22 Electrophilic Ring-Opening Polymerization of New Cyclic Trivalent Phosphorus Compounds Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 13, 2015 | http://pubs.acs.org Publication Date: August 16, 1985 | doi: 10.1021/bk-1985-0286.ch022
A Novel Mechanism of Ionic Polymerization SHIRO KOBAYASHI Department of Synthetic Chemistry, Faculty of Engineering, Kyoto University, Kyoto 606, Japan
This paper describes the e l e c t r o p h i l i c ring-opening polymerization of seven new c y c l i c phosphorus(III) compounds, 1-7. The polymeri zation of f i v e - and six-membered deoxophostones, 1 and 3, and of a benzoxaphosphole 2, produced poly(phosphine oxide)s 11, 35, and 23 v i a Arbuzov type reactions. The polymerization of five and six -membered deoxothiolphostones, 4 and 5, gave poly(phosphine s u l f i d e ) s , 36 and 37, v i a a new type of the C-S bond s c i s s i o n . The polymerization of seven- and eight-membered c y c l i c phosphonites, 6 and 7, produced polyphosphinates, 42 and 46, consisting of a "normal" unit. Kinetic studies of the e l e c t r o p h i l i c polymerization of monomers 1 and 2 led to a new mechanisms of ionic polymerization. E l e c t r o p h i l i c covalent propagating species of 1 (MeI i n i t i a t o r ) showed even greater polymerizability than ionic ones (MeOTf i n i t i a t o r ) . The e l e c t r o p h i l i c polymerization of 2 proceeds only v i a the covalent propagating species l i k e 27, 29, and 30 which are very reactive; stable phosphonium species, 24 being inactive. With MeI i n i t i a t o r , the overall rate of polymerization i s governed by the S i process (k ) to produce an active a l k y l iodide species 32 from a phosphonium iodide 31. Based on a proposed general mechanism, k1 values have been determined and a new c l a s s i f i c a t i o n of ionic polymerization i s given. The present paper reports the ring-opening polymerization of new c y c l i c trivalent phosphorus monomers. Kinetic and mechanistic studies indicate that these monomers polymerize by a novel mechanism that may be called e l e c t r o p h i l i c ring-opening polymeriza t i o n . The monomers include a five-membered deoxophostone (1), a benzoxaphosphole (2), a six-membered c y c l i c phosphonites (6 and 7). Among these, compounds 3,4 and 5 have been prepared for the f i r s t time. Cyclic phosphorus compounds are good starting monomers for preparing functional polymers having phosphorus groups i n the main chain. They are known to undergo ring-opening polymerization v i a cationic, anionic, or thermal processes. U n t i l recently, cationic ( e l e c t r o p h i l i c ) ring-opening polymerizations have been reported for 1,3,2-dioxaphospholanes (8) (1-10) and for 1,3,2-dioxaphosphorianes (9) (7,11,12). Generally, the polymerizations proceed via the Arbuzov reaction to produce polymers containing phosphinate or N
1
0097-6156/ 85/ 0286-0293S06.00/0 © 1985 American Chemical Society
In Ring-Opening Polymerization; McGrath, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.
294
RING-OPENING POLYMERIZATION
Ph-pQ
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 13, 2015 | http://pubs.acs.org Publication Date: August 16, 1985 | doi: 10.1021/bk-1985-0286.ch022
Ph-
Ph-r(^
Ph
p h
-O-0
/
Ph-P V
ο0
6
phosphonate repeating units ("normal" unit, ].Qa). During the polymerizations, however, a side reaction occurs to give an "isomerized" unit, lg£. The amount of the isomerized unit 10b
0 II
-e-pox-h I R R-P
v
10a
X
•P0X0I_. . I I 'n R R
8; X=-f- CHo 9;
10b
X=-f-CH 2
R=alkyl, a r y l , a l k y l o x y ,
aryloxy
depends upon the monomer and the reaction conditions, and i t sometimes exceeds that of normal unit ]J£a, (7_>9) · Therefore, the polymerizaton of these c y c l i c phosphorus(III) monomers i s complicated. Our very recent results of the e l e c t r o p h i l i c r i n g opening polymerizations of monomers 1-7 have shown that they occur cleanly to produce polymers consisting exclusively of normal units without isomerizatlon, and that they are suitable for k i n e t i c analysis.
