Chapter 4: Chemical kinetics

Chapter 4

chemical kinetics

Introduction :

In the thermodynamics, we have studied whether a reaction will take place or not and if it does then upto what extent (chemical equiibrium), In this chapter we will study about how fast a chemical  reaction takes place and what are the different factors affecting this rate of chemical reaction. How to optimise the conditions as to maximse the output in optimum time. The last part of chapter will be dealing with the mechanism of a chemical reaction and catalysis.

Rate/Velocity of chemical reaction

The rate of change of concentration with time of different chemical species taking part in a chemical reaction is known as rate of reaction of that species.

Rate =  =  = mol lit^–1 time^–1  = mol dm^–3 time^–1 

Rate is always defined in such a manner so that it is always a positive quantity.

Types of Rates of chemical reaction :

For a reaction R  P

Average rate = 

   

Instantaneous rate : rate of reaction at a particular instant.

R_{instantaneous} 

Instantaneous rate can be determined by drawing a tangent at time t on curve drawn for concentration versus time.

Initial Rate : Instantaneous rate at ‘t = 0’ is called initial rate [slope of tangent at t = 0].

Relation between reaction rates of different species involved in a reaction :

For the reaction :  N₂ + 3H₂  2NH₃

Rate of reaction of N₂ =

Rate of reaction of H₂ =

Rate of reaction of NH₃ =

These rates are not all equal. Therefore by convention the rate of a reaction is defined as

Rate of reaction 

Note : Rate of reaction value is dependent on the stoichiometric coefficients used in the reaction while rate of any species will be fixed value under given conditions.

Factors affecting rate of chemical reaction :

1. Concentration

2. Temperature 

3. Nature of reactants & products

4.  Catalyst

5. pH of the solution

6. Dielectric constant of the medium.

7. Radiations/light

8. Pressure

9. Electrical & Magnetic field.

The first four factors generally affect rate of almost all reactions while other factors are specific to some reactions only. The common examples of these reactions are :

Concentration : We known from law of mass action that Rate is proportional to concentration of reactants. “ So, generally  rate of reaction decreases with passage of time, since concentration of reactants decreases.

Temperature :     

Nature of reactants & Products : 

(a) Physical state of reactants :

Gaseous state > Liquid state > Solid state

Decreasing order of rate of reaction.

(b) Physical size of reactants : As we decreases the particle size rate of reaction increases since surface area increases.

(c) Chemical nature of reactants :

If more bonds are to be broken, the rate of reaction will be slow.

Similarly bond strength is more, rate of reaction will be slow.

Catalyst :

Presence of positive catalyst lower down the activation energy hence increases the rate of reaction.

presence of negative catalyst increases activation energy hence decreases the rate of reaction.

Radiations/light : Radiation are useful for photochemical reaction.

Pressure :  Pressure is important factor for gaseous reaction.

Rate Law (Dependence of rate on concentration of reactants) : 

The representation of rate of reaction in terms of the concentration of the reactants is called the rate law.

It can only be established by experiments.

Generally rate law expressions are not simple and these may differ for the same reaction on conditions under which the reaction is being carried out.

But for large number of reactions starting with pure reactants we can obtain simple rate laws.

For these reactions :     

Rate µ (conc.)^order 

Rate = K (conc.)^order      – differential rate equation or rate expression

Where K = Rate constant = specific reaction rate = rate of reaction when concentration is unity

 unit of K = (conc)^{1– order} time–1 

Note : Value of K is a constant for a given reaction, depends only on temperature

Order of reaction :

Let there be a reaction   m₁A + m₂B ¾® products.

Now, if on the basis of experiment, we find that

R µ [A]^P [B]^q    Where p may or may not be equal to m1 & similarly q may or may not be equal to m₂.

p is order of reaction with respect to reactant A and q is order of reaction with respect to reactant B and (p + q) is overall order of the reaction.

Note : Order of a reaction can be ‘zero’ or any whole number, can be a fractional number and it can even be negative with respect to a particular reactant. But oveall order is not found to be negative for any reaction till observed.

