REACTION KINETICS

Reaction Rate and Reaction Order:

The rate of a reaction is the velocity with which a reactant or

reactants undergo chemical change, e.g.: The hydrolysis (inversion) of

sucrose to glucose and fructose in aqueous solution, under the catalytic

influence of acid. The equation for the overall reaction is:

C12H22O11 + H2O C6H12O6 + C6H12O6 Sucrose Glucose Fructose

The rate at which the concentration of sucrose, designated c,

decreases with time, t, was found to be proportional to the concentration

of unhydrolyzed sucrose. The change may be expressed in the notation of

calculus by the differential equation.

dc /dt = kc (1)

Where dc /dt is the velocity (rate) with which the concentration

of sucrose decreases as it undergoes hydrolysis, and k is a constant called

the velocity constant or rate constant.

The term reaction order refers to the way in which concentration

of a reactant, or reactants, influences the rate of a chemical reaction.

First-Order Reactions:

When the rate of a reaction is proportional to the first power of the

concentration of a single reactant and may be expressed mathematically

in the form of Eq.1, the reaction is said to be first-order with respect to

the reactant

Eq.1 may be written

= kdt (2) dc

c

Packard bellHighlight

Integrating equation (2) between concentration co at time t = 0 and

concentration c at some later time t, we have

(3)

ln c ln co = k (t 0)

ln c = ln co Kt

Converting to common logarithms yields

Log c = log co Kt/2.303 (4)

Or

(5)

Equation (5) is often written as

(6)

In which the symbol a is customarily used to replace co, x is the

decrease of concentration in time t, and (a x) = c.

The plot of the logarithm of the concentration against the time as

shown in Figure 1 produces a straight line. The slope of the line is

k/2.303 from which the rate constant is obtained and the intercept with

the y-axis equals co. c2 at a later time t2 are used to calculate k by the

following modification of the preceding equation.

Half-life :

The period of time required for a drug to decompose to one half the

original concentration is the half-life, t1/2 for a first order reaction.

Writing Eq 5 in the form

dc/c = K dt

2.303 co K = log t c

2.303 a K = log t a x

(7)

it is apparent that

(8)

In practical studies of rate of reaction it is common to study the

change of concentration with time through two or three half-life periods.

Pseudo First Order Reactions:

If a large excess of one of the reactants in a second order reaction

is present throughout the reaction, then its concentration remains virtually

constant and the rate of concentration change follows the first order law.

Second-Order Reactions:

A second-order reaction is one in which the experimentally

determined rate of reaction is found to be proportional to the

concentration of each of two reactants or to the second power of the

concentration of one reactant. If substances A and B undergo reaction at a

rate proportional to the concentration of each (CA and CB, respectively)

we may write

(9)

If a and b represent the molar concentrations of A and B at the start

of the reaction, and x is the number of moles of each which has reacted at

time t the reaction rate, dx/dt, may be expressed as

2.303 co 0.693 K = log = t1/2 co1/2 t1/2

0.693 t1/2 = K

dcA dcB = = K.cAcB dt dt

dx / dt = K (a x) ( b x) (10)

If the initial concentrations of A and B are equal , Eq 10 simplifies

to

dx /dt = K ( a x)2 (11)

Equation (11) is integrated, employing the conditions that x = 0 at

t = 0 and x = x at t = t .

dx / (a x)2 = K dt 1 1

= Kt a x a 0

x = Kt (12) a (a x) Or 1 x K = (13) at a x

When, in the general case, A and B are not present in equal

concentrations, integration of equation (10) yields.

2.303 b (a x) log = Kt (14)

a b a (b x) Or

2.303 b (a x) K = log (15) t (a b ) a (b x)

It can be seen by reference to equation (12) that when x/a (a x) is

plotted against t, a straight line results if the reaction is second order. The

slope of the line is k. When the initial concentrations, a and b, are not

equal, a plot of log b (a x) (b x) against t should yield a straight line

with a slope of (a b) k/2.303. The value of k can thus be obtained. The

rate constant k in a second-order reaction therefore has the dimensions

liter/mole sec or liter mole-l sec-1.

