User:Knights who say ni/sandbox

From Wikipedia, the free encyclopedia

The Arrhenius equation[1] is a simple, but remarkably accurate, formula for the temperature dependence of a chemical reaction rate, more correctly, of a rate coefficient, as this coefficient includes all magnitudes that affect reaction rate except for concentration. The equation was first proposed by the Dutch chemist J. H. van't Hoff in 1884; five years later, the Swedish chemist Svante Arrhenius provided a physical justification and interpretation for it. Nowadays it is best seen as an empirical relationship [2]. It states that:

k = A e^{\frac{-E_a}{RT}}.



It can be seen that either increasing the temperature or decreasing the activation energy (for example through the use of catalysts) will result in an increase in rate of reaction.

Given the small temperature range in which kinetic studies are carried, it is reasonable to assume that the preexponential factor and the activation energy are not temperature dependant.

Taking the natural logarithm of the Arrhenius equation yields:

\ln(k)= \frac{-E_a}{R}\frac{1}{T} + \ln(A).

So, when a reaction has a rate constant which obeys the Arrhenius equation, a plot of ln(k) versus T -1 gives a straight line, whose slope and intercept can be used to determine Ea and A. This procedure has become so common in experimental chemical kinetics that practitioners have taken to using it to define the activation energy for a reaction. That is the activation energy is defined to be (-R) times the slope of a plot of ln(k) vs. (1/T):

\ E_a \equiv -R \left( \frac{\partial ~ln k}{\partial ~(1/T)} \right)_P

Although some authors state that "it is not feasible to stablish, on the basis of temperature estudies of the rate constant, wether the predicted T½ dependence of the preexponential factor is observed experimentally"[2]. Some others, define a modified Arrhenius equation,[4] that corrects the temperature dependence of the preexponential factor


Contents

[edit] Kinetic theories

Arrhenius argued that in order for reactants to be transformed into products, they first needed to acquire a minimum amount of energy, called the activation energy Ea. At an absolute temperature T, the fraction of molecules that have a kinetic energy greater than Ea can be calculated from the Maxwell-Boltzmann distribution of statistical mechanics, and turns out to be proportional to \ e^{\frac{-E_a}{RT}}. The concept of activation energy explains the exponential nature of the relationship, in one way or another, it is present in all kinetic theories.

The preexponential factor or A factor is a constant specific to a particular reaction and if the reaction is first order it has the units s-1, for that reason it is often called frequency factor.

[edit] Collision theory

One example comes from the "collision theory" of chemical reactions, developed by Max Trautz and William Lewis in the years 1916-18. In this theory, molecules are supposed to react if they collide with a relative kinetic energy along their line-of-centers that exceeds Ea This leads to an expression very similar to the Arrhenius equation, with the difference that the preexponential factor "A" is not constant but instead is proportional to the square root of temperature. This reflects the fact that the overall rate of all collisions, reactive or not, is proportional to the average molecular speed which in turn is proportional to T1/2. In practice, the square root temperature dependence of the preexponential factor is usually very slow compared to the exponential dependence associated with Ea, to the point that some think it can not be experimentally proven.


[edit] Transition state theory

Another Arrhenius-like expression appears in the Transition State Theory of chemical reactions, formulated by Wigner, Eyring, Polanyi and Evans in the 1930's. This takes various forms, but one of the most common is:

\ k = \frac{k_BT}{h}e^{-\frac{\Delta G^\Dagger}{RT}}

where ΔG‡ is the Gibbs free energy of activation, \ k_B is Boltzmann's constant, and h is Planck's constant.

At first sight this looks like an exponential multiplied by a factor that is linear in temperature. However, one must remember that free energy is itself a temperature dependent quantity. The free energy of activation includes an entropy term as well as an enthalpy term, both of which depend on temperature, and when all of the details are worked out one ends up with an expression that again takes the form of an Arrhenius exponential multiplied by a slowly varying function of T. The precise form of the temperature dependence depends upon the reaction, and can be calculated using formulas from statistical mechanics (it involves the partition functions of the reactants and of the activated complex)


[edit] Notes and references

  1. ^ IUPAC Goldbook definition of Arrhenius equation
  2. ^ a b Kenneth Connors, Chemical Kinetics, 1990, VCH Publishers
  3. ^ IUPAC Goldbook definition of Arrhenius activation energy
  4. ^ IUPAC Goldbook definition of modified Arrhenius equation
  • Laidler, K. J. (1997) Chemical Kinetics,Third Edition, Benjamin-Cummings
  • Laidler, K. J. (1993) The World of Physical Chemistry, Oxford University Press
Retrieved from "http://en.wikipedia.org../../../k/n/i/User%7EKnights_who_say_ni_sandbox_ab58.html"