The diagrams in Fig. 16 demonstrate this evaluation: the upper graph shows
the determination of the propagation rate constants from the oligomer distributions
in dependence on time; the lower graph shows the determination from the distributions in dependence on the ratio Ti/Al (the concentration of the active species C*).
In both diagrams, the rate constants determined via ν max(cal) by means of
all calculated distributions result in the same values of 120 L mol
À1 s
À1 . The rate
constants determined via ν max(exp) result, as expected, in lower values. However, the
course of these values extrapolated in the diagrams backwards to zero point aims at
this theoretical ν max value of 120 L mol
À1 s
À1 . Indeed, an excellent agreement!
2.3.4 Kinetic Evaluation of Oligomer Distribution in Dependence
on Different Parameters
From the integrated peak areas of the oligomer distributions, even a total monomer
consumption can be determined, which leads to an “average” but, because of the
very short reaction times, precise polymerization rate. This kinetic evaluation was
carried out according to the equation:
v p ¼
P v
0
m v Á v
p Á t
;
with m v being the chromatogram-determined amount of alkane in mol, ν the kinetic
chain length, p the analytical probe volume, and t the reaction time. It should be
emphasized again that these rates are determined absolutely in the homogeneous
phase of the polymerization course without any influence from precipitating
polyethylene.
The kinetic analysis is shown in Fig. 17 and leads to detail concerning the
concentration of the active species, insight into the superposition of the Al isotherm
and its dependence on the chain length of the starting Ti-alkyl group, and determination of a reaction order equal to one for the monomer dependence.
Figure 17a shows the monomer dependence. The resulting slope leads to tg
α ¼ [C*]k p . Using the k p values from Sect. 2.3.3 (Fig. 16), we can estimate the
concentration of C* at 5% of the charged Ti component at a ratio Al/Ti ¼ 2.
Figure 17b shows the polymerization rates in dependence on the reaction time.
Further evaluation of this diagram follows the equations:
v p ¼ ÀdM=dt ¼ k p ½C
Ã
½M 0 and hence; ln½M 0 =½M t ¼ k p ½C
Ã
t:
Hence, the data plotted according to log[M] 0 /[M] t versus t in Fig. 17c result
in a slope of tg α ¼ k p [C*] Â 0.434. Again using the k p values of Fig. 16,
this estimation leads to 5% of the charged Ti component at a ratio Al/Ti ¼ 2.
The analogous evaluation at a ratio of 4 gives [C*] ¼ 9% of [Ti] 0 , which is in best
agreement with the stirred tank reactor kinetics.
Contributions to the Ziegler–Natta Catalysis: An Anthology
19
the determination of the propagation rate constants from the oligomer distributions
in dependence on time; the lower graph shows the determination from the distributions in dependence on the ratio Ti/Al (the concentration of the active species C*).
In both diagrams, the rate constants determined via ν max(cal) by means of
all calculated distributions result in the same values of 120 L mol
À1 s
À1 . The rate
constants determined via ν max(exp) result, as expected, in lower values. However, the
course of these values extrapolated in the diagrams backwards to zero point aims at
this theoretical ν max value of 120 L mol
À1 s
À1 . Indeed, an excellent agreement!
2.3.4 Kinetic Evaluation of Oligomer Distribution in Dependence
on Different Parameters
From the integrated peak areas of the oligomer distributions, even a total monomer
consumption can be determined, which leads to an “average” but, because of the
very short reaction times, precise polymerization rate. This kinetic evaluation was
carried out according to the equation:
v p ¼
P v
0
m v Á v
p Á t
;
with m v being the chromatogram-determined amount of alkane in mol, ν the kinetic
chain length, p the analytical probe volume, and t the reaction time. It should be
emphasized again that these rates are determined absolutely in the homogeneous
phase of the polymerization course without any influence from precipitating
polyethylene.
The kinetic analysis is shown in Fig. 17 and leads to detail concerning the
concentration of the active species, insight into the superposition of the Al isotherm
and its dependence on the chain length of the starting Ti-alkyl group, and determination of a reaction order equal to one for the monomer dependence.
Figure 17a shows the monomer dependence. The resulting slope leads to tg
α ¼ [C*]k p . Using the k p values from Sect. 2.3.3 (Fig. 16), we can estimate the
concentration of C* at 5% of the charged Ti component at a ratio Al/Ti ¼ 2.
Figure 17b shows the polymerization rates in dependence on the reaction time.
Further evaluation of this diagram follows the equations:
v p ¼ ÀdM=dt ¼ k p ½C
Ã
½M 0 and hence; ln½M 0 =½M t ¼ k p ½C
Ã
t:
Hence, the data plotted according to log[M] 0 /[M] t versus t in Fig. 17c result
in a slope of tg α ¼ k p [C*] Â 0.434. Again using the k p values of Fig. 16,
this estimation leads to 5% of the charged Ti component at a ratio Al/Ti ¼ 2.
The analogous evaluation at a ratio of 4 gives [C*] ¼ 9% of [Ti] 0 , which is in best
agreement with the stirred tank reactor kinetics.
Contributions to the Ziegler–Natta Catalysis: An Anthology
19