LINEAR FREE ENERGY RELATIONSHIP AND TAFT EQUATION

The introduction of substiuents X in the organic molecule may increase or decrease it's reactivity.

                                                A number of quantitative relationships has been suggested to account for to predict the effects of substiuents on the rate constant of reactions such structure - reactivity correlation are known as LFER

                            For example :-

               Hammett equation and Taft equation are, 

MATHEMATICAL REPRESENTATION :-

                         According to ARRT Theory, 

κ =   RT     -   ∆G≠ 

       Nh          eRT

  RT  e = constant

 Nh

lnκ = constant -   ∆G≠  .........(1)

                              RT

Accoding to van't haff reaction Isotherm, 

             ∆G = - RTlnk

           lnk = −   ∆G  ........(2)

                         RT

According to bronsted equation, 

         κ ∝ K∝

         κ = G.k∝

lnk = constant + ∝ lnk ............(3)

Substituting Equation (1) and (2) in (3)

Constant -   ∆G≠  = constant - ∝.   ∆G  

                      RT                                RT

Constant − ∆G≠ = constant − ∝ ∆G ....(4)

This equation is called linear free energy relationship for the given chemical reaction. 

Converting this equation into graph 

TAFT EQUATION :-

                         Taft equation is linear free energy relationship used in physical organic chemistry in the study of reaction mechanism and the development of quantitative structure - reactivity relationships for organic compounds. 

                      It was developed by R.W Taft, it's the modification of Hammet equation while the Hammet equation account for Inductive of mesomeric effects influencing the reaction rates. 

But Taft equation also describes the steric effects of the substiuent. 

The Taft equation is, 

log  κs    = ρ*σ* + δ Es

       κCH₃

Here, 

κs − Rate constant for substiuent reaction 

κCH₃ − Rate constant for the reference reaction 

ρ* − polar sensitivity factor 

σ* − polar substiuent constant. 

δ − Sensitivity factor for the reaction to steric effects 

Es − Steric substiuent constant. 

POLAR SUBSTITUENT CONSTANT :- (σ*) 

             

                       It describes how the constituent influence the reaction rate through polar effect. 

Mathematically σ* is defined as follows. 

                             For the hydrolysis of methyl ester through acid and base catalyst reactions. 

σ* =         1       [loɡ ( κs  ) B − loɡ (  κs  ) A]

           2.48ρ*           κCH₃               κCH₃

A - Acid catalyst 

B - Base catalyst 

ρ* - 1 ( for methyl ester hydrolysis) 

  1    =   correction factor introduced to 

2.48 

make σ* similar to the magnitude of Hammet  σ value. 

STERIC SUBSTITUTENT CONSTANT :- (Es) 

                Es has positive (+) value when the size of substiuents is smaller than CH₃ group. 

                    Es has negative (-) value when the size of substiuent is greater than methyl group. 

Mathematically Es is defined as, 

    Es =  1  log (  κs   )

              δ         κCH₃

 POLAR SENSITIVITY FACTOR (ρ*) 

                  It's describes the susceptibility of a reaction it's to polar effects. 

                  The steric effects doesn't influence the reaction rate. 

log (   κs   ) = ρ* σ*

        κCH₃

ρ* can be obtained by following 

  log (  κs   ) Vs σ. The slope is ρ*

           κCH₃

STERIC SENSITIVITY FACTOR (δ) 

                    It describes to what magnitude the reaction rate is influence by steric effects. 

                            When the reaction is not significantly influenced by polar effect Taft equation becomes, 

log   κs     = δ Es

      κCH₃

A plot of log    κs     Vs Es will give 

                         κCH₃

Straight line with slope δ.

(1) If δ is positive (+), steric effects is greater, therefore decreases the reaction rate. 

(2) If δ is negative (-), steric effect is lesser, therefore increases the reaction rate.