of 712. Figure 1 shows the energy levels and the transitions allowed in quadrupole splitting. The spectra were analyzed by determining the velocity for each transition and fitting these data to the theoretical expression for each transition by the method of Shenoy and Dunlap.s In this method, the nuclear levels are characterized as state I and level I,. Then 4
E(I,IJ = e2QVZZ E aN(I,I,)'tN N-0
(1)
where E(1,IJ is the energy of state I and level I,, Q is the nuclear quadrupole moment for state I, V,, is the electric field gradient (EFG), and uN(I,I,) are the coefficients of a power series in the asymmetry parameter, 7. Values for aN(I,I,) are tabulated by Shenoy and Dunlap.8 The asymmetry parameter is defined by 't
= ( V u - Vyy)/Vzz
(2)
The the resonance energy between the levels I*,I,* (excited state) and I,I, (ground state) is (3) R(Iz*,Iz)= [E(I*,I,*) - E(1,Iz)I + 6 where R(I,*,Iz) is the energy of the resonant transition (see Figure 1) and 6 is the isomer shift. This procedure results in seven equations in three unknowns. (Line 1 (Figure 1)in the lBI Miissbauer spectra has a low intensity and is found at a high negative velocity. This line is normally ignored in the analysis of a spectrum.) These equations were solved by a least-squares algorithm to give values for the isomer shift (a), the quadrupole coupling constant (e2qQlzs/h),and the (8) Shenoy, G. K.; Dunlap, B. D. Nucl. Instrum. Methods. 1969, 71, 285.
1291 Mossbauer Studies of Iodine
Langmuir, Vol. 4, No. 5, 1988 1195
Table I. lZrIMossbauer Transitions for Sorbed IDIa energy, mm/s transition exutl fittedb 2 3 4 5 6 7 8
-16.86 0.44 -4.36 1.95 13.09 10.73 16.45
-16.80 0.41 -4.46 1.75 12.95 10.68 16.69
C
C
Table 11. Mossbauer Parameters for mm/s
Sutcliff-Speakman Charcoal, "A"Pattern. *Derived from fit: 6 = 4.88 mm/s, e2qQl,/h = -2430 MHz, n = 0.09. OMean absolute deviation 0.12 mm/s. I
H0
4
I
1
relative to Zn'qe 1.51 1.36 1.38 1.47 1.45 1.39 1.40
temp, sorbent K exptl GX 176newcarbon 15 5.00 GX-176 K-2 5 4.85 8 4.87 GX-176 14509 GX-176 19740R 12 4.96 8 4.94 GX-176 15021 Sutcliff-Speakman 8 4.88 Witco 10 4.89
0.13 0.05 0.13 0.10 0.06 0.09 0.06
e2qQin/h, MHz -2450 -2450 -2500 -2460 -2490 -2430 -2500
mm/s relative sorbent
.5
Sutcliff-Speakman GX-176 19740R GX-176 K-2 GX-176 15021 Witco
1
1.5 -20
temp, K 8 12 5 8 10
e2qQiw/h,
Zdqe
9
MHz
3.57 3.66 3.82 3.67 3.59
0.08 0.17 0.33 0.18 0.10
0.13 0.24 0.22 0.15 0.24
-1230 -1190 -1270 -1200 -1230
Table IV. Mossbauer Parameters for Sorbed lmIZ (Miscellaneous) -1 0
0
10
20
isomer shift,
mm/s sorbent Ag Mordenite MWA-1 AC-6120 CH31on new carbon (GX-176) Iz in polyethylene GX-176, 15021, Brz treated GX-176, new carbon, Na2S03treated
0
!i
U
i? 2.5
5 -20
to
exptl
Figure 2. lZgIMossbauer spectrum of lZsIsorbed on charcoal.
rpi
TJ
Table 111. Mossbauer Parameters for lz0I2 Sorbed on Carbon (B Spectrum) isomer shift,
0
B
.B
Sorbed on Carbon
(A Spectrum) isomer shift,
-10
0
10
20
Velodty, "/s
Figure 3. lZsIMossbauer spectrum of lZsIsorbed on AC-6120. asymmetry parameter (7). By convention, e2qQlZs/h is converted to ezqQln/hand reported in megahertz to allow comparison with NQR results. A sample fit of experimental data to the values calculated from theory is shown in Table I. The statistical estimation procedure and its derivation are discussed in the Appendix.
