Name: determining activation energy from temperature
Uploaded: 2020-09-24T15:06:50.000Z
Duration: 2 min 34 s
Channel: JoVE

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Arrhenius Plots

Die Geschwindigkeit einer chemischen Reaktion reagiert sehr empfindlich auf Temperaturänderungen. Diese Temperaturabhängigkeit wird mathematisch mit Hilfe der Arrhenius-Gleichung erklärt; , die die Beziehung zwischen der Geschwindigkeitskonstante, der absoluten Temperatur, dem Frequenzfaktor und der Aktivierungsenergie ausdrückt.
Die Aktivierungsenergie und der Frequenzfaktor können auch grafisch bestimmt werden, indem die Arrhenius-Gleichung in eine nicht-exponentielle Form umgewandelt wird.
Unter Verwendung der natürlichen Logarithmen auf beiden Seiten wird eine Gleichung für eine lineare Funktion generiert. Der Steigungswert entspricht dem negativen Wert der Aktivierungsenergie gegenüber der Gaskonstante, und der y-Achsenabschnitt entspricht dem natürlichen Logarithmus des Frequenzfaktors.
Diese Gleichung kann verwendet werden, um ein Diagramm zu erzeugen, das als Arrhenius-Diagramm bezeichnet wird, in dem der natürliche Logarithmus der Geschwindigkeitskonstante als Funktion des Kehrwerts der Temperatur in Kelvin bezeichnet wird.
Kinetische Daten von Experimenten und Reaktionen können mit diesem Arrhenius-Diagramm illustriert und analysiert werden. In diesem Beispiel ergibt das Diagramm eine gerade Linie. Der in Kelvin angegebene Steigungswert wird gleich dem negativen Wert der Aktivierungsenergie über R gesetzt. Nach Zuweisung des Wertes für die Gaskonstante und Auflösen der Aktivierungsenergie erhält man einen Wert von 93,1 kJ/mol.
Außerdem ist der y-Achsenabschnitt von 26,8 gleich dem natürlichen Logarithmus des Frequenzfaktors. Das Auflösen nach A ergibt also den Wert von 4,36 × 1011 mit der Einheit eins über Molarität-Sekunden – die gleiche Einheit wie die Geschwindigkeitskonstante.
Bei begrenzten kinetischen Daten oder Schwierigkeiten mit der grafischen Darstellung kann eine Zwei-Punkt-Form der Arrhenius-Gleichung verwendet werden, um die Aktivierungsenergie auf nicht-grafische Weise zu berechnen.
In solchen Fällen wird die nicht-exponentielle Form der Arrhenius-Gleichung so modifiziert, dass sie Geschwindigkeitskonstanten bei zwei verschiedenen Temperaturen enthält.
Die anschließende Subtraktion und Neuanordnung des Ausdrucks ergibt die Zwei-Punkt-Form der Arrhenius-Gleichung, die zur Berechnung der Aktivierungsenergie aus experimentell erzeugten Geschwindigkeitskonstanten bei zwei verschiedenen Temperaturen verwendet wird. Durch Ersetzen der Werte wird die Aktivierungsenergie für diese Reaktion mit 145 kJ/mol berechnet.