In Ring-Opening Polymerization; McGrath, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.
22.
295
Electrophilic Polymerization: A Novel Mechanism
KOBAYASHI
2-Phenyl-l,2-oxaphospholane
(1)
E l e c t r o p h i l i c Polymerization. The ring-opening polymerization of 2-phenyl-l,2-oxaphospholane, 1, a five-membered deoxophostone i s induced by a cationic i n i t i a t o r to give a white powdery material, poly(phenyltrimethylenephosphine oxide 11 (13). I n i t i a t o r s l i k e
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 13, 2015 | http://pubs.acs.org Publication Date: August 16, 1985 | doi: 10.1021/bk-1985-0286.ch022
/
0 _
1
E
+
if » -f^PCHoCHoCHo-*— £ π
Ph-P >
1
2 —> i-Bu AlH
h
1
2
11
—ePCH CH CH ->2
2
2
12 +
w
e
r
e
MeOS0 CF (MeOTf), Mel, PhCH Br, BF «OEt , and Et 0 BF4~ e f f e c t i v e i n producing polymer No side reactions leading to an isomerized unit took place during the polymerization. A polymer sample having a molecular weight of 3400 was prepared. It melted at 269°C and decomposed at 465°C. This i s the f i r s t example of an e l e c t r o p h i l i c ring-opening polymerization of c y c l i c phosphorus(III) monomer that yields a polymer with a clear-cut structure. Reduction of 11 to polyphosphine 12 was attempted using a HSiCl3/Et3N reagen£%ut resulted i n the formation of a small amount of an unidentified unit (>Ί5%) i n addition to the desired phosphine units 12. A novel clean reduction method, therefore, has been developed (14). The method i s an one-pot reaction, i n which ^ i s f i r s t treated with (C0C1 ) and then with i-Bu AlH, giving rise to the polyphosphine ,12. Macroporous-type crosslinked chloromethylated or iodomethylated polystyrene ^ was used to i n i t i a t e the polymerization of monomer 1. The product poly(styrene-g-phenyltrimethylenphosphine oxide) JL4^ i s a white bead-like resin (n-4.1-10.5), which showed e f f i c i e n t chelating properties toward heavy metal ions such as U0o , Th^ , Hg +, Pd* , and C u (15). The adsorbed ions, eg, U0 *, were readily desorbed by treating the resin with 10% aqueous Na C0 . This adsorption-desorption procedure could be performed repeatedly without reducing the chelating e f f i c i e n c y of 14 and without destroying i t s bead-like structure (15). ^ 2
3
2
2
3
2
3
2
2+
2
+
2 +
+
2
2
2
In Ring-Opening Polymerization; McGrath, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.
3
296
RING-OPENING POLYMERIZATION
Kinetics and Mechanism. Since the ring-opening polymerizations of ^ by MeOTf and Mel are clean reactions, they are amenable to a k i n e t i c study. The k i n e t i c analysis was carried out by monitoring the polymerization reaction using 31p NMR spectroscopy. MeOTf-Initiated System. Figure 1 shows the P NMR spectra of the MeOTf-initiated polymerization system: (a) immediately after the mixing MeOTf with 1 i n PhCN and (b) after 30 min at 70°C (16). The P NMR chemical s h i f t i s r e l a t i v e to external 85% H3PO4 standard. Based on the P NMR signals observed i n Figure 1. which are assigned as c r i c l e d l e t t e r s i n Scheme I, as well as on ^ F j *H NMR spectra of the reaction mixture, the following scheme i s proposed f o r the polymerization. 3 1
3 1
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 13, 2015 | http://pubs.acs.org Publication Date: August 16, 1985 | doi: 10.1021/bk-1985-0286.ch022
3 1
a n (
(a) Β
//-
j
(b)
1 110
Figure 1.