Results The lZgIMossbauer spectra of sorbed lZsIcan be divided into two types: strongly quadrupole-split spectra for charcoal, and single-line spectra for other sorbents such as silver mordenite. The spectra of lZgIsorbed on all charcoals was essentially the same, two overlapping seven-line spectra, despite differences among the charcoals in aging and manufacturers' additives. The two overlapping spectra, labeled A and B in Figure 2, invite the conclusion that iodine is sorbed on two different sites or as two different species. The lZsIMossbauer parameters for Iz9I sorbed on charcoal are listed in Table I1 for the A spectrum and in Table I11 for the B spectrum.
temp,
relative to e2qQ127/h, Znl2qe TJ MHz -0.33 -0.56 0.59 0.31 -410 0.03 0.09 -1750
K 4.2 8 7 14
exptl 3.16 2.93 4.02 3.52
12 12
3.09 5.03
-0.40 1.54
10
3.11
-0.38
0.04 0.03
-1370 -2810
The agreement of the isomer shifts data in Table I1 is considered excellent, since a velocity error of one channel corresponds to 0.17 mm/s. The values for q are small, and considered 0 within the estimated error of fO.lO. The quadrupole splitting constants have an estimated error of f40 Hz. The estimated errors in the parameters for spectrum B (Table 111) are f0.34 mm/s for 6, f O . l for q, and f50 MHz for e2qQl,,/h. Several experiments were attempted with the object of changing the ratio of spectrum A to spectrum B in the Mossbauer spectra of lZsIsorbed on charcoal; either reactions with chemical agents did not change the spectrum, or the sorbed iodine reacted to form a new compound. The lZsI-loadedcharcoal was stirred with (a) hexane, (b) 0.1 M Na2Sz03for 1/2 h, (c) 0.1 M Br, with hexane for h, and (d) 0.1 M Na2S03for l/z h. Hexane and Na2S203did not change the spectra; treatment with Brz resulted in a new spectrum with Mossbauer parameters essentially the same as the literature values for IBr., Treatment with Na2S03 produced the single-line spectrum of an alkali-metal iodine., The Mossbauer parameters of these spectra, together with some miscellaneous spectra on other sorbents, are collected in Table IV. Several experiments attempted an isotopic exchange between 1271 and IZsI. The '%I-loadedcharcoal was stirred
1196 Langmuir, Vol. 4, No. 5, 1988
Hyder et al.
Table V. Comparison of "1 Mossbauer Parameters
charcoal"
12-HMDA (IO) A B
parameter A B 0.17 (7) 1.51 (2) 0.28 (2) 6, mm/s 1.42 (5) e2qQ,,,/h, MHz -2470 (20) -1230 (25) -2582 (20) -1272 (20) tl
0.09 (3)
0.20 (4)
0.19
0.06
Average deviation among samples is shown in the parentheses.
with a saturated solution of 12712 in hexane for 1 h. The spectrum after this treatment was about 50% weaker (1.2% effect versus 2.5%), but the ratio of spectrum A to spectrum B was unchanged. This result could be interpreted as evidence for the binding of iodine as a molecular complex or as evidence that iodine was sorbed on two sites that exchanged at the same rate. Since neither Mossbauer spectrum corresponds to that of an iodine molecule, both spectra must be due to bound iodine. The Mossbauer parameters of silver mordenite (Table IV) and MWA-1 resin agree with the literature parameters for AglmI and an alkali-metal i ~ d i n e .It ~ must be noted that the spectra for silver mordenite and AC-6120 have a broad line (Figure 3) with a satellite resonance at a slightly more positive velocity, considered evidence for very weak quadrupole splitting. The spectrum of AC-6120 has been analyzed as such; the parameters obtained and the parameters for I, diffused into polyethylene do not correspond to any found in the literature.
