# Arrhenius equation

The Arrhenius equation predicts the rate of a chemical reaction at a certain temperature, given the activation energy and chance of successful collision of molecules. It is named after Swedish scientist Svante Arrhenius.

The average amount of thermal energy that molecules possess at a certain temperature is equal to RT, where R is the molar gas constant. The fraction of molecules that have enough energy to overcome the energy barrier—those with energy over the activation energy, E (joule)— depends exponentially on the ratio of the activation to thermal energy. This forms the Arrhenius equation:

[itex]k=Ae^{\frac{-E}{RT}}[itex]

where

k is the rate constant for the reaction
the A factor or the frequency factor A is a constant specific to each reaction that depends on the chance the molecules will collide in the correct orientation.

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.

A more general form of Arrhenius equation is

[itex]k = A'e^{-\frac{\Delta G^\Dagger}{RT}}[itex]

where A' can be determined with statistical mechanics and it depends on the shape of the energy surface of the complex (the dissolved ions). ΔG (joule) is the free energy. For simple cases,

[itex]k = \kappa\frac{k_BT}{h}e^{-\frac{\Delta G^\Dagger}{RT}}[itex]

where κ is the transmission coefficient (a value between zero and unity); [itex]k_B[itex] is Boltzmanns constant; T is the temperature; h is Plancks constant.

Taking the natural logarithm of the Arrhenius equation yields

[itex]\ln(k)= \frac{-E}{R}\frac{1}{T} + \ln(A).[itex]

The modified Arrhenius equation yields an equation of the form y=mx+b, where

[itex]y = \ln(k) \,[itex]
[itex]m = \frac{-E}{R}[itex]
[itex]x = \frac{1}{T}[itex]
[itex]b = \ln(A). \,[itex]

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. Slope and intercept can be used to determine E and A characteristics.

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