Transition State Theory - Physical Chemistry | CHEM 3520, Study notes of Physical Chemistry

Download Transition State Theory - Physical Chemistry | CHEM 3520 and more Study notes Physical Chemistry in PDF only on Docsity! CHEM 3520 Spring 2004 Transition State Theory Transition state theory also called activated-complex theory is a theory of the rate of elementary reactions. The theory focuses on transient species (called activated complex or transition state) locate din the vicinity of the top of the barrier height (activation energy) of a reaction. En er gy Reaction coordinate Products A+B AB‡ En er gy – Consider an elementary reaction: Products BA ⇒+ – The rate law is given by: ]B][A[]P[ k dt dv == – According to the activated complex theory, the model of the reaction is a two-step process: PABBA ‡ →→←+ where AB is the activated complex ‡ – The transition state (activated complex) quantities are denoted by a double dagger sign, ‡ (not ≠). – Another important assumption is that the reactants and the activated complex are in equilibrium with each other, and the equilibrium constant for this equilibrium is: ]B][A[ ]AB[ /]B[/]A[ /]AB[ ‡‡‡ o oo o c cc cKc == where c is the standard-state concentration (this is often 1 ) o 3mol/dm00. – The activated complexes are assumed to be stable within a small region of width δ center at the barrier top. According to the transition-state theory, the rate of the reaction is given by the rate of a unimolecular process, with a rate constant , that transform the activated complex into products: ‡k ][AB][ ‡‡k dt Pd = – The rate constant is proportional to the frequency with which the activated complexes cross over the barrier top ( ‡k cν ), where the proportionality constant (κ ) is called transmission coefficient (and is assumed to be 1 if no other information is available): cc ‡ νκν ≅=k – Because [ ⇒ ocKc /]B][A[]AB ‡‡ = ]B][A[/]B][A[][ ‡ c‡ c o o c K cK dt Pd c c ν ν == CHEM 3520 Spring 2004 – Compare to ]B][A[]P[ k dt d ==v ⇒ oc Kc ‡ cυ=k (k has units of ) 11 sconc −− ⋅ – The equilibrium constant written in terms of partition functions: )/)(/( )/( ]B][A[ ]AB[ BA ‡‡ ‡ VqVq cVqcKc oo == where q , q , and q are the partition functions of A, B and . A B ‡ ‡AB – Assume that the motion of the reacting system over the barrier top is a one-dimensional translational motion. – The translational partition function, , for this one-dimensional motion: transq δπ h Tkmq 2/1 B ‡ trans )2( = ; ( is the mass of activated complex) ‡m – Write the partition function of the activated complex as a product of the partition function of the motion of passing over the energy barrier and a partition function for the rest of degrees of freedom of the activated complex: ‡inttrans ‡ qqq = )/)(/( )/()2( BA ‡ int 2/1 B ‡ ‡ VqVq cVq h TkmKc o δ π = ⇒ )/)(/( )/()2( BA ‡ int 2/1 B ‡ c VqVq cVq hc Tkmk o o δ π ν= – The quantities cν and δ are hard to determine directly but the product of them is a speed to which activated complexes crosses the barrier for which we consider the average speed cac δν>=< u . Use the Maxwell-Boltzmann distribution to calculate the average speed: 2/1 ‡ B 0 2/ B ‡ 0 ac 22 )( B 2‡      =        =>=< ∫∫ ∞ − ∞ m Tkduue Tk mduuufu Tkum ππ – The transition-state theory expression for the rate constant became: ⇒ ‡B BA ‡ intB )/)(/( )/( K hc Tk VqVq cVq hc Tkk o o o == where ‡K is the equilibrium constant for the formation of the transition state from the reactants but with one degree of freedom (the motion along the reaction coordinate) excluded from the activated complex partition function.