The rate of a reaction is the amount or concentration of a degradation product formed or the reactant lost per unit time. The rate of a reaction is described by a rate equation.

**Reaction
rate and order**

The
*rate* of a reaction is the amount or
concentration of a degradation product formed or the reactant lost per unit
time. The rate of a reaction is described by a *rate equation*. For example, for a hypothetical reaction,

*a*A* *+* b *B* *→* m*M* *+* n*N (7.1)

where:

A
and B are the reactants

M
and N are the products

*a*,* b*,*
m*, and* n *are the* stoichiometric coefficients *(number of
moles partici-pating in the reaction) for the corresponding reactant or product

The
rate of this reaction can be described in terms of rate of disappearance of A
or B or the rate of appearance of M or N, which are all interrelated. Thus,

Where,
d*C*_{A} is the change in the
concentration of reactant A over a period of time *dt*, and respectively, for all reactants and products in the
equation.

The
rate equation, describing the rate of formation of a product or the rate of
disappearance of a reactant, for this reaction can be written as:

where
*k* is the rate constant.

The
negative sign associated with the reactants indicates their rate of
dis-appearance from the system; that is, the concentration after time period, *t*, is lower than the starting
concentration. The rate equation for the products does not carry the negative
sign, since it indicates the rate of formation or appearance of products in the
system; that is, the concentration after time period, *t*, is higher than the starting concentration. The rate constant, *k*, is positive in both rate equations.

The
*order* of a reaction is the sum of
powers to which the concentra-tion terms of the reactants are raised in the
rate equation. The order of a reaction can also be defined with respect to a
given reactant. Hence, in the aforementioned example, the order of the reaction
with respect to reactant A is *a*,
order with respect to reactant B is *b*,
and the overall order of the reaction is *a*
+ *b*.

Identifying
the order of a reaction helps understand the dependence of the reaction on the
concentrations of starting materials. Thus, reactions can be of zero order
(indicating independence to reactant concentrations), first order (indicating
that reaction rate is proportional to the first power of one of the reactants),
second order (indicating that reaction rate is propor-tional to the first power
of two of the reactants or the second power of one of the reactants), or higher
order.

For
example, for the base-catalyzed hydrolysis of an ester in the reaction,

CH
_{3}COOC_{2}H _{5} + NaOH → CH_{3}COONa + C_{2} H_{5}OH

Reaction
rate is defined as:

The
reaction rate equation is:

Rate
= *k*[ CH_{3}COOC_{2}H_{5}
][ NaOH] (7.6)

The
order of this equation is 1 + 1 = 2, since both reactants are raised to the
power of one in the rate equation.

The
number of molecules taking part in a reaction is called the *molecularity* of a reaction. The
molecularity of a reaction is determined by the mechanism of a reaction and is
expressed in the reaction equation. The order of a reaction may or may not be
same as the *molecularity* of a
reaction. In the aforemen-tioned example, the molecularity of the reaction is
2, which is same as the order of the reaction.

The
observed order of a reaction may sometimes be different than the sum of
stoichiometric coefficients of reactants. The rate of a reaction may sometimes
be independent of the concentration of one of the reactants, even though this
reactant is consumed during the reaction. For example, if one of the two
reactants is the solvent in which the other reactant is dis-solved at low
concentration, such as an aqueous solution of a hydrolytically sensitive drug,
the order of the reaction may be independent of the solvent concentration—the
reactant present in a significantly higher concentration. Such reactions are
termed as *pseudo-nth order* reactions.
Thus, a truly sec-ond-order reaction, such as equimolar reaction of an ester
compound with water in an aqueous solution, that presents itself as a
first-order reaction is termed a *pseudo-1st
order* reaction.

For
example, for the hydrolysis of a dilute solution of ethyl acetate,

CH_{3}COOCH_{2}CH_{3}
+ H_{2}O → CH_{3}COOH + CH_{3}CH_{2}OH

Reaction
rate is defined as:

The
reaction rate equation is:

Rate
= *k*[ CH_{3}COOC_{2}H_{5}
] (7.8)

The
order of this equation is 1, since only one of the reactants is involved in the
rate equation. Nevertheless, the molecularity of this reaction is 2, since two
molecules are involved in the mechanism of the reaction and are expressed in
the reaction equation. Pseudo-order reactions are typically the cases where the
molecularity of a reaction is not the same as the order of a reaction.

The
order of a reaction is determined experimentally, while molecularity of a
reaction–which determines the equation–is determined by a thorough
understanding of the reaction mechanism. The order of a reaction can be
experimentally determined by one of the several methods:

1.
Initial rate method. Initial rate of a reaction is measured for a series of
reactions with varying concentrations of reactants to determine the power to
which the reaction rate depends on the concentration of each reactant. Only the
initial rate is measured to ensure that the reac-tant’s concentration is the
predominant influence on the reaction rate.