1 100
// 1
I
1 40
» 30 (ppm)
3 1
P {^-H} NMR spectra of the cationic ring-opening polymerization of 1 i n i t i a t e d with MeOTf i n PhCN([M]° = 1.25 mol/L and [ I ] = 0.125 mol/L): (a) before heating; (b) after 30 min at 70°C. Reproduced with permission from Ref. 16. Copyright 1984, American Chemical Society. Q
In Ring-Opening Polymerization; McGrath, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.
22.
Electrophilic Polymerization: A Novel Mechanism
KOBAYASHI
297
Scheme I Initiation
MeOTf
0-
Me
0—1
• TfO'
+
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 13, 2015 | http://pubs.acs.org Publication Date: August 16, 1985 | doi: 10.1021/bk-1985-0286.ch022
Ph
®
15
Propagation 0 II Me—PCrl-
TfO'
ph
®
16 0 II
0 II Me—PCH, Ph
Ah
+/
2
®
2
17
•TfO"
> P h -
®
The propagating ends i n this polymerization are c y c l i c phosphonium ions such as l^ç\}^ which are opened by nucleophilic attack of monomer 1 to form P-phenyltrimethylenephosphine oxide units 17 v i a an Arbuzov-type reaction. The propagation rate constant" ( k ) i s obtained by the following equation, assuming as S 2 reaction: N
p
ÎÎÎÎI
kp[P*][M]
dt where [P*] and [M] are the concentrations of propagating species and monomer. It was found that i n i t i a t i o n i s very rapid, that the concentration of propagating phosphonium species i s equal to that of the i n i t i a t o r charged, and that this remained constant throughout the polymerization. Therefore, the integrated form of the above equation i s given by In
{[M1X/IMJ21 =
kp[P*](t2-ti)
(1)
Plots of ln{[M] /[M]2> versus t£-ti showed a linear relationship, from which k values were obtained. Arrhenius plots of k values at four temperatures gave a straight line whose slope led to the a c t i v a t i o n parameters (Table I) (16). 1
p
p
In Ring-Opening Polymerization; McGrath, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.
RING-OPENING POLYMERIZATION
298
Table I . Propagation Rate Constants and Activation Parameters i n the Polymerization of \^
k
χ 10*
Initiators PhCH Br
MeÔtf
Iter
3·52(50°Ο
4.53 (50°C)
PhCH^Cl
2
e
2.13(70°C)
1.88(130 C)
4.34(80°C) 7.99(90 C) 15.0(100°C)
3.89(140 C) 5.45(150 C) 10.5 (160 C)
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 13, 2015 | http://pubs.acs.org Publication Date: August 16, 1985 | doi: 10.1021/bk-1985-0286.ch022
Ρ (L/mol.sec)
e
9.27(60 C) 24.3 (70 C) 40.3 (80°C) e
e
k_x 10*(50 C) tL/mol.sec)
11.5 25.0 43.8
3.52
ΔΗ±(50°Ο (kJ/mol) e
AS±(50 C) (J/K.mol)
(60° C) (70°C) (80 C) e
4.53
e
(0.475)
b
e
e
e
(0.00538)
73.3
66.7
63.5
74.3
-84.8
-103
-132
-136
b
a) [M] =1.25 mol/L and [l] » 0.125 mol/L i n PhCN solvent. e
e
b) Calculated values.
J i
Alkyl Halide Initiated Systems. Figure 2 shows the P NMR spectra of the polymerization system i n i t i a t e d by Mel i n PhCN. Peak assignments were made as c i r c l e d l e t t e r s i n Scheme II and led to the generalpolymerization mechanism (Scheme I I ) . No signal due to phosphonium species was detected under the polymerization conditions employed. This i s i n sharp contrast to the MeOTf-initiated system. Intermediates ^ and ^ are unstable. The stable propagating ends
Scheme I I Initiation RX slow R=Me, PhCH
/
\_
Ph
18 2
> R-PCH CH2CH X ^ 2
fast
p
©
h
19
X=I, Br, CI
In Ring-Opening Polymerization; McGrath, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.
