Transition State Theory

Transition state theory is also called as "Activated-Complex Theory" or "Theory Of Absolute Rate Of Reactions" or "Absolute Rate Theory". 

This theory was proposed by Henry Erying in 1935 and further modification is done by Merrideth G. Evans and Michael Polanyi. This theory is used (or provide) as an alternative of the "Collision Theory and Arrhenius Law". Since, it is used as an alternative to the Collision theory or Arrhenius law, this theory provides a greater understanding of the activation energy and thermodynamics properties. 

.........."The basic assumption of Transition-State theory is the existence of activated complex in which chemical bonds are partially broken and partially formed"........

        

Transition State Theory (or TST) tells us that there exists an equilibrium state between the state where all molecules are reactants and the state where all molecules are products, known as the "Transition State". 

In this equilibrium state, all the reactant molecules are combined into a new form of molecule, called as "Activated Complex". This transition state of molecule is the state at which the energy of the reactant molecule is higher in the Reaction-energy diagram. 

Fig:- Recation-Energy Diagram 

Activated complexes are highly unstable in nature and is held together by loose bonds (& have short lifetime or finite lifetime) and due to their instability, they decompose easily into products to achieve stability. Activated complex and the reactant molecules are in chemical equilibrium.                      

☛ Based on three major factors, this theory also tells us whether a reaction will occur even if there is a collision between the reacting molecules. These factors are;

1. Concentration of activated complex
2. The rate of decomposition of activated complex 
3. The way of decomposition of an activated- complex, whether it is decomposed in products or decomposed into reactants. 

Thus, we can say that the rate of the reaction depends on the number of decompositions of the activeted complex to form the product. The greater the number of decompositions of the activeted complexes, the higher the rate of the reaction. And the lower the number of decompositions of the activeted complexes, the lower the rate of the reaction.

How A Chemical Reaction Takes Place According To TST   

Let's take a picture to understand how a chemical reaction take place according to transition state theory;

Fig:- Reaction Energy Diagram

In the above figure, the energy of the reactant molecules is taken on the Y-axis, while the distance along reaction path (or reaction co-ordinate) is taken on the X-axis. Overview of the above figure, 

★Activation energy - The minimum amount of energy that must possess by the reacting molecules. 

★∆Hᵣ - Represent heat of the reaction which is define as the difference in energies of reactant and product molecules. 

★Transition state - Represent the highest energy level on the reaction energy diagram. 

As in the above figure, first the reactant molecule takes the activation energy and reaches the higher energy level i.e. the transition state, where the reactant molecule forms active complexes. And further these active complexes decompose into product molecules. Thus, a chemical reaction proceeds.

Derived Transition State Theory Formula (or relation)

Let us see how the transition state theory describes whether or not a chemical reaction will occur. Consider a chemical reaction;

                  A  +  B   ➝  AB

                             -------------------(1) 

But the actual reaction is taking place in the following manner, 

               A  +  B  ⇌  AB*  ➝  AB 

                                     ---------------------(2) 

Where,   

    AB* represent the activated complex which is highly unstable and easily decompose into product or may be convert back into reactant. Here, the rate constant for the formation of activated complex (or forward reaction) is 'k₁' and the rate constant for backward reaction is 'k₂' and the rate constant for the decomposition of  "AB* into AB" is 'k₃'.

The rate of formation of activated complex is faster as compare to the rate of decomposition of activated complex into products therefore the rate of decomposition of the activated complex is the rate determining step which controls the rate of reaction. Hence, 

➩ Rate, r = k₃[AB*]

---------------------(3) 

Here, the concentration of activated complex [AB*] is not measurable so we are replacing this by a measurable concentration terms. 

According to the transition state theory, 

               ➩ k₃ = kT/h     ---------------(4) 

   Where, 

         k- Represent the "Boltzmann Constant"

         h- Represent the "Plank's Constant"

As we know that for a case of reversible reaction, we define an equilibrium constant which is nothing but the ratio of rate constants of forward and backward reactions and represented by symbol 'Kc' or 'Keq'. Kc  generally measured in "Moles Per Liter". 

Consider a reaction;

                       aA + bB ⇌ cC + dD

The equilibrium constant for the above reaction is given as, 

     ➩ 'Kc' or 'Keq' = kf/kb = [C]ᶜ[D]ᵈ / [A]ᵃ[B]ᵇ

Thus, the equilibrium constant for a chemical reaction given by the equation (1) can be written as, 

➩ 'Kc' or 'Keq' = k₁/k₂ = [AB*]/[A][B]

➩    [AB*] = Kc[A][B]

--------------------(5) 

Now substituting the value from equation (5) in equation (3) we get, 

➩ Rate, r = k₃[AB*] = (kT/h) (Kc.[A].[B]) 

-----------------(6) 

From thermodynamics, we know that Gibb's Helmholtz equation, 

➩  ∆G* = ∆H* - T ∆S* = - RT ln(Kc) 

➩   ∆H* - T ∆S* = - RT ln(Kc) 

➩   Kc = exp{(-∆H*/RT)+(∆S*/RT)}

------------------(7) 

Therefore the equation (6) becomes;

➩ r = (kT/h)×exp{(-∆H*/RT)+(∆S*/RT)}×[A]×[B]

-----------------(8) 

Here, the term '∆S*/RT' shows less temperature dependency that means entropy of a system is unaffected by the temperature. 

Therefore we can say that;

➩ Rate constant, k ∝ T exp(-∆H*/RT) 

----------------(9)

Equation (9) is not the result because it contains a non-measurable term '∆H*'. So, we have to eliminate the term '∆H*' by using an empirical relationship. As we know that, For a case of liquids,

➩ Activation energy,  Eₐ = ∆H* - RT

And, Chemical reaction equilibrium says that, 

➩ ∆H* = ∆H₁(forward) - ∆H₂(backward) 

--------------------(10) 

For forward and backward reactions, 

➩ k ∝ T₁ exp(-∆H₁*/RT) ∝ T₁ exp(-E-RT/RT)

&

➩ k ∝ T₂ exp(-∆H₂*/RT) ∝ T₂ exp(-E-RT/RT)

In general we can say that, 

➩ k ∝ T exp{(-Eₐ- RT)/RT} 

➩ k ∝ T exp{(-Eₐ /RT)-1} 

➩ k ∝ T exp{-Eₐ/RT}.exp(-1) 

(∵ exp(-1)=1/e, is a constant value) 

➩ [ k ∝ T¹.exp(-Eₐ/RT) ]

------------------(11) 

Equation (11) is a consequence of the transition state theory. This theory shows the exponential relationship between the rate constant, activation energy and the temperature at which a reaction occurs. 

This theory tells us that the rate constant is directly depend on the first power of temperature at which a reaction is occurred and also depend exponentially on the activation energy. 

Hope you have found this article helpful!!

Let me know what you think about TRANSITION STATE THEORY. Feel free to comment if you have any queries.!!mm