An example of a second-order reaction in which two reactants are

involved is the saponification of an ester, such as ethyl acetate, in alkaline

solution.

CH3COOC2H5 + OH- CH3COO- + C2H5OH When only one reactant is involved in a second-order reaction, Eqs

12, and 14 apply. A reaction of this type is the decomposition of

hydrogen iodide, which in the gaseous state undergoes the reaction.

2HI H2 + I2 The reaction rate in this case is expressed by

d cHI / dt = k. cHI2 (16)

The half-life of a second -order reaction has significance only

when a single reaction, are involved. In both these cases the time, t 1/2,

required for half of the reactant or reactants ( at equal concentration ) to

undergo reaction may be calculated from Eq 13 to be:

1 T1/2 = (17)

Ka It varies inversely with the initial concentration of reactant.

Third Order Reactions:

A third-order reaction is one in which the experimentally

determined rate of reaction is found to be:

- proportional to the concentration of each of three

reactants,

- or proportional to the concentration of one of two

reactants and to the second power of the concentration of the

other,

- or proportional to the third power of the concentration

of a single reactant.

For the case of a reaction between one molecule each of A, B, and

C, at molar concentrations, a, b, and c, respectively, the rate equation is:

dcA dcB dcC = = = K. cA. cB. cC (18)

dt dt dt

Using the notation of Eq 10, this becomes

dx = K ( a x) ( b x) (c x) (19)

dt

When a = b = c

dx

= K ( a x)3 (20) dt

1 1 1 K = (21)

2t ( a x)2 a2

Such reactions are rare and their analysis is complex. Reactions of even higher orders are unlikely to occur.

3 t1/2 =

2 a2 K Zero-Order Reactions:

The rate expression for the change of concentration with time is:

dC = Ko (22) dt The velocity of reaction is seen to be constant and independent of

the concentration reactant. The rate equation may be integrated between

the initial concentration Co at t = 0, and Ct, the concentration after t hours:

Ct Co = Kot

Or

Ct = Co Kot (23)

When this linear equation is plotted with C on the vertical axis

against t on the horizontal axis, the slope of the line is equal to K.

The half-period, or half-life is given by the following equation:

t 1/2 = (24)

Co 2 K

dC = Ko dt

Apparent zero order

Example: Suspensions

Determination of Order:

The order of a reaction may be determined by several methods.

1. Substitution Method.

2. Graphical Method.

3. 3. Half-life Method:

In general, the half-life of a reaction in which the concentrations of

all reactants are identical is

1

t1/2 (25) an 1

in which n is the order of the reaction. Thus if two reactions are run

at different initial concentrations, a1 and a2 the half-lives t1/2(1) and t1/2(2)

are related as follows:

t1/2 (1) (a2)n 1 a2 n 1 = = (26) t1/2(2) (a1)n 1 a1

or in logarithmic form

t1/2 (1) a2 log = ( n 1 ) log (27) t1/2(2) a1

and finally

log ( t1/2(1) / t1/2(2)) n = + 1 (28) log (a2/a1)

Effects on Reaction Rate:

A number of factors other than concentration may affect the

reaction velocity. Among these are temperature, solvents, catalysts, and

light.

1- Temperature:

The rate of most solvolytic reactions of pharmaceuticals is

increased roughly 2- to 3- fold by a 10C increase in temperature in the vicinity of room temperature.

Arrhenius noted, in 1889, that the variation with temperature of the

rate constant of chemical reactions could be expressed by the equation.

K = Se E a/RT (29)

Where Ea is the Arrhenius activation energy (the difference

between the average energy of reactive molecules and the average energy

of all molecules), e E a/RT is the Boltzmann factor which represents the

fraction of molecules having the energy Ea, S is a constant called the

frequency factor, R is the gas constant (1.987 cal/K-mole), T is the absolute temperature.

