The Arrhenius equation 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.[1] The equation was first proposed by the Dutch chemist J. H. van 't Hoff in 1884; five years later in 1889, the Swedish chemist Svante Arrhenius provided a physical justification and interpretation for it. Nowadays it is best seen as an empirical relationship.[2]

Overview

In short, the Arrhenius equation is an expression that shows the dependence of the rate constant k of chemical reactions on the temperature T (in Kelvin) and activation energy Ea, as shown below:.[3]

.

where A is the pre-exponential factor or simply the prefactor and R is the gas constant. The units of the pre-exponential factor 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. When the activation energy is given in molecular units, instead of molar units, e.g. joules per molecule instead of joules per mol, the Boltzmann constant is used instead of the gas constant. 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 approximate the activation energy as being independent of 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

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

Some authors define a modified Arrhenius equation,[4] that makes explicit the temperature dependence of the pre-exponential factor. If one allows arbitrary temperature dependence of the prefactor, the Arrhenius description becomes overcomplete, and the inverse problem (i.e. determining the prefactor and activation energy from experimental data) becomes singular. 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 T½ dependence of the pre-exponential factor is observed experimentally".[2].. but 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.

Taking the natural logarithm of the Arrhenius equation yields:

.

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):

[edit] Kinetic theories interpretation of Arrhenius equation

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 . The concept of activation energy explains the exponential nature of the relationship, and in one way or another, it is present in all kinetic theories:

[edit] Collision theory

Main article: 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 pre-exponential 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 1930s. This takes various forms, but one of the most common is

where ΔG‡ is the Gibbs free energy of activation, kB 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, which is multiplied by the absolute temperature, as well as an enthalpy term. Both of them 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 involving the partition functions of the reactants and of the activated complex.

The Arrhenius equation

What the various symbols mean

Starting with the easy ones . . .

Temperature, T

To fit into the equation, this has to be meaured in kelvin.

The gas constant, R

This is a constant which comes from an equation, pV=nRT, which relates the pressure, volume and temperature of a particular number of moles of gas. It turns up in all sorts of unlikely places!

Activation energy, EA

This is the minimum energy needed for the reaction to occur. To fit this into the equation, it has to be expressed in joules per mole - not in kJ mol-1.

Note:  If you aren't sure about activation energy, you should read the introductory page on rates of reaction before you go on. Use the BACK button on your browser to return to this page.

And then the rather trickier ones . . .

e

This has a value of 2.71828 . . . and is a mathematical number, a bit like pi. You don't need to worry exactly what it means, although if you have to do calculations with the Arrhenius equation, you may have to find it on your calculator. You should find an ex button - probably on the same key as "ln".

The expression, e-(EA

/ RT)

For reasons that are beyond the scope of any course at this level, this expression counts the fraction of the molecules present in a gas which have energies equal to or in excess of activation energy at a particular temperature. You will find a simple calculation associated with this further down the page.

The frequency factor, A

You may also find this called the pre-exponential factor or the steric factor.

A is a term which includes factors like the frequency of collisions and their orientation. It varies slightly with temperature, although not much. It is often taken as constant across small temperature ranges.

By this time you've probably forgotten what the original Arrhenius equation looked like! Here it is again:

You may also come across it in a different form created by a mathematical operation on the standard one:

"ln" is a form of logarithm. Don't worry about what it means. If you need to use this equation, just find the "ln" button on your calculator.

Using the Arrhenius equation

The effect of a change of temperature

You can use the Arrhenius equation to show the effect of a change of temperature on the rate constant - and therefore on the rate of the reaction. If the rate constant doubles, for example, so also will the rate of the reaction. Look back at the rate equation at the top of this page if you aren't sure why that is.

What happens if you increase the temperature by 10°C from, say, 20°C to 30°C (293 K to 303 K)?

The frequency factor, A, in the equation is approximately constant for such a small temperature change. We need to look at how e-(E

A /

RT) changes - the fraction of molecules with energies equal to or in excess of the activation energy.

Let's assume an activation energy of 50 kJ mol-1. In the equation, we have to write that as 50000 J mol-1. The value of the gas constant, R, is 8.31 J K-1 mol-1.

At 20°C (293 K) the value of the fraction is:

By raising the temperature just a little bit (to 303 K), this increases:

You can see that the fraction of the molecules able to react has almost doubled by increasing the temperature by 10°C. That causes the rate of reaction to almost double. This is the value in the rule-of-thumb often used in simple rate of reaction work.

Note:  This approximation (about the rate of a reaction doubling for a 10 degree rise in temperature) only works for reactions with activation energies of about 50 kJ mol-1 fairly close to room temperature. If you can be bothered, use the equation to find out what happens if you increase the temperature from, say 1000 K to 1010 K. Work out the expression -(EA / RT) and then use the ex button on your calculator to finish the job.

The rate constant goes on increasing as the temperature goes up, but the rate of increase falls off quite rapidly at higher temperatures.