As
a reaction proceeds, the reaction rate can be influenced by changes in reaction
conditions, such as accumulation of product or by-product of a reaction. Thus,
the measurement of only the initial rate of a reac-tion provides a robust way
to quantitate the dependence of reaction rate on the concentration of
reactant(s).

2. Integrated rate law method. The concentration–time data
of a reaction can be used to assess how the rate of a reaction changes as a
function of reactant concentration. This plot is compared to theoretical
predic-tions made by integrated rate equations, discussed later in this
chapter, to infer reaction order.

3. Graph method. Similar to the integrated rate law method,
this method plots the concentration–time profile of a reaction graphically to
check fit to different reaction order kinetics.

4. Half-life method. The dependence of half-life, the time
it takes for the reactant concentration to reach half of the measured initial
concentration, on the initial concentration of the reactant is different for
reactions of different orders. Half-lives of reactants can be determined
experimentally and compared to theoretical predictions to determine reaction
order.

A
zero-order reaction is one in which the reaction rate is independent of the
concentration(s) of the reactant(s). The rate of change of concentration of
reactant(s) or product(s) in a zero-order reaction is constant and indepen-dent
of the reactant concentration. Many decomposition reactions in the solid phase
or in suspensions follow zero-order kinetics.

The
reaction rate for most zero-order reactions depends on some other factor, such
as absorption of light for photochemical reactions or the inter-facial surface
area for heterogeneous reactions (i.e., reactions that happen at the
solid–liquid, liquid–gas, or solid–gas interface). Thus, the slowest or the
rate-determining factor of the reaction is different than the concentration(s)
of reactant(s) for zero-order reactions.

__Rate equation__

Reaction
rate of a zero-order reaction is constant and independent of the reactant
concentration. Thus, the rate expression for the change in reactant
concentration, *C*, with time, *t,* for a zero-order reaction (Figure 7.1) can be written as:

*Figure 7.1** Zero-order kinetics. Plot of
concentration, C, versus time, t.*

where
*k*_{0} is the rate constant
for a zero-order reaction. Thus, the change in the concentration of the
reactant depends only on the time multiplied by a constant value, *k*_{0}.

Integrating
this equation from concentration *C*_{0}
at time = 0 to concentra-tion = *C _{t}*
at time =

*C _{t} *−

Thus,
the rate equation (Figure 7.1) is:

*C _{t} *=

This
is a linear equation of the form, *y* =
*mx* + *c*. Thus, a plot of concentra-tion, *C _{t}*, on the

__Half-life__

Scientists
are frequently interested in the time required for the reduction of a given
proportion of starting drug concentration. For example, a drug’s shelf life is
defined in terms of the time taken for the reduction of labeled drug
concentration to its 90% level. The half-life (*t*_{1/2}) of a reaction is defined as the time required for
one-half of the material to decompose. Thus, con-centration of a reactant at
its half-life (*C _{t}*

*C _{t}*

Thus,
for a zero-order reaction,

*Ct *_{1/2}* *=* C*_{0}* *−* k*_{0}*t*_{1 /2} (7.15)

*C _{0}/ 2 = C*

After
rearranging, the half-life expression (Figure 7.1)
for a zero-order reaction is:

A
first-order reaction is one in which the rate of reaction is directly
proportional to the concentration of one of the reactants. Many decomposition
reactions in the solid phase or in suspensions follow first-order kinetics. In
a first-order reaction, concentration decreases exponentially with time, with
the reaction rate slowing down progressively as the reactant is consumed in the
reaction.

__Rate equation__

The
rate of disappearance of the reactant is the concentration of the reactant
multiplied by a constant. Thus, the first-order rate equation (Figure 7.2) is:

*Figure 7.2** First-order kinetics. Plot of
concentration, C, against time, t (A), and plot of natural logarithm of the
concentration, C, against time, t.*

Where,
*C* is the reactant concentration at time
*t*, and *k* is the first-order rate constant.

This
equation can be rearranged as:

Integrating
this equation from concentration *C*_{0}
at time = 0 to concentration = *C _{t}
*at time

Solving
this integral,

ln
*C* − ln*C*_{0} = −*k*(*t*
− 0) = −*kt *(7.21)

This
equation can also be expressed as:

Thus,
the rate equation (Figure 7.2) is:

ln
*C* = ln*C*_{0} –
*kt *(7.22)

Or,
in the exponential form, the rate equation can be expressed as:

*C *=* C*_{0}e^{−}* ^{kt }*(7.23)

Alternatively,
by converting ln to the log base 10 (log_{10}),

*C *=* C*_{0}10 ^{kt/}^{2.303} (7.24)

Thus,
in a first-order reaction, the concentration decreases exponentially with time
(Figure 7.2). A plot of log or ln of
concentration against time is a straight line, whose slope provides the rate
constant, *k*.