In logarithmic form the Arrhenius equation becomes

Ea log k = log S (30) 2.303 RT

If this equation is valid, and it is for a large number of reactions, a

straight line is obtained on plotting log k against the reciprocal of the

absolute temperature. Ea may be calculated from the slope of the line,

which is Ea/2.303 R; the intercept on the log k axis is log S (Fig. 2).

On differentiating the natural logarithm form of Eq 32 with respect

to temperature.

d ln k / dT = Ea / RT2 (31)

and on integration between the limits k2 and k1 at temperatures T2

and T1.

K2 Ea T2 T1 Log = = (32) K1 2.303 R T1T2

which equation makes it possible to calculate Ea for a reaction

when the rate constants are known at two temperatures or to calculate the

rate constant at one temperature if Ea and the rate constant at another

temperature are known.

Activation energy:

When two molecules undergo chemical interaction, it is reasonable

to suppose that they must first come close enough to each other to effect a

collision and then, if conditions are right, undergo a rearrangement of

certain electrons to form the bonds characteristic of new molecules. The

colliding molecules must possess at least the amount of energy Ea before

reaction can occur. The greater this energy requirement is, the smaller the

proportion of colliding molecules that will have the necessary energy, and

the slower will be the reaction.

In the Arrhenius equation, S is a factor related to frequency of

collisions, and e E a/RT is the probability that at temperature T a collision

will occur with sufficient energy to provide a successful collision.

It would appear that in the reaction.

A + B [AB] * products

There is an activated complex or transition state, represented by

[AB]*, which is not a complex in the usual sense but involves a critical

intermediate geometric configuration in which the bonds of the reactants

are weakened and the bonds of the products begin to form; if the complex

has sufficient energy to effect the changes, the reaction proceeds; if it has

not, it reverts to the state of the separate reactants.

The concept of energy of activation, in relationship to the energy of

the reactants and of the products, is illustrated in Fig. 3.

2- Specific Acid and Specific Base Catalysis:

The term specific acid catalysis refers to catalysis by the hydrated

proton or hydrogen ion, and specific base catalysis refers to catalysis by

the hydroxide ion.

If the rate of hydrolysis of an ester such as ethyl acetate is studied

at constant pH in a strongly buffered solution, the rate of disappearance

of intact ester will be an apparent first-order reaction. If the reaction is

studied in solutions buffered at several different pH values in a

sufficiently acid pH region, a different apparent first-order rate constant

will be observed for each pH value and a rate-pH profile for the reaction

is obtained. The observed rate depends on the concentration of both the

ester and hydrogen ion and is actually a second-order reaction, although

at a constant hydrogen-ion concentration it is an apparent first-order

reaction.

k observed = k1 (H + ) (33)

The observed apparent first-order rate constant determined in

buffered solution is therefore proportional to hydrogen-ion concentration.

The variation in observed rate constant with pH can be illustrated by

taking logarithms of Eq 35.

log k observed = log k1 + log (H + ) (34)

log k observed = log k1 p H (35)

Thus a plot of logarithm Kobserved, vs pH should be linear with a

slope of 1..

Similarly, if the same hydrolysis reaction is studied in buffered

solution at several pH values in a sufficiently alkaline region of the pH

scale, the observed apparent first-order rate constants will be found to

vary with hydroxide-ion concentration.

k observed = k2 (OH ) (36)

Kw log k observed = log k2 + log (OH ) = log k2 + (37)

(H+) log k observed = log k2 + log Kw + p H (38)

A plot of logarithm Kobserved vs pH would be a straight line with a slope

of +1.

The complete rate expression for hydrolysis of the compound

described above at all pH values would therefore be

dc / dt = k1 (H+) + k2 (OH) c (39)

At any specified pH value

k observed = k1 (H +) + k2 (OH ) (40)

The complete logarithm Kobserved vs pH profile would be similar to

that illustrated in Fig. 4 for the hydrogen-ion and hydroxide-ion (specific

acid and specific base) catalyzed hydrolysis of the ester atropine. The pH

at which the minimum rate of hydrolysis is observed is a function of the

relative magnitude of the specific rate constants k1 and k2.