Discussion Mossbauer spectra of iodine sorbed on charcoals The indicate that there are two different iodine atoms on the charcoals, and the isotope-exchange experiments can be interpreted as showing that the two iodine atoms sorb as a molecule. The possibility that iodine is sorbed on two different sites that exchange at the same rate cannot be ruled out, however, although it must be considered less probable. The lZgIMossbauer spectra of iodine sorbed on charcoals are remarkably similar to the spectrum of the charge-transfer complex of I, with hexamethylenetetramine (HMTA).1° The lZgIMossbauer spectrum of the I,-HMTA complex has two overlapping quadrupole-split spectra, like 1, sorbed on charcoals, and the Mossbauer parameters of the I,-HMTA complex and I, sorbed on charcoal compared quite closely (Table V). The structure of the I,-HMTA charge-transfer complex has been shown by single-crystal X-ray crystallography to have one iodine atom attached to an HMTA nitrogen and a N- -1A-IB bond angle very close to 18Oo.l1 The 1-1 bond distance increases from 2.70 A in free I, to 2.83 A in the I,-HMTA complex, indicating a strong N- -IAbond. The similarity of the Mossbauer spectra for lBIz sorbed on charcoals to the spectra of I,-HMTA suggests that I, is bound to the charcoal surface the same way; the I, molecule attaches to the carbon by IA, and the IA (bridging) and IB (terminal) iodines have spectrum A and spectrum B, respectively. The difference in intensities between the spectra was explained'O as due to the difference in the recoilless fraction for the two iodines; the bridging iodine is held more firmly and would be expected to have a larger recoilless fraction than the terminal iodine. The lZ9IMossbauer parameters of 1291z sorbed on charcoal can be used to estimate the 5p electron charge dis(9) Bancroft, G. M.; Platt, R. H. Adu. Inorg. Chem. Radiochem. 1972, 15, 192. (LO) Ichiba, S.; Sakai, H.; Negita, H.; Maeda, Y. J. Chem. Phys. 1971, 54, 1627. (11) Pritzkow, H. Acta Cystallogr., Sect. E: S t r u t . Crystallogr. Cryst. Chem. 1975, B31, 1589.
tribution of the sorbed Iz. The method, due to Townes and Dailey,12depends upon a combination of theoretical considerations and empirical correlations from the lZ9I Mossbauer parameters of a large number of iodine-containing compounds7 to estimate the electron population of each of the three iodine 5p orbitals. Nominally, the occupation number, Vi, is U, = U, = 2 and U, = 1 for the I, molecule. The expressions used in the calculation of the hole population (bpi = Vi - 2) are 6 = 1.5hP- 0.54 mm/s (4) where 0.54 nm/s is the empirically determined shift6J for pure unhybridized a-bonds and h,, the hole population, is (5) h, = hP* + hPy + hPc The reduced quadrupole coupling constant, Up,is defined by Up = eqEol/eqat = hP2- (hp, h,)/2 (6)
+
and the asymmetry of the EFG is
vmol = (hp, - hp)/(h,
- hp/3)
(7)
Q127 units, eqat is a constant, -2293 MHz. Average values from Table V were substituted in eq 4-7 and the equations solved for the hole populations of the 5p orbitals for IA and IB. The results, converted to occupation number, are U, = 1.89, U, = 1.95, and U, = 0.85 for IA and U, = 1.94, Uy= 2.10, and U, = 1.49 for IB. Thus, I, is positive, with an electron configuration of 5s25p4.69, and IB is negative, with the configuration 5s25p5.53.The charcoal sorbent contributes 0.22e to the iodine molecule.
In
Acknowledgment, The information contained in this article was developed during the course of work under Contract No. DE-AC09-76SR00001 with the U.S. Department of Energy. Appendix. Statistical Estimation Procedure Equation 3 represents a system of equations Ri = A[aoi + aiiv
i
+ a,iq2 + a3iq3 + a4iv4] + 6 = 1, 2,
(Al)
..., 7
where Ri is the experimental line position (mm/s), A = e2q/h, the a's are coefficients that were obtained from Shenoy and Dunlap: q is the asymmetry parameter, and 6 is the isomer shift. This model is inherently nonlinear (since the derivatives dRi/dq are functions of v), but it is only so in the parameter q. It is inherently linear in the parameters A and 6, which are the slope and intercept, respectively, of a straight line in the derived variable [aoi aliq a,iq, + a3i773 + a4iq41. In general, the least-squares estimate of a nonlinear parameter cannot be obtained in explicit analytic form; it must be found by implicit numerical iteration. But numerical instabilities can arise during the course of such iterations, leading to difficulties in convergence, sensitivity of parameter estimates to miniscule changes in the data, etc. In the particular case at hand, when v approaches 0 (one of its conceptual limits), the model becomes "overparameterized" by virtue of its inability to distinguish A from 6. The resultant association of A with 6 determines that the residual sum-of-squares surface achieves its minimum diffusely along a trough line, not sharply at a
+
+
~~
(12) Townes, C.-H.; Dailey, B. P. J . Chem. Phys. 1949, 17, 482.