2
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 13, 2015 | http://pubs.acs.org Publication Date: August 16, 1985 | doi: 10.1021/bk-1985-0286.ch022
22.
Electrophilic Polymerization: A Novel Mechanism
KOBAYASHI
Figure 2.
299
31p {lfl} NMR spectra of the cationic ring-opening polymerization of 1 i n i t i a t e d with MeI([M] * 1.25 mol/L and [ I ] = 0.125 mol/L): (a) after 1 min at 70°C; (b) after 10 min at 70°C; (c) after 20 min at 70°C. Reproduced with permission from Ref. 15. Copyright 1984, American Chemical Society. Q
0
®
Ph
&p, slow
20
«
0
0
II
II
* Me — PCH CH CH -ePCH CH CH - -)—PCH CH CH I / I /*\ n/l I I I fast 2
®
P
h
9
2
9
2
0
®
2
P
0
2
0
2
0
9
0
0
h
22 are a l k y l iodide species l i k e ^ M , and The rare-determining steps of both i n i t i a t i o n and propagation are the dipole-dipole S 2 reactions between and a l k y l iodide and monomer 1 producing transient phosphonium species such as 18 and 21, which are converted rapidly into covalent a l k y l iodide species ^ N
In Ring-Opening Polymerization; McGrath, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.
300
RING-OPENING
POLYMERIZATION
and ^ v i a nucleophilic attack of the iodide counteranion. From the plots of the second-order kinetics of Equation 1 k values were obtained. S i m i l a r l y , benzyl bromide and benzyl chloride were also found to proceed via a l k y l halide propagating ends as stable species. These k i n e t i c results are given i n Table I (16) . The k values are very much dependent upon the counter-anion derived from i n i t i a t o r . The r e l a t i v e r e a c t i v i t i e s at 50°C are in the following order; MeOTf : Mel : PhCH Br : PhCH Cl = 654 : 842 :88.3 : 1.0. The polymerization of proceeds via two different mechanisms which i s due to differences in the n u c l e o p h i l i c i t y of the anions TF0~ and Χ" (X = I, Br, CI), which affect the r e l a t i v e s t a b i l i t y of the phosphonium species. The difference i s reflected by AS* values; the reduced polymerizability of the PhCl^Br or PhCI^Cl system i s attributed to the less favorable entropy term i n comparison with the MeOTf system. This can be interpreted i n terms of solvationdesolvation phenomena from the i n i t i a l state to the t r a n s i t i o n state (16). In ring-opening polymerizations, c y c l i c onium propagating species are usually more reactive than covalently bonded ones, eg, superacid macroestertype species in c y c l i c ether polymerizations (17,18) and a l k y l halide type species i n 2-oxazoline polymerizations (19). It i s to be emphasized, however, that the Mel-initiated polymerization of ^. proceeds even faster than the MeOTf-initiated system, although the difference i s small. This i s the f i r s t case of covalent ( e l e c t r o p h i l i c ) propagating species showing a higher r e a c t i v i t y than an ionic propagating one (16). p
p
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 13, 2015 | http://pubs.acs.org Publication Date: August 16, 1985 | doi: 10.1021/bk-1985-0286.ch022
2
2
l-Phenyl-3H-2,1-benzoxaphosphole (2) Ring-Opening Polymerization. Monomer 2 i s a five-membered c y c l i c phosphinite, an analogue of dexophostone ^. ^ was prepared according to the reported procedure (20). The ring-opening polymerization of 2 produced white powdery materials of poly(phosphine oxiâe ) J ^ , whose structure was determined by IR, 31p, 1 and C NMR spectroscopy as well as by elemental analysis. The polymerization results are given i n Table II (21). H>
0
R
23
+
O - i
24(R=Me, E t ; X=BF , TfO) 4
In Ring-Opening Polymerization; McGrath, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.
