2009_Book_FoodChemistry

132 2 Enzymes
For the reaction rate follows:
A
k = M · = M · k1 = M · K≠
A≠ k −1
with
M = K B · T
h
= R · T
N A · h
(2.86)
(2.87)
(K # equilibrium constant, k B Boltzmann constant,
h: Planck constant, N A : Avogadro number).
For the equilibrium constant follows:
K≠ = e −G≠/RT
(2.88)
Resulting for the equilibrium constant in:
k = k B · T
h
e−ΔG≠/RT
(2.89)
and for the free activation enthalpy:
ΔG≠ = −RT ln k · h
(2.90)
k B · T
If k is known for any temperature, ΔG≠ can be
calculated according to equation 2.90. Furthermore,
the following is valid:
ΔG≠ = ΔH≠ − TΔS≠
(2.91)
A combination with equation 2.90 results in:
−RT ln k · h
k B · T = ΔH≠ − TΔS≠ (2.92)
and
log k T = −log h − ΔH≠
k B 2.3RT + TΔS≠ (2.93)
2.3R
It is possible to determine ΔH≠ graphically based
on the above equation if k is known for several
temperatures and logk/T is plotted against 1/T.
If ΔG≠ and ΔH≠ are known, ΔS≠ can be calculated
from equation 2.91.
The activation entropy is contained in the Arrhenius
factor A as can be seen by comparing
the empirical Arrhenius equation 2.84 with equation
2.89 which is based on the transition state
hypothesis:
k = A · e −E a
/RT
k = k B
h · e−S≠/R · T · e −H≠/RT
(2.94a)
(2.94b)
Activation energy E a and activation enthalpy
H≠ are linked with each other as
follows:
dlnk
dT
= E a
RT 2 (2.95)
dlnk
dT
= 1 T + H≠ RT +H≠
RT 2 =
RT 2 (2.96)
E a = H≠ + RT (2.97)
Using plots of logk against 1/T, the activation energy
of the Arrhenius equation can be determined.
For enzyme catalyzed reactions, E a is 10–60, for
chemical reactions this value is 50–150 and for
the inactivation of enzymes, the unfolding of proteins,
and the killing of microorganisms, 250–
350 kJ/mol are required.
For enzymes which are able to convert more
than one substrate or compound into product,
the activation energy may be dependent on the
substrate. One example is alcohol dehydrogenase,
an important enzyme for aroma formation
in semiripened peas (Table 2.13). In this case the
activation energy for the reverse reaction is only
slightly influenced by substrate.
Under consideration of the temperature dependence
of the rate constant k in equation 2.80, the
implementation of the expression from Arrhenius
equation 2.84 leads to:
c 1 = c 0 · e −k 0·t·e −Ea/RT (2.98)
For a constant effect follows:
c t
= const. = e −k 0·t·e −Ea/RT (2.99)
c 0
Table 2.13. Alcohol dehydrogenase from pea seeds: activation
energy of alcohol dehydrogenation and aldehyde
reduction
Alcohol E a Aldehyde E a
(kJ · mole −1 ) (kJ · mole −1 )
Ethanol 20
n-Propanol 37 n-Propanal 20
2-Propenol 18
n-Butanol 40 n-Butanal 21
n-Hexanol 37 n-Hexanal 18
2-trans-
2-transhexenol
15 Hexenal 19
2-trans-
Heptenal 18