   Examples showing different  values of order of reactions :  

  The reaction (2) does not take place in one single step. It is almost impossible for all the 12 molecules of the reactants to be in a state of encounter simultaneously. Such a reaction is called complex reaction and takes  places in a sequence of a number of elementary reactions. For an elementary reaction the sum of stoichiometric coefficients = order of the reactions. But for complex reactions order is to be experimentally calculated.

Integrated rate laws

(a) Zero Order Reactions :

   For a zero order reaction

   General rate law is, Rate = k [conc.]º = constant

   If C0 is the initial concentration of a reactant and Ct is the

   concentration at time ‘t’ then

   Rate = k =    or   kt = C₀ – C_t or  C_t = C₀ – kt            

   Unit of K = same as that of Rate = mol lit–1 sec–1.

   Time for completion =

   t1/2 (half life period)   at t_1/2 , Ct  = , so  kt_1/2 = Þ t_1/2 =

 t_1/2  µ C₀  

Examples of zero order reactions :

Generally decomposition of gases on metal surfaces at high concentrations follow zero order kinetics.

 

(b) First Order Reactions :

     

   If any substance is growing/increasing following first order kinetics then :

   

where a is initial concentration of the substance and x is the increment in its concentration after time t. 

Half life time (t1/2) 

Half life period for a 1st order reaction is a constant quantity.

Graphical Representation :

   

First order growth reaction :

For bacteria multiplication or virus growth use following concept Consider a growth reaction

 or

Generation time :

At  ¾® t = generation time , x = a

  t =

Examples of 1st order reactions :

1. H₂O₂    ¾®    H₂O + O₂

2. NH₄ NO₂ ¾®   2H₂O + N₂

3. Radiactive decay       

All radioactive decays are always first order kinetics.

     

(c) Second order reaction :

2nd order Reactions

Two types

(d) Psuedo first order reaction :

A second order (or of higher order) reactions can be converted into a first order reaction if the other  reactant is taken in large excess. Such first order reactions are known as psuedo first order reactions.

 For A + B ¾® Products   [Rate = K [A]¹ [B]¹]

   

Now if ‘B’ is taken in large excess b > > a.

 

‘b’ is very large can be taken as constant

Þ  kb = log    Þ     k¢ =  log 

k¢ is psuedo first order rate constant

K’ will have units of first order.

K will have units of second order.

Examples of Pseudo 1st order reactions :

(a)    Hydrolysis of canesugar

C₁₂H₁₂O₁₁   +   H₂O  ¾¾®  C₆H₁₂O₆ + C₆H₁₂O₆

    sucrose            excess         

(b)    Hydrolysis of esters

CH₃COOCH₃  + H2O       CH₃COOH + CH₃OH

                          (excess)

Table : Characterstics of First-and Second-Order Reactions of the Type A Products

Graphical comparison of different orders 

(A)     integrated rate law method :

It is method of hit and trial. By checking where the kinetic data (experimetal data) best fits into which integrated rate law ,  we determine the order. It can also be done graphically.

(B)  Method of half lives :

The half lives of each order is unique so by comparing half lives we can determine order

(C)  Ostwald’s isolation method : 

This method is useful for reaction which involve a large number of reactants. In this method, the concentration of all the reactants are taken in large excess exception that of one, so if

rate = k [A]^a [B]^b [C]^c = k₀ [A]^a 

Then value of ‘a’ can be calculated by previous methods and similarly ‘b’ and ‘c’ can also be calculated

Methods to monitor the progress of the reaction :

(A)  Pressure measurement :

Progress of gaseous reaction can be monitored by measuring total pressure at a fixed volume & temperature.

This method can applied for those reaction also in which a gas is produced because of decomposition of a solid or liquid. We can get an idea about the concentration of reacting species at a particular time by measuring pressure.

 The pressure data can be given in terms of

 (i) Partial pressure  of the reactant

(ii) Total pressure of the reaction system

(iii)  Pressure at only some points of time  

(b)  Volume measurement :

(i)  By measuring the volume of product formed we can monitor the progress of reactions.   

(c) Optical rotation measurment :

It is used for optically  active sample. It is applicable if there is atleast one optically active species involved in chemical reaction.