arrhenius-plots

Die Arrhenius-Gleichung setzt die Aktivierungsenergie und die Geschwindigkeitskonstante k für chemische Reaktionen in Beziehung. In der Arrhenius-Gleichung ist k = Ae−Ea/RT, R ist die ideale Gaskonstante, die einen Wert von 8,314 J/mol·K hat, T ist die Temperatur auf der Kelvin-Skala, Ea ist die Aktivierungsenergie in J/mol , e ist die Konstante 2,7183 und A ist eine Konstante namens Frequenzfaktor, die mit der Häufigkeit von Kollisionen und der Ausrichtung der reagierenden Moleküle zusammenhängt. Die Arrhenius-Gleichung kann verwendet werden, um die Aktivierungsenergie einer Reaktion aus experimentellen kinetischen Daten zu berechnen. Ein bequemer Ansatz zur Bestimmung des Ea für eine Reaktion besteht in der Messung von k bei zwei oder mehr verschiedenen Temperaturen. Es verwendet eine modifizierte Version der Arrhenius-Gleichung, die die Form einer linearen Gleichung annimmt:  <img alt="Eq1" src="/files/ftp_upload/11380/11380_Equation_1.png" style="height: 200px;"> Eine Auftragung von ln k gegen 1/T ist linear mit einer Steigung gleich −Ea/R und einem y-Achsenabschnitt gleich ln A. Betrachten Sie die folgende Reaktion:  <img alt="Eq2" src="/files/ftp_upload/11380/11380_Equation_2.png" style="height: 50px;"> Die Aktivierungsenergie dieser Reaktion kann bestimmt werden, wenn die Variation der Geschwindigkeitskonstante mit der Temperatur aus den reaktionskinetischen Daten bekannt ist, wie gezeigt. <table style="margin-left: auto; margin-right: auto;"> <tbody> <tr> <td style="color: black; border: 1px solid black; text-align: center;">Temperature (K)</td> <td style="color: black; border: 1px solid black; text-align: center;">Rate constant (L/mol/s)</td> </tr> <tr> <td style="color: black; border: 1px solid black; text-align: center;">555</td> <td style="color: black; border: 1px solid black; text-align: center;">3.52 × 10–7</td> </tr> <tr> <td style="color: black; border: 1px solid black; text-align: center;">575</td> <td style="color: black; border: 1px solid black; text-align: center;">1.22 × 10–6</td> </tr> <tr> <td style="color: black; border: 1px solid black; text-align: center;">645</td> <td style="color: black; border: 1px solid black; text-align: center;">8.59 × 10–5</td> </tr> <tr> <td style="color: black; border: 1px solid black; text-align: center;">700</td> <td style="color: black; border: 1px solid black; text-align: center;">1.16 × 10–3</td> </tr> <tr> <td style="color: black; border: 1px solid black; text-align: center;">781</td> <td style="color: black; border: 1px solid black; text-align: center;">3.95 × 10–2</td> </tr> </tbody> </table> Die bereitgestellten Daten können verwendet werden, um die Werte des Kehrwerts der Temperatur (1/T) und des natürlichen Logarithmus von k (ln k) abzuleiten. <table style="margin-left: auto; margin-right: auto;"> <tbody> <tr> <td style="color: black; border: 1px solid black; text-align: center;">1/T (K–1)</td> <td style="color: black; border: 1px solid black; text-align: center;">ln k</td> </tr> <tr> <td style="color: black; border: 1px solid black; text-align: center;">   1.80 × 10–3    </td> <td style="color: black; border: 1px solid black; text-align: center;">     –14.860     </td> </tr> <tr> <td style="color: black; border: 1px solid black; text-align: center;">1.74 × 10–3</td> <td style="color: black; border: 1px solid black; text-align: center;">–13.617</td> </tr> <tr> <td style="color: black; border: 1px solid black; text-align: center;">1.55 × 10–3</td> <td style="color: black; border: 1px solid black; text-align: center;">–9.362</td> </tr> <tr> <td style="color: black; border: 1px solid black; text-align: center;">1.43 × 10–3</td> <td style="color: black; border: 1px solid black; text-align: center;">–6.759</td> </tr> <tr> <td style="color: black; border: 1px solid black; text-align: center;">1.28 × 10–3</td> <td style="color: black; border: 1px solid black; text-align: center;">–3.231</td> </tr> </tbody> </table>  <img alt="Image1" src="/files/ftp_upload/11380/11380_Image 1.png" style="height: 350px;"> Beim Auftragen der abgeleiteten Datenpunkte mit ln k gegen 1/T wird ein Liniendiagramm erzeugt, das eine lineare Beziehung zwischen ln k und 1/T zeigt, wie gezeigt. Die Steigung der Linie, die der Aktivierungsenergie entspricht, kann anhand von zwei beliebigen experimentellen Datenpaaren geschätzt werden.  <img alt="Eq3" src="/files/ftp_upload/11380/11380_Equation_3.png" style="height: 350px;"> Ein alternativer Ansatz zur Ableitung der Aktivierungsenergie beinhaltet die Nutzung der Geschwindigkeitskonstanten bei zwei verschiedenen Temperaturen. Bei diesem Ansatz wird die Arrhenius-Gleichung in eine praktische Zweipunktform umgestellt:  <img alt="Eq4" src="/files/ftp_upload/11380/11380_Equation_4v2.png" style="height: 75px;"> Durch Umstellen der Gleichung entsteht ein Ausdruck für die Aktivierungsenergie.  <img alt="Eq5" src="/files/ftp_upload/11380/11380_Equation_5.png" style="height: 75px;"> Durch Ersetzen zweier beliebiger Datenpaare und weiterer Berechnung erhält man den Wert für die Aktivierungsenergie in Joule pro Mol oder Kilojoule pro Mol.  <img alt="Eq6" src="/files/ftp_upload/11380/11380_Equation_6.png" style="height: 135px;"> Dieser alternative Zwei-Punkte-Ansatz liefert das gleiche Ergebnis wie der grafische Ansatz. In der Praxis liefert der grafische Ansatz jedoch typischerweise zuverlässigere Ergebnisse, wenn mit tatsächlichen experimentellen Daten gearbeitet wird. Dieser Text ist adaptiert von <a href="https://openstax.org/books/chemistry-2e/pages/12-5-collision-theory">Openstax, Chemistry 2e, Section 12.5: Collision Theory</a>

