3. Integrated rate equations

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