Rate Expression for Unimolecular Gas-phase Reaction

R. Manjunath

#16/1, 8th Main Road, Shivanagar, Rajajinagar, Bangalore - 560 010, India

DOI : http://dx.doi.org/10.13005/msri/090213

ABSTRACT:

The inherent goal of this article is to establish a rate equation for unimolecular gas- phase reaction.

KEYWORDS: Unimolecular gas- phase reaction; transition state theory; rate equation; compressibility factor

Copy the following to cite this article: Manjunath R. Rate Expression for Unimolecular Gas-phase Reaction. Mat.Sci.Res.India;9(2)

Copy the following to cite this URL: Manjunath R. Rate Expression for Unimolecular Gas-phase Reaction. Mat.Sci.Res.India;9(2). Available from: http://www.materialsciencejournal.org/?p=1144

Introduction

Any unimolecular gas- phase reaction can in general be represented by the equation

A → Products

This reaction occurs through the following
steps

A + A ↔ A* +A

A* → Products

where:
A represents inactive and A* activated molecules.

The equilibrium constant for the formation of active molecules is given by the expression

K *= e -ΔG*/RT = n*n / n2 = n*/ n

where:

ΔG* is the standard Gibbs free energy of
activation

n* and n represent the number of moles of A and A* respectively

n*= n e -ΔH*/RT e ΔS*/R

where

ΔH* is the standard enthalpy of activation ΔS* is the standard entropy of activation

But

standard enthalpy of activation (ΔH*) is approximately equal to energy of activation (Ea)
and, consequently,

n*= n e –Ea / RT e ΔS*/R

The fraction of molecules activated is given by the expression

n*/n0 = number of moles activated / total number of moles = e -Ea/RT From this it follows that
n0 = n e ΔS*/R

The fraction of gas molecules reacted to form products represents the degree of molecular reactivity and it is denoted by the symbol α α = nr / n0 = number of moles reacted to form products/ total number of moles Now we can write

n r =α n e ΔS*/R

According to transition state theory, the rate of unimolecular gas- phase reaction is given by the expression

ν = υ* n r = (k B T /h) n r

where k B is the Boltzmann constant, n r is the number of activated moles crossing forward to form products and h is the Planck constant. Substituting the value of n r we get

ν = (k B T /h) α n e ΔS*/R

We know that

P V= Z n RT

or
n = P V/ Z RT

where Z is the compressibility factor. Substituting the value of n we get

ν = (P V /Z N h) α e ΔS*/R

“We have thus established the rate expression for unimolecular gas- phase reaction”.

References

1. Laidler, K. J., Chemical Kinetics: “Theories of Reaction Rates”, In, McGraw-Hill Book Company, Inc., New York, 1950
2. Samuel H. Maron and Carl F. Prutton, Principles of physical chemistry: “Kinetics of Homogeneous Reactions”, fourth edition, Oxford & IBH Publishing Co. Pvt. Ltd
3. “Lindemann Mechanism” by W. R. Salzman at the University of Arizona, 2004. Access date 8 December 2007.
4. Properties of Natural Gases. Includes a chart of compressibility factors versus reduced pressure and reduced temperature (on last page of the PDF document).

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