Arrhenius-Plots

Die Arrhenius-Gleichung setzt die Aktivierungsenergie und die Geschwindigkeitskonstante k für chemische Reaktionen in Beziehung. In der Arrhenius-Gleichung ist k = Ae−Ea/RT, R ist ...

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determining activation energy from temperature

The Arrhenius equation links activation energy and the rate constant, k, for chemical reactions. It is written as k = Ae−Ea/RT. In this equation, R is the ideal gas constant with a value of 8.314 J/mol·K, T is the temperature on the kelvin scale, Ea is the activation energy in J/mole, e is the constant 2.7183, and A is the frequency factor. The frequency factor depends on collision frequency and the orientation of reacting molecules.

Experimental kinetic data can be used to find the activation energy of a reaction. One practical method is to measure k at two or more temperatures. The Arrhenius equation can then be rewritten in a linear form. A plot of ln k versus 1/T gives a straight line. The slope equals −Ea/R, and the y-intercept equals ln A.

For example, a reaction with known rate constants at 555 K, 575 K, 645 K, 700 K, and 781 K can be analyzed this way. The rate constants are 3.52 × 10−7, 1.22 × 10−6, 8.59 × 10−5, 1.16 × 10−3, and 3.95 × 10−2 L/mol/s, respectively. These values can be converted to 1/T and ln k. The resulting data set can then be plotted to show the linear relationship between ln k and 1/T.

The slope of the line gives the activation energy. It can be estimated from any two experimental data pairs on the graph. Another option is the two-point form of the Arrhenius equation. This rearranged equation uses rate constants at two temperatures to calculate Ea directly. Both methods give the same result. In practice, the graphical method is often more reliable when working with real experimental data.

Determining Activation Energy from Temperature

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arrhenius plots

The Arrhenius equation links activation energy and the rate constant, k, for chemical reactions. It is written as k = Ae−Ea/RT. In this equation, R is the ideal gas constant with a value of 8.314 J/mol·K, T is the temperature on the kelvin scale, Ea is the activation energy in J/mole, e is the constant 2.7183, and A is the frequency factor. The frequency factor depends on collision frequency and the orientation of reacting molecules. Experimental kinetic data can be used to find the activation...

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Learn how the Arrhenius equation uses temperature and rate constants to find activation energy with a graph or a two-point calculation.

Determining Activation Energy from Temperature | JoVE K12

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Determining Activation Energy from Temperature

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