Integral And Differential Method

Both methods are used to calculate the reaction order in a reaction or to analysis the kinetic data. The "Differential Method" directly deals with the rate law written for a given reaction. While the "Integral Method" is a hit and trial method in which we substitute the values of the concentrations and time and then we check whether the rate equations are satisfied with these values or not. 

The integrated rate law equation expresses the concentration of the reactant as a function of time. And the order and rate constant of the reaction can be found from the integrated rate law equation. This is done by substituting the concentration versus time data in various integrated rate law equations, checking whether all calculated "k" values ​​are the same.

Either the "Differential And Integral Rate Law Equation" is used to determine the reaction order. 


Let Us Define Both Methods One By One-
Differential Rate Law
"The differential rate law is a relation that describes the dependency of reaction rate on the reactants concentration". This means that the differential rate law is similar to the actual rate law written for a reaction (means both the differential rate law and the actual rate law are the same). Thus, the differential method of analysis deals directly with the rate laws or differential rate equations.
Differential rate law is calculated at a particular instant of time (or called instantaneous rate law). Hence, differential rate law represents the slope of the tangent at any point in the "Concentration Vs Time" graph.

Integrated Rate Law
"The integrated rate law is a relationship between the reaction rate constant and the concentration of reacting species (or reactants)".
Integrated rate law is very difficult to determine as it is a hit and trial method. So, we will find out the differential rate law by guessing first. Now, we integrate this differential rate law to determine the reaction order and obtain the relationship between the concentration and the reaction rate constant. Then, we check whether this relation is true or not. If true, this would be our "Integrated Rate Law" otherwise we would guess another differential rate law again. And then repeat the whole method again.

To check, whether an obtained Integrated rate law is valid or not we plot the data on "Concentration Vs Time" graph and if a reasonably a good straight line is obtained then the rate equation is said to satisfactorily fit the data. For different orders of reactions, we observe different integrated rate equations.


Integrated And Differential Rate Laws
1) Differential method is useful in complicated cases whereas Integrated method is easy to use and is recommended for testing specific mechanism.
2) Differential method requires large and more accurate data whereas Integrated method requires small amount of data.
3) Integrated method is a hit and trial method whereas differential method is not a hit and trial method.
4) Differential method can be used for fractional order reaction whereas integrated method cannot be used.
5) When compared on the basis of accuracy, the differential method is not as accurate as the integrated method.
......."Generally, the integrated method is attempted first and if it is not successful then we go for using differential method".......


Let us understand the integral and differential rate equations by an example
Consider a chemical reaction,
                      A + B ➝ products 
 Reactions rate,
                        ➪ rA = k[A][B]⁰
Since the rate of reaction is define as the rate of disappearance of reactant molecule therefore,
                  ➪ rA = -d[A]/dt = k[A][B]⁰
                        ➪ - d[A]/dt = k[A] 
                                          ------------------(1)
Here,
      Equation (1) is known as differential rate law. While if we integrate the above equation (1) over time period then we have a rate equation,
                      ∫-(1/[A]) d[A] = k∫dt
                                      --------------------(2)
                    Since,
                           At t=0 , [A] = [A]₀
                           At t =t , [A] = [A]
                    And,
                         ∵ ∫-(1/[A]) d[A] = log([A])
Therefore, after applying the boundary condition in the above equation (2), we get-
                ➪ - ( log [A]- log [A]₀) = k (t–0)
                ➪ log [A]₀ - log [A] = kt
                ➪ log ( [A]₀/[A] ) = kt
                ➪ k = (1/t) log ([A]₀/[A])
                                             -----------------(3)
Here equation (3) is known as integrated rate law.

Points To Be Remember
☛ In general, it is suggested that integral analysis be attempted first and if not successful then we used or attempted the differential method to analysis the given kinetic data.
☛ Integral method is easy to use and is recommended when testing specific mechanism or relatively simple rate expressions or when the data are so scattered.
☛ The differential method is useful in more complicated situations and requires more accurate or larger amount of kinetic data.



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