In the atropine example the minimum rate of hydrolysis is at pH

3.7. If for a specific chemical species k1 = k2, the expected minimum rate

of reaction for that species would be expected to occur at pH7 (for a

temperature of 25, at which pKw = 14, and (H+) = (OH-) at pH 7.0).

At pH values below the minimum in the plot of logarithm k

observed versus pH, the hydrogen-ion-catalyzed reaction is must more

significant than the hydroxide-ion-catalyzed reaction, and the plot has a

slope of-1. At pH values above the minimum in the plot, the hydroxide-

ion-catalyzed reaction is the much more important reaction and the plot

has a slope of +1.

If a reaction is catalyzed not only by hydrogen ion and hydroxide

ion, but also by the solvent (also called the uncatalyzed reaction), the

logarithm Kobserved versus pH plot might appear as in Fig. 5, indicating a

flat region where the rate of reaction is apparently not pH-dependent. In

this region the solvent reaction is much more important than that of either

the hydrogen ion or hydroxide ion. The apparent first-order rate constant

for such a reaction is

k observed = ko + k1 (H +) + k2 (OH ) (41)

At low pH, the term k1 (H +) is greater than k0 or k2 (OH ) because

of the greater concentration of hydrogen ions, and specific hydrogen ion

catalysis is observed. Similarly, at high pH at which the concentration of

(OH) is greater, the term k2 (OH ) outweighs the k0 and k1 (H +) terms,

and specific hydroxyl ion catalysis is observed. When the concentrations

of H+ and OH- are low, or if the products k1 (H +) and K2 (OH ) are

small in value, only k0 is important, and the reaction is said to be solvent

catalyzed. If the pH, of the reaction medium is slightly acidic, so that k0 and k1 (H+) are important and K2 (OH) is negligible, both solvent and

specific hydrogen ion catalysis operate simultaneously. A similar result is

obtained when the pH of the medium is slightly alkaline, a condition that

could allow concurrent solvent and specific hydroxide ion catalysis.

3- General Acid-Base Catalysis:

In most systems of pharmaceutical interest, buffers are used to

maintain the solution at a particular pH. Often, in addition to the effect of

pH on the reaction rate, there may be catalysis by one or more species of

the buffer components. The reaction is then said to be subject to general

acid or general base catalysis depending, respectively, on whether the

catalytic components are acidic or basic.

The rate-pH profile of a reaction that is susceptible to general acid-

base catalysis exhibits deviations from the behavior expected on the basis

of equations (35) and (38). For example, in the hydrolysis of the

antibiotic, streptozotocin, rates in phosphate buffer exceed the rate

expected for specific base catalysis. This effect is due to a general base

catalysis (Fig. 6). The alkaline branch of the rate-pH profile for this

reaction is a line whose slope is different from one.

Verification of a general acid or general base catalysis may be

made by determining the rates of degradation of a drug in series of

buffers that are all at the same pH (i.e., the ratio of salt to acid is constant)

but that are prepared with an increasing concentration of buffer species.

Windheuser and Higuchi, using acetate buffer, found that the degradation

of thiamine is unaffected at pH 3.90, where the buffer is principally acetic

acid. At higher pH values, however, the rate increases in direct proportion

to the concentration of acetate. In this case, acetate ion is the general base

catalyst.

Webb et al. demonstrated the general catalytic action of acetic acid,

sodium acetate, formic acid, and sodium formate in the decomposition of

glucose. The equation for the overall rate of decomposition of glucose in

water in the presence of acetic acid HAc and its conjugate base Ac can

be written:

dG = ko [G] + kH [H+] [G] + kA [Hac] [G] (42) dt + kOH [OH] [G] + kB [Ac] [G]

in which [G] is the concentration of glucose, k0 is the specific

reaction rate in water alone, and the other k values, known as catalytic

coefficients, represent the specific rates associated with the various

catalytic species. The overall first-order rate constant k, which involves

all effects, is written as follows:

dG / dt k = = ko + kH [H+] + kA [Hac] (43) [G]

+ kOH [OH] + kB [Ac] k = ko + k i c i (44)

In which Ci is the concentration of the catalytic species i and ki is

the corresponding catalytic coefficient. In reactions in which only specific

acid-base effects occur, i.e., in which only [H+] and [OH-] act as

catalysts, the equation is

k = ko + kH (H +) + kOH (OH ) (45)

4- Ionic Strength:

In general the effects of increasing concentrations of electrolytes

on reaction rate can be predicted by consideration of the influence of

ionic strength on interionic attraction. Increased ionic strength would be

expected to decrease rate of reaction involving interaction between

oppositely charged ions, and increase rate of reaction between similarly

charged ions.

