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The Arrhenius Equation
The Arrhenius equation is a formula for the dependence of a reaction rate on temperature:
Learning Objectives

Convert between the exponential and nonexponential forms of the Arrhenius equation

Explain the Arrhenius equation and the meaning of the variables contained within it
Key Points
 The equation relates k, the rate constant for a given chemical reaction, with the temperature, T, the activation energy for the reaction, E_{a}_{ }, the preexponential factor A, and the universal gas constant, R.
 High temperature and low activation energy favor larger rate constants, and therefore speed up the reaction.
 The equation is a combination of the concepts of activation energy and the MaxwellBoltzmann distribution.
Term

Exponential Decay
When a quantity decreases at a rate proportional to its value.
Full Text
The Arrhenius equation is a simple but remarkably accurate formula for the temperature dependence of the reaction rate constant, and therefore, the rate of a chemical reaction. The equation was first proposed by Svante Arrhenius in 1884. Five years later, in 1889, Dutch chemist J. H. van 't Hoff provided physical justification and interpretation for it. The equation combines the concepts of activation energy and the Boltzmann distribution law into one of the most important relationships in physical chemistry:
In this equation, k is the rate constant, T is the absolute temperature, E_{a} is the activation energy, A is the preexponential factor, and R is the universal gas constant.
Take a moment to focus on the meaning of this equation, neglecting the A factor for the time being. First, note that this is another form of the exponential decay law. What is "decaying" here is not the concentration of a reactant as a function of time, but the magnitude of the rate constant as a function of the exponent –Ea /RT.
What is the significance of this quantity? If you recall that RT is the average kinetic energy, it will be apparent that the exponent is just the ratio of the activation energy, E_{a}, to the average kinetic energy. The larger this ratio, the smaller the rate, which is why it includes the negative sign. This means that high temperatures and low activation energies favor larger rate constants, and therefore these conditions will speed up a reaction. Since these terms occur in an exponent, their effects on the rate are quite substantial.
Plotting the Arrhenius Equation in NonExponential Form
The Arrhenius equation can be written in a nonexponential form, which is often more convenient to use and to interpret graphically. Taking the natural logarithms of both sides and separating the exponential and preexponential terms yields:
Note that this equation is of the form
Plot of ln(k) versus 1/T for the decomposition of nitrogen dioxide
The slope of the line is equal to Ea/R.
This affords a simple way of determining the activation energy from values of k observed at different temperatures. We can plot ln(k) versus 1/T, and simply determine the slope to solve for E_{a}.
The PreExponential Factor
Let's look at the preexponential factor A in the Arrhenius equation. Recall that the exponential part of the Arrhenius equation (
If the fraction were unity, the Arrhenius law would reduce to k = A. Therefore, A represents the maximum possible rate constant; it is what the rate constant would be if every collision between any pair of molecules resulted in a chemical reaction. This could only occur if either the activation energy were zero, or if the kinetic energy of all molecules exceeded E_{a}—both of which are highly unlikely scenarios. While "barrierless" reactions, which have zero activation energy, have been observed, these are rare, and even in such cases, molecules will most likely need to collide with the right orientation in order to react. In reallife situations, not every collision between molecules will be an effective collision, and the value of
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Key Term Reference
 activation energy
 Appears in these related concepts: Factors that Affect Reaction Rate, The Collision Theory, and Activation Energy
 chemical reaction
 Appears in these related concepts: Biomolecules, Linear Combination of Atomic Orbitals (LCAO), and Atomic Theory of Matter
 concentration
 Appears in these related concepts: Calculating Equilibrium Concentrations , Molarity, and Diffusion
 decay
 Appears in these related concepts: Radioactive Decay Series: Introduction, Models Using Differential Equations, and Discovery of Radioactivity
 energy
 Appears in these related concepts: Energy Transportation, Surface Tension, and Introduction to Work and Energy
 fraction
 Appears in these related concepts: SI Unit Prefixes, Separable Equations, and Fractions
 gas
 Appears in these related concepts: Osmotic Pressure, Bond Order, and Three States of Matter
 kinetic
 Appears in these related concepts: Friction: Static, The Kinetic Molecular Theory of Matter, and Sculpture
 kinetic energy
 Appears in these related concepts: Inelastic Collisions in Multiple Dimensions, Friction: Kinetic, and Types of Energy
 kinetics
 Appears in these related concepts: Particle in a Box, The de Broglie Wavelength, and Organic Reactions Overview
 logarithm
 Appears in these related concepts: Converting between Exponential and Logarithmic Equations, Special Logarithms, and Odds Ratios
 molecule
 Appears in these related concepts: Molecules, Levels of Organization of Living Things, and Chemical Reactions and Molecules
 ratio
 Appears in these related concepts: Benchmarking, Basic Descriptive Statistics, and Schedules of Reinforcement
 reactant
 Appears in these related concepts: Physical and Chemical Changes to Matter, The Law of Conservation of Mass, and Writing Chemical Equations
 reaction rate
 Appears in these related concepts: ZeroOrder Reactions, The Equilibrium Constant, and Measuring Reaction Rates
 reduce
 Appears in these related concepts: New Energy Sources, Standard Reduction Potentials, and Predicting if a Metal Will Dissolve in Acid
 temperature
 Appears in these related concepts: Viscosity, The Halogens (Group 17), and Temperature
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Source: Boundless. “The Arrhenius Equation.” Boundless Chemistry. Boundless, 21 Jul. 2015. Retrieved 11 Feb. 2016 from https://www.boundless.com/chemistry/textbooks/boundlesschemistrytextbook/chemicalkinetics13/activationenergyandtemperaturedependence100/thearrheniusequation4233686/