The optically active species may be present in reactant or product.

It is found that    

(a = concentration , x = amount consumed)

where arer₀, r_t, r_¥    are  angle of optical rotation at time    t = 0, t = t  and t = ¥

Effect of temperature on rate of reaction 

In early days the effect of temperature on reaction rate was expressed in terms of temperature coefficient  which was defined as the ratio of rate of reaction at two different temperature differing by 100C(usually these temperatures were taken as 25⁰C and 35⁰ C) 

 T.C. = 2 to 3 ( for most of the reactions)

For some reactions temperature coefficient is also found to be less than unity. for example

2NO + O₂ ® 2NO₂   rate of reaction decreases on increasing temperature.

But the method of temperature coefficient was not exact and to explain the effect of temperature on reaction rate new theory was evolved

Arrhenius theory of reaction rate

It was developed by max Trautz and William lewis.

It gives insight in to the energetics and mechanistic aspects of reactions.

It is based upon kinetic theory of gases.   

Arrhenius proposed a theory of reaction rate which states as follows :

A chemical reaction takes palce due to the collision among reactant molecules. The number of collisions taking place per second per unit volume of the reaction mixture is known as collision frequency (Z).

Every collision does not bring a chemical change. The collision that actually produce the products are effective collision. For a collision to be effective the following two barriers are to be effective the following two barriers are to be cleared.

Energy barrier :

The minimum amount of energy which the colliding molecules must posses as to make the chemical reaction to occur is known as threshold energy.

“The minimum amount of extra energy required by reactant molecules to pariticipate in a reaction is called activaiton energy (E_a)”

Orientation barrier :

Energy alone does not determine the effectiveness of the collision. The reacting molecules must collide in proper direction to make collision effective. Following diagrams can explain importance of suitable direction for collision.

Collision to be effective the colliding molecules must posses some certain minimum energy called threshold energy of the reaction.

Reactant molecules having energy equal or greater than the threshold are called active molecules and those having energy less than the threshold are called passive molecules.

Concept of energy of activation (E_a)

he extra amount of energy which the reactant molecules (having energy less than the threshold) must acquire so that their mutual collision may lead to the breaking of bond(s) and hence the energy is known as energy of activation of the reaction. It is denoted by the symbol Ea. Thus,

E_a = Threshold energy – Actual average energy,

Ea is expressed in kcals mole^–1 or kJ mole^–1.

The essence of Arrhenius Theory of reaction rate is that there exists an energy barrier in the reaction path between reactant(s) and product(s) and for reaction to occur the reactant molecules must climb over the top of the barrier which they do by collision. The existence of energy barrier and concept of Ea can be understood from the following diagram.

Collision frequency is the number of collisions per unit  

volume per unit time. It is denoted by the symbol Z. Z is directly proportional to . By 10ºC rise in temperature, so it is the fraction of the total number of effective collision that increases markedly resulting into marked increase in the reaction rate.

Arrhenius equation                        

Integrating equation 4 assuming Ea to be constant we get,

lnk = –         or       ln = –  or       k = 

This is integrated form of Arrhenius equation.

Where, Constant A = pre–exponential factor it is a constant for a given reaction.

From this equation it is evident that as T ® ¥, k ®  A. Thus, the constant A is the rate constant of reaction at infinity temperature. The rate constant goes on increasing with temperature.

So, when T approaches inifinty, k will be maximum. That is to say, A is the maximum rate constant of a reaction.

The expontential term i.e. e–Ea/RT measures the fraction of total number of molecules in the activated state or fraction of the total number of effective collisions.

 

Where  

nEa  =  no. of molecules of reactant in the activated state   

n = total no. of molecules of the reactant in the reaction

From Arrhenius Equation we have,     

log k = 

So from this it is evident that a plot of log k versus  will be a straight line of the slope equal to and intercept equal to log A as shown below :

Thus, from this plot Ea and A both can be determined accurately.

If k₁ and k₂ be the rate constant of a reaction at two different temperature T₁ and T₂ respectively, then we have

log k₁ =    and

log k₂ = 

Subtracting we get  

General characteristics of catalyst

A catalyst does not initiate the reaction. It simply fastens it.

