Introduction

Chemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes. It includes investigations of how different experimental conditions can influence the speed of a chemical reaction. Chemical kinetics deals with the experimental determination of rates from which rate laws and rate constants are derived. Relatively simple rate laws exist for zero-order reactions, first-order reactions, and second-order reactions, and can be derived for others. The main factors that influence the reaction rate include: the physical state of the reactants, the concentrations of the reactants, the temperature at which the reaction occurs, and whether or not any catalysts are present in the reaction.

8.2 The Rate of Reaction

The rate of a chemical reaction is the change in concentration of the reactants or products as a function of time. With the passage of time, the concentration of reactants decreases while the concentration of products increases as shown in Figure 8.1.

Figure 8.1

For example consider a simple reaction:

A’B

Rate of reaction= -(d[A])/dt=(d[B])/dt

The negative sign is present because the concentration of the reactant, A, decrease with time. From the plot it is clear that the rate depends on the particular values of time, and is called the “instantaneous rate.” A more complicated example considering stoichiometry:

2 ‘NO’_((g)) + O_(2(g)) ‘ 2 ‘NO’_(2(g))

The rate of reaction at any given time will depend upon the concentration of the reactants at that time. As the reaction progresses, the concentration of reactants keeps on falling with time. The rate of reaction will be

Rate of reaction= -1/2 (d[NO])/dt=-(d[O_2])/dt=1/2 (d[‘NO’_2])/dt

Any one of the three derivatives can be used to define the rate of the reaction.

Reactions rate has the units of concentration divided by time. We express concentrations in moles per liter and time in any convenient unit second (s), minutes (min), hours (h), days (d) or possible years. Therefore, the units of reaction rates may be:

mole/liter sec or mol 1’1 s

mole/liter min or mol 1’1 min’1

mole/liter hour or mol 1’1 h’1 and, so on

8.3 Molecularity and Order of a Reaction

The molecularity of an elementary reaction is defined as the number of reactant molecules involved in a reaction. Thus the molecularity of an elementary reaction is 1, 2, 3, etc., according as one, two or three reactant molecules are participating in the reaction. For example:

‘CH’_3 COOC_2 H_5+ H_2 O’ CH_3 COOH + C_2 H_5 OH

The molecularity of this reaction is 2.

The order of a reaction is defined as the sum of the powers of concentrations in the rate law.

Let us consider the example of a reaction

mA+nB ‘Product

The rate law is:

rate = k ‘[A]’^m ‘[B] ‘^n

The reaction order with respect to A is m and with respect to B it is n. The overall order of reaction (m + n) may range from 1 to 3 and can be fractional.

Reactions may be classified according to the order. If in the rate law (1) above

m + n= 1, it is first order reaction

m + n= 2, it is second order reaction

m + n= 3, it is third order reaction

8.3.1 Zero Order Reactions

In a zero order reaction, rate is independent of the concentration of the reactions. Let us consider

a zero-order reaction of the type

A ‘Products

Initial conc. a 0

Final conc. a ‘ x x

Rate of reaction

(-d[A])/dt=k_0 ‘[A]’^0

(-dx)/dt=k_0 ‘(a-x)’^0

On integrating we get

k_0= x/t

x=k_0 t”.(8.1)

Wherek0 is the rate constant of a zero-order reaction, the unit of which is concentration per unit time. In zero order reaction, the rate constant is equal to the rate of reaction at all concentrations.

8.3.2 First Order Reactions

Let us consider a first order reaction

A ‘product

Initial conc. a 0

Final conc. a ‘ x x

We know that for a first order reaction, the rate of reaction, dx/dt, is directly proportional to the concentration of the reactant. Thus,

dx/dt=k(a-x)

dx/(a-x)=kdt

On integrating above equation

-ln'(a-x)=kt+I ”.(8.2)

Where I is the integration constant. The constant k may be evaluated by putting t= 0 and x = 0.

Thus,

I= -lna

Substituting for I in equation (7.2)

-ln”a/(a-x)’=kt

k= 1/t ln”a/(a-x)”’.(8.3)

8.3.3 Second Order Reactions

Let us take a second order reaction of the type

A+A ‘products

Initial conc. a a 0

Final conc. a-x a-x x

We know that for such a second order reaction, rate of reaction is proportional to the square of the concentration of the reactant. Thus,

dx/dt=k'(a-x)’^2”..(8.4)

Where k is the rate constant, rearranging equation (7.4), we have

dx/((a’-x)’^2 )=kdt

On integration, it gives

1/((a-x))=kt+I”..(8.5)

Where I is the integration constant. I can be evaluated by putting x = 0 and t = 0. Thus,

I=1/a”'(8.6)

Substituting the value of I from equation (7.6) into equation (7.5) we get,

1/((a-x))=kt+1/a

kt=1/((a-x))-1/a

k=1/t.x/(a(a-x))”.(8.7)

This is the integrated rate equation for a second order reaction.

