Technical Implications of Organic Syntheses in Ball Mills - Ball Milling Towards Green Synthesis: Applications, Projects, Challenges - page 262

through a maximum. 29,88 As this is typical for a quadratic function it is
reasonable from mathematical point of view, but it is also explainable from
chemical viewpoint. High energy impacts may distort the chemical equi-
librium, favouring unwanted side reactions, resulting in a decrease of the
mentioned variables. On the other hand, this high energy impact may also
open reaction routes to side or consecutive reaction products, lowering
the performance of a specific process. 89,90 This effect is also possible for
a change of the ball mill type to apparatus that realize higher energy
densities. 66

10.4.2 Milling Time
In addition to operating frequency, statistical analysis of solid-state reactions
in ball mills identifies the reaction time (t) as the main variable determining
the performance of a chemical reaction. 10,72,73 This is readily understandable
on looking at the physics on the one hand and at chemical kinetics on the
other hand. Direct proportionality between energy (E)andtime(t) is a physical
fact (Eq. 10.12). Hence, prolonged ball milling goes along with an increase of
collisions between the moving parts of the mill and of the grinding beads with
the mill chamber or feed material. More energy is distributed to the mill
charge, although the energy density is kept constant:

E ¼ Pt (10.12)

Chemical kinetics, however, also prove the importance of the time for
reactions carried out in ball mills. T he reaction rate (r) of a specific chemical
reaction indicates that more moles can be converted at increased reaction
time, t (Eq. 10.13):

dt ¼ k ðÞ Y
j

r i ¼ 1
n i

dn i

n n j
educt ; j

(10 : 13)

where

r i is the reaction rate for product i,
n i (mol): molar amount,
t (s): reaction time,
k(T): temperature-dependent rate constant;,
n: stoichiometric coecient,
index i ¼ product; index j ¼ educt(s).

The exponential correlation between r, the rate constant (k), and tem-
perature (T) expressed by the Arrhenius (Eq. 10.14) or Eyring-Polanyi
equation (Eq. 10.15) limits the infinite increase of conversion or yield:

k ¼ Ae E a þ 0 : 1D V ðÞ (10 : 14)

RT

k ¼ k B T
h

RT ¼ k B T
h

e DG z e DH z

RT þ DS z R

(10 : 15)