BLBS102-c38 BLBS102-Simpson March 21, 2012 14:17 Trim: 276mm X 219mm Printer Name: Yet to Come
38 Thermal Processing Principles 727is the minimum energy required to initiate the collision, and the
statistical probability for collisions between certain molecules
that possess an adequate energy level for the reaction to occur at
a given temperature. To measure a reaction rate, it is necessary to
monitor the concentration of one of the reactants or products as a
function of time. Therefore, the rate of reaction can be expressed
as a rate of change in concentration to change in time.
On the basis of this concept, the rate law is an expression
relating the rate of a reaction to the concentrations of the chem-
ical species present, which may include reactants, products, and
catalysts. Many food reactions follow a simple rate law, which
takes the formr=k[A]a[B]b[C]c (1)that is, the rate (r) is proportional to the concentrations of the
reactants (A, B, C) each raised to some power. The constant
of proportionality,k, is called the rate constant. The power a
particular concentration is raised is the order of the reaction
with respect to that reactant. Note that the orders do not have to
be integers. The sum of the powers in Equation 1 is called the
overall reaction order.Zero-Order ReactionA zero-order reaction is independent of the concentration of the
reactants. A higher concentration of reactants will not speed up
the zero-order reaction. This means that the rate of the reaction is
equal to the rate constant,k, of that reaction. Zero-order reaction
is described asRate=r=−dA
dt=k[A]^0 (2)After separating variables and integrating both sides of Equa-
tion 2
∫AA 0dA=−∫tt 0kdt (3)This provides the integrated form of the rate law[A]=[AO]−kt (4)wherekis the rate constant of the reaction,Ais concentration at
timet.In a zero-order reaction, when concentration data [A]is
plotted versus time (t), the result is a straight line.First-Order ReactionA first-order reaction is one where the rate depends on the con-
centration of the species to the first power. Most of the reactions
involved in the processing of foods are of first-order reactions.
For a general unimolecular reaction, the decrease in the concen-
trationAover timetcan be written asRate=r=−dA
dt=k[A]^1 (5)Rearranging the equation−dA
A=kdt (6)Integrate both sides of the equation
∫AA 0dA
A=−∫t 0t 0kdt (7)and the linear for of Equation 7 is:ln[A]=ln[A 0 ]−kt (8)where [A] is the concentration at timet,[A 0 ] is the concentration
at timet=0, andkis reaction rate constant (s−^1 ).
Plotting ln [A] with respect to time (t) for a first-order reaction
gives a straight line with the slope of the line equal to−k,where
the rate constant is calculated.Second-Order ReactionA second-order reaction is one where the rate depends on the
concentration of the species to the second power. The reaction
rate expression for a unimolecular second-order reaction is
dA
dt=−k[A]^2 (9)Separation of variables and integration of both sides of equa-
tions will give us
dA
[A]^2=−kdt (10)
∫AA 0A
[A]^2=−∫tt 0kt (11)1
[A]=1
[A 0 ]+kt (12)Second-order unimolecular reaction is characterized by a hy-
perbolic relationship between concentration of the reactant or
product and time. A linear plot will be obtained if 1 /Ais plotted
against time.
Second-order bimolecular reactions may also follow the fol-
lowing rate equation:A+B→PRate=dA
dt=−k[A][B] (13)whereAandBare the reactants. After separating variables, the
differential equation may be integrated by holdingBconstant to
give
dA
[A]=−k[B]dt (14)
∫AA 0dA
A=−∫t 0t 0k′dt (15)1
[A]=1
[A 0 ]−k′t (16)k′is a pseudo-first-order rate constant:k′=kB.
A second-order bimolecular reaction will yield a similar
plot of the concentration of the reactant against time as a
first-order unimolecular reaction, but the reaction rate constant
will vary with different concentrations of the second reactant
(Table 38.1).