8.4 Rate Laws

A rate law is an equation that tells us how the reaction rate depends on the concentrations of the chemical species involved. The rate law may contain substances which are not in the balanced reaction and may not contain some things that are in the balanced equation (even on the reactant side). Consider a reaction between two species A and B

aA+bB=pP

then the rate laws takes the form for this reaction,

r=k'[A]’^a ‘[B]’^b

Where a and b are small whole numbers or simple fractions and k is called the rate constant. The sum of a+b is called the order of the reaction.

The differential and integrated rate law, units of rate constant and half-life is summarized in table 8.1.

Table 8.1 Summary of empirical rate law relationships for general reactions of type

A B.

Reaction Order Differential Rate Law Integrated Rate Law Units of Rate Constant Half Life

Zero

First

Second

-d[A]/dt = k

-d[A]/dt + k[A]

-d[A]/dt = k[A]2 [A] = [A0] ‘ kt

[A] = [A0] e-kt

[A] = [A0]/1+kt[A0] Mole L-1 Sec-1

Sec-1

Mole-1 L Sec-1

[A0]/2k

0.693/k

1/k[A0]

8.5 Method for Determining Rate Laws

Different experimental methods used for the determination of rate law are briefly described below:

Initial Rate Method (Isolation method)

The Initial Rates Method can be used irrespective of number of reactants involved, suppose one is studying a reaction with the following stoichiometry:

aA + bB ‘ cC

In this method the rate of reaction is measured by taking known concentrations of reactants (A and B). Now the concentration of one of the reactants (A) is changed taking same concentrations of other reactant (B) as before. The rate of reaction is now determined again, this will give the rate of reaction with respect to reactant A. Now change the concentration of other reactant (B) by taking same concentration of reactant B. This will give the rate expression with respect to reactant B. Total order of reaction can be determined by combining both rate expressions. For example the kinetics of synthesis of HI from H2 and I2 is pseudo first order with respect to H2 in the presence of large excess of I2 and also pseudo first order w.r.t. I2 in presence of large excess of H2, so the overall order of reaction is second order.

Integrated Rate Expression Method

This is the most common method used for determination of rate law of various reactions. In this method differential rate equations of different order reactions are integrated. With the help of these integrated equations we can find the value of k. This method can be used analytically or graphically.

In analytic method we assume the order of reaction and with the help of integrated equation of that order reaction we can calculate the value of k, the constant value of k suggests order for the reaction. If values of k are not constant, then we assume a different order for the reaction and again calculate the value of k, see if value of k is constant.

In graphical method the integral rate expression for first order reaction can be utilized to know if the reaction is first order.

The integrated form of firs order reaction is:

t= 1/k ln”a/(a-x)’

According to first order integrated rate law, if the plot of lna/a-x versus time is a straight line, then the reaction is of first order. Similarly the integrated rate expression for second order reaction can be used to determine rate law.

The Half-Life Method

The half-life (t1/2) of a reaction is the time taken for the initial concentrations of the reactants to decrease by half. The half-life of the first order reaction is directly proportional to initial concentration of reactant.

t_(1’2)'[a]

For first order reaction half-life is independent of temperature

t_(1’2)’1/'[a]’^0

For second order reaction it is inversely proportional to initial concentration

t_(1’2)’1/'[a]’^1

So in general we can say that for nth order reaction

t_(1’2)’1/'[a]’^(n-1)

If a1 and a2 are the initial concentrations in two different experiments and t1 and t2 are the time for a definite fraction of the reaction to be completed, then

Taking log of the above equation we get

”..(8.8)

So by using this equation we can determine the order of reaction.

Differential Rate Method

This method was given by Van’t Hoff, according to this method the rate of nth order reaction is:

r = k_n C_n

Taking log on both side of the equation

ln r = lnk_n+ n ln C

For two different concentrations C1 and C2of the reactants, we have

ln r_1 = lnk_n + n ln C_1

ln ‘r ‘_2= lnk_n + n ln C_2

Subtracting equations, we have

ln ‘(r’_1-r_2) = n (ln C_1- ln C_2)

n = ln ‘(r’_1-r_2)/(ln C_1- ln C_2) ”..(8.9)

From this equation by putting the values of r1 and r2 at two different concentrations C1 and C2 respectively, we can calculate the order of reaction.