__Half-life__

Half-life,
*t*_{1/2}, is defined as the
time for the drug concentration to get to half of its original concentration (*C*_{0}); that is, *C _{t}* =

Thus,
the half-life of the reactant in a first-order reaction is given by:

Hence,
the half-life expression (Figure 7.2) is:

*t*_{1/2} = 0.693/ *k *(7.32)

For
a reaction observing first-order kinetics, the half-life, t_{1/2}, or
the time to any proportional reduction in concentration (e.g., t_{0.9},
i.e., time to 90% of initial concentration), is a constant number and
independent of the initial reactant concentration, *C*_{0}.

Bimolecular
reactions, reactions involving two different molecules A and B, involve
reactions of two molecules.

A
+ B → Products

The
rates of bimolecular reactions are frequently described by a second-order
equation. The rate of change in the concentrations of products and reactants in
second-order reactions is proportional either to the second power of the
concentration of a single reactant or to the first powers of the concentrations
of two reactants.

When
the speed of the reaction depends on the concentrations of A and B, with each
term raised to the first power, the rate of decomposition of *A* is equal to the rate of decomposition
of B and both are proportional to the product of the concentrations of the
reactants. This can be expressed as the rate equation:

__Rate equation__

Assuming
that the initial concentrations of A and B are same, that is, *C*_{0}, and their concentration
after time, *t*, is *C*, the rate equation can be written as:

Or,
using their concentration value, *C*,
the *rate expression* (Figure 7.3) is:

**Figure 7.3*** Second-order kinetics. Plot of the reciprocal of the concentration, C,
against time, t.*

Integrating,

Or,
the *rate equation* (Figure 7.3) is:

__Half-life__

In
a second-order reaction, the time to reach a certain fraction of the initial
concentration (such as t_{1/2} or t_{0.90}) is dependent on the
initial concentration. The half-life is defined as:

Thus,
the *half-life expression* (Figure 7.3) is:

Hence,
for a second-order reaction, t_{1/2} decreases with increasing initial
concentration of the two reactants. This is consistent with the molecular
mechanism of intermolecular reactions, where the two molecules must col-lide
and react with each other for the reaction to happen. Thus, higher initial
concentration increases the probability of collision and reaction between the
molecules of two different types, increasing the probability and rate of the
reaction.

Often,
a drug undergoes more than one chemical reaction or a series of reactions in
the same environment. Such complex reactions can be exemplified by reversible,
parallel, or consecutive reactions. In such cases, the experimental methods for
detection of reaction rates usually have limi-tations in that each reaction
intermediate and product may not be detected or accurately quantitated. Thus,
one may be quantitating a product whose concentration is impacted by the
starting material or drug concentration in a complex manner. Understanding how
such complexities might be linked to the kinetics of reactions can help
delineate the mechanisms of degradation of drugs. This section describes the
predicted kinetics of com-plex reactions.

Reversible
reactions are bidirectional; that is, the product can convert back to the
reactant. The rate constant of the forward reaction can be designated by *k*_{1}, and the rate constant of
the reverse reaction can be des-ignated by *k*_{−1}.

Assuming
first-order reaction in either direction, the rate of the forward reaction at
equilibrium is described by:

The
rate of the forward reaction may not equal the rate of the reverse reaction. In
fact, the rate of the forward and the reverse reactions may be affected by
different environmental conditions. Equilibrium reactions are characterized by
a constant ratio of the concentration of reactants and products, without regard
to their absolute concentration.

Parallel
reactions involve two or more simultaneous reaction pathways for a reactant.
For example,

Assuming
first-order kinetics for both reactions, the individual reaction rates and the
rates of formation of individual products are defined by the individual rate
equations:

The
overall rate of degradation of a reactant is given by:

Rate
= − d[A] / *dt *= *k*_{1}[A] + *k*_{2}[A] = (*k*_{1} + *k*_{2})[A] = *k*_{obs}[A] (7.47)

where
*k*_{obs} is the observed rate
of degradation of the reactant A.

The
concentration of reactant A at any time *t*
is given by the exponential first-order equation:

[*A*] = [*A*_{0} ]e^{−}* ^{k}*obs

Thus,
the overall or observed rate of degradation is a sum of the rates of
degradation of all individual parallel reactions that occur simultaneously.

Consecutive
reactions involve the formation of an intermediate, which is transformed into
the final product.

The
rate equations for this mechanism can be written as:

The
concentration time profiles for all species in this reaction can be obtained by
simultaneously solving the above differential equations.

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