# Arrhenius equation

The Arrhenius equation , named after Svante Arrhenius , approximately describes a quantitative temperature dependence in physical and, above all, chemical processes in which an activation energy has to be overcome at the molecular level . The Arrhenius equation describes a phenomenological relationship and applies to a great many chemical reactions. The Arrhenius equation is related to the Eyring equation , which is a connection between the microscopic interpretation.

## Arrhenius equation in chemical reaction kinetics

Arrhenius graph : decay of NO 2
according to first-order kinetics

In chemical kinetics, the Arrhenius equation describes the quantitative dependence of the reaction rate constants on the temperature for the special case of monomolecular reactions : ${\ displaystyle k}$

${\ displaystyle k = A \ cdot \ mathrm {e} ^ {- {\ frac {E _ {\ mathrm {A}}} {R \ cdot T}}}}$

With

• ${\ displaystyle A}$Pre-exponential or frequency factor , according to the impact theory corresponds to the product of the impact number  Z and the orientation  factor P :${\ displaystyle A = Z \ cdot P}$
• ${\ displaystyle E _ {\ mathrm {A}}}$ Activation energy (unit: J mol −1 ),
• ${\ displaystyle R}$ universal gas constant (8.314 J K −1 mol −1 ),
• ${\ displaystyle T}$absolute (thermodynamic) temperature (unit: K ).

The Arrhenius graph is a reciprocal representation in which the logarithmic rate constant is plotted against the reciprocal value of the temperature ( see Fig. ):

${\ displaystyle \ Leftrightarrow \ ln (k / A) = - {\ frac {E _ {\ mathrm {A}}} {R}} \ cdot {\ frac {1} {T}} = f \ left ({\ frac {1} {T}} \ right)}$

### Temperature dependence of the frequency factor

However, the Arrhenius equation is not exactly valid, because it is also temperature-dependent and often the law ${\ displaystyle A}$

${\ displaystyle A = u ^ {*} \ cdot {\ sqrt {T}}}$

follows. Thus the pre-exponential factor also increases slightly with increasing temperature ( root function ). However, its temperature dependence is significantly less than that of the exponential term. In this case a modified Arrhenius equation can be used:

${\ displaystyle k = B \ cdot T ^ {n} \ cdot \ mathrm {e} ^ {- {\ frac {E _ {\ mathrm {A}}} {R \ cdot T}}}}$

With the exponent combined to form the Arrhenius number ${\ displaystyle \ gamma}$

${\ displaystyle \ gamma = {\ frac {E _ {\ mathrm {A}}} {R \ cdot T}}}$

the Arrhenius equation is also represented as follows:

${\ displaystyle k = B \ cdot T ^ {n} \ cdot \ mathrm {e} ^ {- \ gamma}}$

## Arrhenius equation in other processes

The temperature dependence of the viscosity of liquids , the charge carrier density with intrinsic conduction in semiconductors and the diffusion coefficient in solids is also described by an Arrhenius equation.

## Calculation of the activation energy

By measuring two rate constants , and two temperatures of the same reaction, the activation energy can be calculated by setting up the Arrhenius equation for the two measurements as follows (assuming that A does not depend on temperature): ${\ displaystyle k_ {1}}$${\ displaystyle k_ {2}}$${\ displaystyle T_ {1}, T_ {2}}$

{\ displaystyle {\ begin {aligned} (1) && k_ {1} & = A \ cdot \ mathrm {e} ^ {\ frac {-E_ {A}} {RT_ {1}}} \;, \\ ( 2) && k_ {2} & = A \ cdot \ mathrm {e} ^ {\ frac {-E_ {A}} {RT_ {2}}} \;. \ End {aligned}}}
${\ displaystyle {\ frac {k_ {2}} {k_ {1}}} = {\ frac {{\ bcancel {A}} \ cdot \ mathrm {e} ^ {\ frac {-E_ {A}} { R \ cdot T_ {2}}}} {{\ bcancel {A}} \ cdot \ mathrm {e} ^ {\ frac {-E_ {A}} {R \ cdot T_ {1}}}}} = \ mathrm {e} ^ {- {\ frac {E_ {A}} {R \ cdot T_ {2}}} + {\ frac {E_ {A}} {R \ cdot T_ {1}}}}}$

Taking the natural logarithm and introducing a main denominator yields:

${\ displaystyle \ ln \ left ({\ frac {k_ {2}} {k_ {1}}} \ right) = - {\ frac {E_ {A}} {R \ cdot T_ {2}}} + { \ frac {E_ {A}} {R \ cdot T_ {1}}} = {\ frac {E_ {A}} {R}} \ cdot {\ frac {T_ {2} -T_ {1}} {T_ {1} \ cdot T_ {2}}} \ ;.}$

Changing after finally results in: ${\ displaystyle E_ {A}}$

${\ displaystyle E_ {A} = R \ cdot \ ln \ left ({\ frac {k_ {2}} {k_ {1}}} \ right) \ cdot {\ frac {T_ {1} \ cdot T_ {2 }} {T_ {2} -T_ {1}}} \ ;.}$

An increase in temperature leads to an increase in the reaction rate. A rule of thumb , the so-called reaction rate temperature rule (RGT rule) , predicts a doubling to quadrupling of the reaction rate when the temperature rises . The factor by which the reaction rate changes with a temperature increase of 10 K is referred to as the Q 10 value. ${\ displaystyle T_ {2} -T_ {1} = 10 \ \ mathrm {K}}$

For a -fold higher reaction speed, the following applies: ${\ displaystyle {\ boldsymbol {n}}}$

${\ displaystyle {\ boldsymbol {n}} = {\ frac {k_ {2}} {k_ {1}}}}$

and thus:

${\ displaystyle E_ {A} = R \ cdot \ ln ({\ boldsymbol {n}}) \ cdot {\ frac {T_ {1} \ cdot T_ {2}} {T_ {2} -T_ {1}} } \ ;.}$

## Individual evidence

1. ^ Jacobus Henricus van't Hoff: Études de dynamique chimique. Frederik Muller & Co., Amsterdam 1884, pp. 114-118.
2. ^ Svante Arrhenius, Z. Phys. Chem. 1889, 4 , pp. 226-248.
3. Entry on Arrhenius equation . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.A00446 Version: 2.3.1.
4. Entry on modified Arrhenius equation . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.M03963 Version: 2.3.1.
5. ^ Charles E. Mortimer, Ulrich Müller, Chemistry - The basic knowledge of chemistry, page, 11th edition, ISBN 978-3-13-484311-8 , page 266f
6. M. Binnewies, Allgemeine und Anorganische Chemie, 1st edition 2004. ISBN 3-8274-0208-5 , pp. 299f.