Reactions between ions and dipolar molecules, and reactions

between neutral molecules, are generally less sensitive to ionic stregth

effects than are reactions between ionic compounds. However, reactions

which result in formation of oppositely charged ions as products may

exhibit considerable increase in rate with increasing ionic strength.

5- Dielectric Constant of Solvent:

Reactions involving ions of opposite charge are accelerated by

solvents of low dielectric constant. The rate of hydrogen-ion-catalyzed

hydrolysis of sulfate esters, for example, is much greater in low-

dielectric-constant solvents such as methylene chloride than in water.

Reactions between similarly charged species are favored by high-

dielectric-constant solvents. Reactions between neutral molecules which

produce a highly polar transition state, such as reaction of trithylamine

with ethyl iodide to produce a quaternary ammonium salt, will also be

enhanced by high-dielectric-constant solvents.

6- Influence of light:

Light energy, like heat, may provide the activation necessary for a

reaction to occur. Photochemical reactions do not depend on temperature

for activation of the molecules; therefore, the rate of activation in such

reactions is independent of temperature. The initial photochemical

reaction may often be followed by thermal reactions.

Examples of photochemical reactions of interest in pharmacy and

biology are the irradiation of ergosterol and the process of

photosynthesis. When ergosterol is irradiated with light in the ultraviolet

region, vitamin D is produced. In photosynthesis, carbon dioxide and

water are combined in the presence of a photosensitizer, chlorophyll.

Chlorophyll absorbs visible light, and the light then brings about the

photochemical reaction in which carbohydrates and oxygen are formed.

7- Catalysis:

The rate of a reaction is frequently influenced by the presence of a

catalyst. Although the hydrolysis of sucrose in the presence of water at

room temperature proceeds with a decrease in free energy, the reaction is

so slow as to be negligible. When the hydrogen ion concentration is

increased, by adding a small amount of acid, however inversion proceeds

at a measurable rate.

A catalyst is therefore defined as a substance that influences the

speed of a reaction without itself being altered chemically. When a

catalyst decreases the velocity of a reaction, it is called a negative

catalysts. Actually, negative catalyst often may be changed permanently

during a reaction, and these should be called inhibitors rather than

catalysts.

Catalysis is considered to operate in the following way. The

catalyst combines with the reactant known as the substrate and forms an

intermediate known as a complex, which then decomposes to regenerate

the catalyst and yield the products. Alternatively, a catalyst may act by

producing free radicals such as CH3. which brings about fast chain

reactions.

Catalytic action may be homogenous or heterogeneous and may

occur in either the gaseous or liquid state. Homogenous catalysis occurs

when the catalyst and the reactants are in the same phase, e.g.: acid-base

catalysis.. Heterogeneous catalysis occurs when the catalyst and the

reactants form separate phases in the mixture. The catalyst may be a

finely divided solid such as platinum, or it may be the walls of the

container. The catalysis occurs at the surface of the solid and is therefore

sometimes known as contact catalysis. The reactant molecules are

adsorbed at various points or active centers on the rough surface of the

catalyst. Catalysts may be poisoned by extraneous substances, that are

strongly adsorbed at the active centers of the catalytic surface where the

reactants would normally be held during reaction. Carbon monoxide is

known to poison the catalytic action of copper in the hydrogenation of

ethylene. Other substances, known as promoters, are found to increase the

activity of the catalyst. The promoter is thought to change the properties

of the surface so as to enhance the adsorption of the reactants and thus

increases the catalytic activity.