Arrhenius equation

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Arrhenius equation

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From Wikipedia, the free encyclopedia

In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates. The equation was proposed by Svante Arrhenius in 1889, based on the work of Dutch chemist Jacobus Henricus van 't Hoff who had noted in 1884 that the van 't Hoff equation for the temperature dependence of equilibrium constants suggests such a formula for the rates of both forward and reverse reactions. This equation has a vast and important application in determining the rate of chemical reactions and for calculation of energy of activation. Arrhenius provided a physical justification and interpretation for the formula.[1][2][3][4] Currently, it is best seen as an empirical relationship.[5]: 188  It can be used to model the temperature variation of diffusion coefficients, population of crystal vacancies, creep rates, and many other thermally-induced processes/reactions. The Eyring equation, developed in 1935, also expresses the relationship between rate and energy.

Equation[edit]

In almost all practical cases, ��≫��

 and k increases rapidly with T.

Mathematically, at very high temperatures so that ��≪��

, k levels off and approaches A as a limit, but this case does not occur under practical conditions.

The Arrhenius equation gives the dependence of the rate constant of a chemical reaction on the absolute temperature as

�=��−�a��,

where

• k is the rate constant (frequency of collisions resulting in a reaction),
• T is the absolute temperature (in Kelvin or degree Rankine),
• A is the pre-exponential factor. Arrhenius originally considered A to be a temperature-independent constant for each chemical reaction.[6] However more recent treatments include some temperature dependence - see Modified Arrhenius equation below.
• Ea is the activation energy for the reaction (in the same units as RT),
• R is the universal gas constant.[1][2][4]

Alternatively, the equation may be expressed as

�=��−�a�B�,

where

• Ea is the activation energy for the reaction (in the same units as kBT),
• kB is the Boltzmann constant.

The only difference is the energy units of Ea: the former form uses energy per mole, which is common in chemistry, while the latter form uses energy per molecule directly, which is common in physics. The different units are accounted for in using either the gas constant, R, or the Boltzmann constant, kB, as the multiplier of temperature T.

The units of the pre-exponential factor A are identical to those of the rate constant and will vary depending on the order of the reaction. If the reaction is first order it has the units: s−1, and for that reason it is often called the frequency factor or attempt frequency of the reaction. Most simply, k is the number of collisions that result in a reaction per second, A is the number of collisions (leading to a reaction or not) per second occurring with the proper orientation to react[7] and �−�a/(��)

 is the probability that any given collision will result in a reaction. 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 of kinetic studies, it is reasonable to approximate the activation energy as being independent of the temperature. Similarly, under a wide range of practical conditions, the weak temperature dependence of the pre-exponential factor is negligible compared to the temperature dependence of the exp⁡(−�a/(��))

 factor; except in the case of "barrierless" diffusion-limited reactions, in which case the pre-exponential factor is dominant and is directly observable.

With this equation it can be roughly estimated that the rate of reaction increases by a factor of about 2 or 3 for every 10°C rise in temperature.

The term �−����

 denotes the fraction of molecules with energy greater than or equal to ��

.[8]

Arrhenius plot[edit]

Main article: Arrhenius plot

Arrhenius linear plot: ln k against 1/T.

Taking the natural logarithm of Arrhenius equation yields:

ln⁡�=ln⁡�−�a�1�.

Rearranging yields:

ln⁡�=−�a�(1�)+ln⁡�.

This has the same form as an equation for a straight line:

�=��+�,

where x is the reciprocal of T.

So, when a reaction has a rate constant that obeys the Arrhenius equation, a plot of ln k versus T−1 gives a straight line, whose gradient 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):

�a≡−�[∂ln⁡�∂(1/�)]�.

Modified Arrhenius equation[edit]

The modified Arrhenius equation[9] makes explicit the temperature dependence of the pre-exponential factor. The modified equation is usually of the form

�=����−�a/(��).

The original Arrhenius expression‎ above corresponds to n = 0. Fitted rate constants typically lie in the range −1 < n < 1. Theoretical analyses yield various predictions for n. It has been pointed out that "it is not feasible to establish, on the basis of temperature studies of the rate constant, whether the predicted T1/2 dependence of the pre-exponential factor is observed experimentally".[5]: 190  However, if additional evidence is available, from theory and/or from experiment (such as density dependence), there is no obstacle to incisive tests of the Arrhenius law.

Another common modification is the stretched exponential form[citation needed]

�=�exp⁡[−(����)�],

where β is a dimensionless number of order 1. This is typically regarded as a purely empirical correction or fudge factor to make the model fit the data, but can have theoretical meaning, for example showing the presence of a range of activation energies or in special cases like the Mott variable range hopping.

Theoretical interpretation of the equation