Only a small amount of catalyst can catalyse the reaction. 

A catalyst does not alter the position of equilibrium i.e. magnitude of equilibrium constant and hence DGº. It simply lowers the time needed to attain equilibrium. This means if a reversible reaction in absence of catalyst completes to go to the extent of 75% till attainment of equilibrium, and this state of equilibrium is attained in 20 minutes then in presence of a catalyst also the reaction will go to 75% of completion before the attainment of equilibrium but the time needed for this will be less than 20 minutes.

A catalyst drives the reaction through a low energy path and hence Ea is less. That is, the function of the catalyst is to lower down the activation energy.

E_a = Energy of activation in absence of catalyst.    

E’_a = Energy of activation in presence of catalyst.    

E_a – E’_a = lowering of activation energy by catalyst.

Comparision of rates of reaction in presence and absence of catalyst :

If k and kcat be the rate constant of a reaction at a given temperature T, and Ea and E’a are the activation energies of the reaction in absence and presence of catalyst, respectively, the

Derivation of a suitable rate law with the help of a suitable mechanism :

Molecularity and Order :

The number of molecules that react in an elementary step is the molecularity of the elementary reaction. Molecularity is defined only for the elementary reactions and not for complex reactions.

No elementary reactions involving more than three molecules are known, because of very low probability of near-simultaneous collision of more than three molecules.

The rate law for the elementary reaction      

aA + bB¾® products

rate = k[A]^a[B]^b, where a + b = 1, 2 or 3.

For an elementary reaction, the orders in the rate law equal the coefficients of the reactants.

While, the order is defined for complex as well as elementary reactions and is always experimentally calculated by the mechanism of the reaction, usually by the slowest step of the mechanism known as rate determining step of the reaction.

Mechanism of a reaction :

Reactions can be divided into

Elementary / simple / single step

Complex / multi-step

ELEMENTARY REACTION :

 These reaction take place in single step without formation of any intermediate

For elementary reaction we can define molecularity of the reaction which is equal to no of molecules which make transition state or activated complex because of collisions in proper orientation and with sufficient energy

molecularity will always be a natural no

1 =  unimolecular one molecule gets excited (like radioectivity)

2.= bimolecular

3 = trimolecular

Molecularly 3 because the probabilty of simaltaneous collision between 4 or more molecules in proper orientation is very low. For elementary reaction there is only single step and hence it is going to be rate determining step so order of an elementary reaction is its molecularity

Order of elementary reaction w.r.t. reactant = stoichiometric co-efficient of the reactant

   H₂ + I₂    2HI ® Simple reaction         rate = k [H₂] [I₂]

   2H₂ + 2I₂   4HI          (not elementary )

reaction obtained by multiplying an elementary reaction with some no will not be of elementary nature

   H₂ + Cl₂   2HCl         order = 0

COMPLEX  REACTION :

Reaction which proceed in more than two steps. or having some mechanism. ( sequence of elementary reaction in which any complex reaction procceds)

For complex reaction each step of mechanism will be having its own molecularity but molecularity of net complex reaction will not be defined.

Order of complex reaction can be zero fractions whole no, even negative w.r.t. some species.

Order of reaction or rate law of reaction is calculated with the help of mechanism of the reaction generally using Rate determine step (R.D.S) if given.

Rate law of a reaction is always written in terms of conc. of reactant, products or catalysts but  never in terms of conc. of interimediates.

The mechanism of any complex recation is always written in terms of elementary steps, so molecularity of each of these steps will be defined but net molecularity of complex reaction has no meaning.

The mechanism of most of the reaction will be calculated or predicted by using mainly the following approximations.

COMPLICATIONS IN 1ST ORDER REACTION

PARALLEL 1st ORDER REACTION

Frequently a species can react in different ways to give a variety of products. For example, toluene can be nitrated at the ortho, meta, or para positions, We shall consider the simplest case, that of two competing irreversible first-order reactions :

A  B and A C  

where the stoichiometric coefficients are taken as unity for simplicity. The rate law is