22. KOBAYASHI
Electrophilic Polymerization: A Novel Mechanism
Table I I . Ring-Opening Polymerization of 2
Initiators (mol % f o r 2)
Solvents
E t 0 BF "(17)
CH C1
MeOTf(19)
CDC1
Time (ht)
a
Polymer Yield(%) Mol. wt.
35
48
0*
3
35
24
0
Mel(21)
CH CN
35
24
91
Mel(18)
CDC1
35
24
100
Mel(1.4)
CHCI3
35
1440
63
MeOTf(5.0)
CH C1
50
24
0
Mel(5.0)
PhCN
50
3
53
Mel(10)
CH C1 2
2
70
16
67
none
CH C1 2
2
70
16
0
none
CH C1
2
130
14
54
3
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 13, 2015 | http://pubs.acs.org Publication Date: August 16, 1985 | doi: 10.1021/bk-1985-0286.ch022
Temp. («c)
4
2
2
3
3
2
2
2
301
b
4500
b
2400
2260
a) 2(0.3 g) i n 1 mL of solvent i n a sealed pressure under nitrogen. b) Stable phosphonium species 24 was formed. Among the cationic i n i t i a t o r s examined, only Mel i s active f o r the polymerization. Et30 BF4~ and MeOTf produced stable phosphonium s a l t s 24^ which did not induce the polymerization of 2. Anionic and r a d i c a l i n i t i a t o r s were inactive. At higher temperatures, eg, above 70°C, 2 starts to show polymerizability without i n i t i a t o r to give polymer of the same structure as 23. The mechanisms of this "thermal" polymerization i s not wêîl understood, but, at present, a zwitterion intermediate ^ derived from two molecules of ^ i s considered to be responsible for the production of polymer 23. The +
"0-P
7-=-r
CH,—P.
—2îL*
25 formation of 25^ requires that one molecule of ^ acts as a nucleophile and the other behaves as an e l e c t r o p h i l e . This "amphip h i l i c " nature of £ has already been confirmed i n the copolymerizations, i n which 2 was copolymerized without i n i t i a t o r with an e l e c t r o p h i l i c monomer l i k e a c r y l i c acid and with a nucleophilic one l i k e 2-methyl-2-oxazoline (22).
In Ring-Opening Polymerization; McGrath, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 13, 2015 | http://pubs.acs.org Publication Date: August 16, 1985 | doi: 10.1021/bk-1985-0286.ch022
302
RING-OPENING POLYMERIZATION
Kinetics of E l e c t r o p h i l i c Ring-Opening Polymerization of 2^: A New Mechanism i n Ionic Polymerizations. The kinetic analysis of the polymerization i n i t i a t e d with Mel was carried out by * l p NMR spectroscopy at 35°C, at which the thermal polymerization does not take place at a l l . Figure 3 shows a ^ l p NMR spectrum of the polymerization system i n ΟΗβΟΝ, i n which a r e l a t i v e l y large amount of the i n i t i a t o r was employed for the kinetic analysis. Peak A at 119 ppm i s assigned to monomer 2. Peaks due to phosphonium species appear over a wide chemical s h i f t range: peak A at 95 ppm i s attributed to the phosphonium iodide and peaks C around 67 and 83 ppm are due to phosphonium iodides l i k e 28^ ^1, and Peaks D are ascribed to various phosphine oxide units including propagating ends such as 27^ and 30^, i f any (Scheme I I I ) .
A B
1
— —/ / — 120 Figure 3.
C
1
D
«—-y μ
90
70
1—
40
30
(ppm)
3 1
P {^-H} NMR spectrum of the Mel-initiated polymerization of 2 after 490 min at 35°C i n C H 3 C N . [M] « 1.0 mol/L and [ I ] = 0.20 mol/L. Q
0
Scheme I I I
Initiation 0 II
Me-P-
-CH I 2
k 27(P
x c l
)
In Ring-Opening Polymerization; McGrath, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.