lZ9I Mossbauer Studies of Iodine
Langmuir, Vol. 4, No. 5, 1988 1197
point. Along this trough line, many different combinations of A and 6 give about the same fit to the data, thereby precluding precise estimation. The pathologies noted above were experienced when all three of the parameters were treated iteratively. An alternative is to iterate only on q and to solve explicitly for A and 6 by linear least squared3 at each iteration. This improves the conditioning of the system and, in so doing, enables more precise estimation of the parameters. A t each iteration (say the jth), least-squares values A and 6 are computed by using the current value qj as follows:
which has the scalar mean FBARO’) = SUM[FETAO‘)]/n The above are conventions of the SAS system, wherein * represents matrix multiplication, / represents scalar division, and SUM[Q] represents the sum of all elements of Q. Further, define the (n X 1)vector [R] of values Ri; its mean is the scalar RBAR = SUM[R]/n Form the vector of cross-products FEBO‘) and of squares FEO’) FEBO‘) = ([FETAG)] - FBARG)) # ([R] - RBAR)
FEG) = ([FETAO‘)]- FBARO’)) # ([FETAG)] - FBARO’)) where xi(qjj)= aoi _+ uliq’ + aziqF + aziq! + aziqf, i?(qj) = [ C ~ i ( ~ j ) l /and n, R = [ b i l / n . When vi has converged to a value qm such that no further reduction‘can be effected in the residual sum-of-squares, qm becomes the least-squares estimate of q; the corresponding values A(qm),6(qm) then become conditional (on qm) least-squares estimates of A and 6. In the SRL computing environment, SAS PROC MATRIXl4provides a convenient capability to implement this combined “explicit-iterative” algorithm. The standard Gauss-Newton algorithm effects the nonlinear iteration, while the linear estimates are obtained algebraically within each iteration. The details of this approach follow: 1. To Start (j= 0). Define the (5 X 1)vedor PARMG), the transpose of which is [PARMWI’ = [I, qj, q t , and the (n
X
v!,
5) matrix A
A=
Ip:i
all
.
RH
%1
a22
a32
qfl
$l
Then compute the (n X 1) vector [FETAG)] = [A]*[PARMG)] (13) Lawton, W. H.; Syvestre, E. A. Technometrics 1971, 13, 461. (14) SAS Institute Inc., C a y , North Carolina.
where # indicates elementwise multiplication. Then the (linear) least-squares estimates of a and 6 (conditional upon qj) are ALPHAHATO’) = SUM[FEBO’)]/SUM[FE(j)] DELTAHATO’) = RBAR - (ALPHAHATO’))# FBARO‘) The resulting vector of fitted values, [RHATV)], is [RHATO’)] = (ALPHAHATO’))[FETAG)] + DELTAHATO‘) with corresponding residual sum-of-squares RESSO’) RESSO‘) = [R - RHAT(j)]’*[R - RHATO’)] 2. To Iterate. Form the (n X 1) vector [ X ( j ) ]of derivatives (wrt q ) of the model [XO’)] = (ALPHAHATO‘))# ([Al][PARMlO’)]) where Al =
[:7 a21
%1
a42-
a2n
a3n
a4n
and [PARMlU)]’ = (1,2q., 3qf, 4$). Then qj+l = qj + DEL, where DEL = INV[i’(j)*X(j)]*XrO’)*[R - RHATG)] and INV(Q) represents the matrix operation of inversion. Registry No. lzeI, 15046-84-1; 12, 7553-56-2.
