An Overview of Chemical Reaction Analysis - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 134

The reaction rate constants are functions of temperature only for a given system. The rate
dependence on temperature is governed by the Arrhenius law or

e E=RT k ð T Þ¼ k (3.53)

0

where E is called the activation energy for the reaction and k 0 is called (unimaginatively) the pre-
exponential factor. For reversible reactions, the difference between the activation energy of the
forward and backward reactions is the heat of reaction. The ratio of the forward reaction rate
constant over the backward reaction rate constant is directly related to the equilibrium constant.
The degree of reduction is defined as the number of electrons a molecule can potentially
give off when reacting with other molecules. The generalized degree of reduction for a carbon
and hydrogen containing molecule is given as

g DR ¼ 4n C þ n H 2n O (3.92)

where v C , v H , and v O are the numbers of carbon, hydrogen, and oxygen atoms, respectively,
in the molecule. The degree of reduction for oxygen (O 2 ) gas is set at zero, although a 2 is set

for oxygen atom in a molecule as shown in Eqn (3.92) .
The heats and Gibbs free energy of combustion (producing gaseous CO 2 and liquid H 2 O)
have been found to relate to the degree of reduction via
DG C ¼ DG g g DR ¼

112 g DR kJ

= (3.93)

mole

and

DH C ¼ DH g g DR ¼ 110 : 9 g DR kJ = mole (3.94)

0
C

0
C

Where DG are the Gibbs free energy and heat of combustion, respectively. The

combustion products are in their elemental form if other than C and H 2 .
Yield is an important parameter in reactor analysis. The yield is defined as

and DH

#-moles of R in all the P formed
#-moles of R in all the A initially

Y P = A = R ¼ (3.67)

where R is a key element of group of elements present entirely in both P and A.
In bioreactions, the natural reference (R) is carbon (C) as most cases we use a carbon source
and carbon is the key component of interest. In this case, Eqn (3.67) can be rewritten as

#-C-moles of P formed
#-C-moles of A initially

Y P = A = C ¼ (3.68a)

or simply

#-C-moles of P formed
#-C-moles of A initially

Y P = A ¼ (3.68b)

as carbon is implied. Yield is dependent on the extent of reaction and thus is not a thermody-
namic (or equilibrium) parameter.
Yield factor is the maximum fractional yield and is defined as

#-C-moles of P that could be formed
#-C-moles of A

¼ jn P j
jn A j

YF P = A ¼ (3.69)