22. KOBAYASHI
Electrophilic Polymerization: A Novel Mechanism
303
Figure 4 shows the time-conversion curves for monomer £ and the t o t a l concentrations of active species [P*]. The following [P*] = [P*i] + [P* ] c
r e l a t i o n s i p holds where [P*i] and [P* ] are the concentrations of phosphonium species and of covalent a l k y l iodide species, respectively. The i n i t i a t i o n finished at an early stage of the reaction. Under the polymerization conditions i n CH3CN [P*i] reached a constant" value that was almost equal to the concentration of the i n i t i a t o r charged, i e , [P*] = [P*i] · After [P*i] became constant, the apparent rate constant of propagation (kp(ap)) obtained based on Equation 1. The values are very mucn dependent upon the solvent employed (Table III) (23). In a highly polar solvent (eg, CH3CN) kp(ap) value i s at least 102 times less than that i n a less polar solvent (eg, PhCl).
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 13, 2015 | http://pubs.acs.org Publication Date: August 16, 1985 | doi: 10.1021/bk-1985-0286.ch022
c
w
a
s
1.0
£N 0.8 —.
Monomer
Ο
Ε ο.β ϋ § 0.4
ϋ 0.2 100
200
300
400
500
Time(min) Figure 4.
Time-conversion curves for monomer 2 and t o t a l active species [P*] (=[P*il) i n the polymerization of 2 with Mel i n i t i a t o r at 35°C i n C H 3 C N . [M] = 1.0 mol/L and [ I ] = 0.20 mol/L. Q
0
These observations lead to the propagation mechanism shown i n Scheme I I I . The phosphonium species ^ and ^ are reasonably considered as "not r e a l l y active", which i s supported by the fact that phosphonium salt did not i n i t i a t e the polymerization.
Table ΠΕ. Values of k , > with Mel I n i t i a t o r i n Three Solvents at 35°C ρ Cap) Solvents k , ,(L/mol«sec) p(apj CH CN
7.3 x 10"
5
PhCN
7.6 x 10"
4
PhCl
>1
2
3
x 10"
In Ring-Opening Polymerization; McGrath, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.
304
RING-OPENING POLYMERIZATION
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 13, 2015 | http://pubs.acs.org Publication Date: August 16, 1985 | doi: 10.1021/bk-1985-0286.ch022
Propagation
32
Me-PI Ph
Ο
p h
'
CH I 2
-» 23
27(really active)
In Ring-Opening Polymerization; McGrath, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.
RING-OPENING POLYMERIZATION
306
General Scheme of the New E l e c t r o p h i l i c Ring-Opening Polymerization. The new polymerization i s given as the following general scheme (Scheme IV).
Scheme IV
+ monomer S 2 type propagation
-> Ρ
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 13, 2015 | http://pubs.acs.org Publication Date: August 16, 1985 | doi: 10.1021/bk-1985-0286.ch022
N
"Pro-active species
"Real a c t i v e " species
1
S i type r e a c t i o n N
(
k) 2
In order to evaluate for the S i reaction, an attempt was made to perform kinetic analyses based on Scheme IV. The rate equations for the monomer consumption and the production of [P* ] are given by Equations 3 and 4. N
c
d[M] — - - kp(c)[P*cl[M] at
(3)
d[P* ] c
kl[P*i] " k
dt
p ( c )
(4)
[P* ][M] c
In a highly polar solvent the relationship (5) w i l l be v a l i d after the i n i t i a t o r i s completely consumed [P* ] + [P*i] = [P*] = [ I ] c
(5)
0
where [ I ] denotes the concentration of the i n i t i a t o r charged* I t was impossible to determine [P* ] precisely. I t i s observed i n Figure 4 that the polymerization stage where [P*i] • constant i s present. Therefore, an assumption that [P* ] i s constant i s made. Then, d[P* ]/dt = 0, leading to [P* ] - k [ I ] / ( ^ p ( c ) [ l + l ) which i s v a l i d at a limited stage of polymerization. Equation 3 i s then expressed as d[M] Mk [I] [M] Q
c
c
M
c
c
p ( c )
d t
x
k
0
0
+
M O M
k
l
The integrated form of this equation i s given as [M]
t2
In
+ k
p ( c )
[I] (t -t ) 0
2
1
k
x
= kp
( c )