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With the failure of the collision theory, Polanyl and Eyring suggested an alternative approach to explain the differences in the rates of similar reactions. The theory is based on following postulate:
1. The reacting molecules with sufficient energy must approach one another and must pass on to an energy barrier,
2. An energy barrier is an unstable, intermediate state where an activated complex is formed which is having higher energy than the reactants and products,
3. The activated complex state also known as Transition State though unstable has a temporary existence and it is in equilibrium with the reactants.
4. At this transition state the reacting molecules are loosely held and rearrangement of valency bonds and energy takes place.
5. The rate of reaction is given by the rate of decomposition of the activated complex and it finally gives the product formation. $$ \underset { \left( Reactants \right) }{ A + B } \leftrightharpoons \underset { \left( Activated \ Complex \right) }{ \left[ { X }^{ * } \right] } \longrightarrow \underset { \left( Products \right) }{ C + D } $$
Applying statistical thermodynamics to this idea, Eyring derived an expression for the specific rate of reaction above. The rate constant (k) for a reaction is given by the equation: $$ k = \frac { RT }{ Nh } { K }^{ * } \qquad ...(1)$$ where R is a gas constant, T is absolute temperature, N is Avogadro number, h is Planck’s constant and \({ K }^{ * }\) is the equilibrium constant for the formation of the activated complex from the reactants.
The relation between free energy change and equilibrium constant is given by $$ \Delta { G }^{ * } = -RT\ln { { K }^{ * } } $$ $$ \therefore \ln { { K }^{ * } } = \frac { -\Delta { G }^{ * } }{ RT } $$ $$ { K }^{ * } = { e }^{ -\Delta { G }^{ * }/RT } \qquad ...(2) $$ where \(\Delta { G }^{ * }\) is free energy change which is also given by equation $$ \Delta { G }^{ * } = \Delta { H }^{ * }-T\Delta { S }^{ * }.$$ Substituting the value of \(\Delta { G }^{ * }\) in equation (2), we get $$ { K }^{ * } = { e }^{ -\left( \frac { \Delta { H }^{ * }-T\Delta { S }^{ * } }{ RT } \right) }$$ $$ { K }^{ * } = { e }^{ \left( \frac { T\Delta { S }^{ * }-\Delta { H }^{ * } }{ RT } \right) }$$ $$ { K }^{ * } = { e }^{ \left( \frac { T\Delta { S }^{ * } }{ RT } - \frac { \Delta { H }^{ * } }{ RT } \right) }$$ $$ { K }^{ * } = { e }^{ \left( \frac { \Delta { S }^{ * } }{ R } - \frac { \Delta { H }^{ * } }{ RT } \right) }$$ $$ { K }^{ * } = { e }^{ \left( \frac { \Delta { S }^{ * } }{ R } \right) } \times { e }^{ \left( \frac { -\Delta { H }^{ * } }{ RT } \right) } \qquad ...(3)$$ where \(\Delta { H }^{ * }\) and \(\Delta { S }^{ * }\) are the enthalpy change and entropy change of the activated complex.
Now, substituting the value of \({ K }^{ * }\) in the first equation, we get $$ k = \frac { RT }{ Nh } \left[ { e }^{ \left( \frac { \Delta { S }^{ * } }{ R } \right) } \times { e }^{ \left( \frac { -\Delta { H }^{ * } }{ RT } \right) } \right] $$ or $$ \boxed { k = \left[ \frac { RT }{ Nh } { e }^{ \left( \frac { \Delta { S }^{ * } }{ R } \right) } \right] { e }^{ \left( \frac { -\Delta { H }^{ * } }{ RT } \right) } } \qquad ...(4)$$ this equation is similar to Arrhenius equation as, $$ \boxed { k = A{ e }^{ \left( \frac { -{ E }_{ a } }{ RT } \right) } } $$ we can equate $$ A = \frac { RT }{ Nh } { e }^{ \left( \frac { \Delta { S }^{ * } }{ R } \right) } \qquad ...(5)$$ where \(\frac { RT }{ Nh }\) is constant at a definite temperature and is independent of the nature of the reactants and products involved in the reaction. Similarly, $$ { e }^{ \left( \frac { -{ E }_{ a } }{ RT } \right) } = { e }^{ \left( \frac { -\Delta { H }^{ * } }{ RT } \right) } $$ which on simplification gives $$ { E }_{ a } = \Delta { H }^{ * } \qquad ...(6)$$ On taking natural log of equation (4), we get $$ \boxed { \ln { k } = \ln { \frac { RT }{ Nh } } + \frac { \Delta { S }^{ * } }{ R } - \frac { \Delta { H }^{ * } }{ RT } } $$ This is the fundamental equation of this theory.
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