The effect of a catalyst

A catalyst will provide a route for the reaction with a lower activation energy. Suppose in the presence of a catalyst that the activation energy falls to 25 kJ mol-1. Redoing the calculation at 293 K:

If you compare that with the corresponding value where the activation energy was 50 kJ mol-1, you will see that there has been a massive increase in the fraction of the molecules which are able to react. There are almost 30000 times more molecules which can react in the presence of the catalyst compared to having no catalyst (using our assumptions about the activation energies).

It's no wonder catalysts speed up reactions!

Other calculations involving the Arrhenius equation

If you have values for the rate of reaction or for the rate constant at different temperatures, you can use these to work out the activation energy of the reaction. Only one UK A' level Exam Board expects you to be able to do these calculations. They are included in my chemistry calculations book, and I can't repeat the material on this site.

Life Cycle Calculator

Aluminum Electrolytic CapacitorsThe operating conditions directly affect the life of an aluminum electolytic capacitor.  The ambient temperature has the largest effect on life.  The relationship between life and temperature follows a chemical reaction formula called Arrhenius' Law of Chemical Activity.  The law put simply says that life of a capacitor doubles for every 10 degree Celsius decrease in temperature.

Voltage derating also increases the life of a capacitor but to a far lesser extent, as compared to temperature deratings.  Internal heating caused by the applied ripple current, reduces the projected life of an aluminum electrolytic capacitor.

The relationship between capacitor life and the operating conditions is expressed by the following equation:

L1 - Load Life Rating

Vr - Maximum Rated Voltage

Vo - Operating Voltage

Tm - Max.Temp.Rating of Capacitor

TA - Ambient Temp.

 

Film CapacitorsThe operating conditions affect the life of a film capacitor in a very similar manner to aluminum electrolytic capacitors.  Voltage derating has a greater effect on the life as compared to an aluminum electrolytic capacitor.  The life expectancy formula for film

capacitors is expressed by the following equation:

L1 - Load Life Rating

Vr - Maximum Rated VoltageVo - Operating Voltage

Tm - Max.Temp.Rating of Capacitor

Ta - Ambient Temp.

 

Ceramic CapacitorsThe life of a ceramic capacitor at the operating conditions is expressed by the following equation:

L1 - Load Life Rating

Vr - Maximum Rated Voltage

Vo - Operating Voltage

Tm - Max.Temp.Rating of Capacitor

Ta - Ambient Temp. 

Best Answer - Chosen by Asker

*Please read below which may help:

Insulation breakdown, termed as ‘faults’ or ‘shorts’ within this paper, include contamination, arc tracking,

thermal aging and mechanical faults. Each type of fault carries a common factor: The resistive and

capacitive properties of the electrical insulation change.

Contamination, in particular water penetration, increases the insulation conductivity. The water tends to

collect in insulation fractures and inclusions within the insulation system. The electrical fields cause changes

to the contaminants, including expansion, which further break down the insulation system. Other

contaminants, including gasses, vapors, dust, etc., can attack the chemical makeup of the insulation system.

Once the insulation system is completely bridged the system is then considered shorted. This normally will

occur first between conductors, where the insulation system is weakest. Key fault areas include the

nonsecured portion of the coil, such as the end turns of a rotating machine (which also is the highest

electrical stress point of the windings), and the highest mechanical stress point, such as the point the coils

leave the slots of a rotating machine.

Arc tracking of insulation systems occur where high current passes between conductors across the surface

of the insulation system. The insulation at those points carbonize, changing the capacitive and resistive

components of the electrical insulation system. Arc tracking is often the result of: Strong electrical stresses;

Contamination; or, both. This type of fault primarily occurs between conductors or coils and normally ends

with a short.

Thermal aging of an insulation system occurs as electrical insulation systems degrade as a result of the

Arrhenius Chemical Equation. The generally accepted “rule of thumb” is that the thermal life of the insulation

system halves for every 10 C increase in operating temperature. The insulation will quickly degrade and

carbonize once it obtains the temperature limit for the insulation system.

Other environmental factors also impact the thermal life of the insulation system including: Winding

contamination, including oil, grease, dust, etc.; Moisture, in particular contaminated water such as salts, etc;

Electrolysis; and, other electrical stresses.

A newly common electrical stress comes from the application of variable frequency drives. The high carrier

frequency (2.5 to 18 kHz) of modern pulse-width-modulated inverters reduces the partial discharge inception

voltage of the motor insulation system. Partial discharge involves small gas bubbles in the winding insulation

system. A charge builds across the void, then discharges at a level that depends upon the severity and

chemical makeup of the void. The result is ozone, which degrades the surrounding insulation material.

Eventually, an ionized electrical path develops which allows electrical stresses (fast rise-time spikes) to

cross the boundary and short. The tendency is for a few turns to short in the end-turns of the motor

windings.

Mechanical faults in the electrical insulation system include stress cracking, vibration, mechanical incursion,

and mechanical faults. The forces within a coil during various operations, will cause mechanical movement

and may end in the fracturing of insulation materials. Electrical and mechanical vibration cause undue stress

on the insulation system resulting in stress fractures and looseness of the insulation system. Mechanical

incursion includes the movement of materials into the insulation system either between conductors and/or

insulation system to ground. Mechanical faults include failures such as bearing faults that cause the bearing

to come apart and pass through the moving components of the system. These faults may end as shorts

between conductors, coils or coils to ground.