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

The rate of a chemical reaction is highly sensitive to changes in temperature. This temperature dependence is mathematically explained using the Arrhenius equation; which expresses the relationship between the rate constant, the absolute temperature, the frequency factor, and the activation energy.
The activation energy and frequency factor can also be determined graphically by converting the Arrhenius equation into a non-exponential form.
Using the natural logarithms on both sides, an equation for a linear function is generated. The slope value corresponds to the negative value of activation energy over gas constant, and the y-intercept corresponds to the natural log of the frequency factor.
This equation can be used to generate a graph called the Arrhenius plot, in which the natural log of the rate constant is denoted as a function of the inverse of temperature in kelvin.
Kinetic data of experiments and reactions can be illustrated and analyzed using this Arrhenius plot. In this example, the graph yields a straight line. The slope value given in kelvin is set equal to the negative value of activation energy over R. After assigning the value for the gas constant and solving for the activation energy, a value of 93.1 kJ/mol is obtained.
Besides, the y-intercept of 26.8 is equal to the natural log of the frequency factor. Thus, solving for A gives the value of 4.36 &#215; 1011 with the unit one-over-molarity-seconds &#8212; the same unit as the rate constant.
In cases of limited kinetic data or difficulties with graphical representation, a two-point form of the Arrhenius equation can be utilized to calculate the activation energy in a non-graphical manner.
In such cases, the non-exponential form of the Arrhenius equation is modified to include rate constants at two different temperatures.
Subsequent subtraction and rearrangement of the expression yield the two-point form of the Arrhenius equation, which is used to calculate the activation energy from experimentally-generated rate constants at two different temperatures. By substituting the values, the activation energy for this reaction is calculated to be 145 kJ/mol.

arrhenius plots

The Arrhenius equation relates the activation energy and the rate constant, k, for chemical reactions. In the Arrhenius equation, k = Ae&#8722;Ea/RT, R is the ideal gas constant, which has a value of 8.314 J/mol&#183;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 a constant called the frequency factor, which is related to the frequency of collisions and the orientation of the reacting molecules.
The Arrhenius equation can be used to compute the activation energy of a reaction from experimental kinetic data. A convenient approach to determining the Ea for a reaction involves the measurement of k at two or more different temperatures. It uses a modified version of the Arrhenius equation that takes the form of a linear equation:
 <img alt="Chemical reaction mechanism diagram, illustrating amination reaction steps with nitrogen gas evolution." src="/files/ftp_upload/11380/11380_Equation_1.png" style="height: 200px;" data-altupdatedat="1754904545000" data-alt="Chemical structure of serotonin, neurotransmitter diagram, molecular formula C10H12N2O." />
A plot of the ln k versus 1/T is linear with a slope equal to &#8722;Ea/R and y-intercept equal to ln A.
Consider the following reaction:
 <img alt="Chemical equilibrium reaction diagram, sulfur dioxide conversion, SO2+O2&#8652;SO3 process." src="/files/ftp_upload/11380/11380_Equation_2.png" style="height: 50px;" data-altupdatedat="1754904545000" data-alt="Chemical structure of aspirin, diagram with molecular formula C9H8O4, key functional groups." />
The activation energy of this reaction can be determined if the variation in the rate constant with temperature is known from the reaction kinetic data, as shown.
<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 &#215; 10&#8211;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 &#215; 10&#8211;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 &#215; 10&#8211;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 &#215; 10&#8211;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 &#215; 10&#8211;2</td>
		</tr>
	</tbody>
</table>
The provided data can be used to derive the values of the inverse of temperature (1/T) and the natural log of k (ln k).
<table style="margin-left: auto; margin-right: auto;">
	<tbody>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">1/T (K&#8211;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 &#215; 10&#8211;3 </td>
			<td style="color: black; border: 1px solid black; text-align: center;"> &#8211;14.860 </td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">1.74 &#215; 10&#8211;3</td>
			<td style="color: black; border: 1px solid black; text-align: center;">&#8211;13.617</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">1.55 &#215; 10&#8211;3</td>
			<td style="color: black; border: 1px solid black; text-align: center;">&#8211;9.362</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">1.43 &#215; 10&#8211;3</td>
			<td style="color: black; border: 1px solid black; text-align: center;">&#8211;6.759</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">1.28 &#215; 10&#8211;3</td>
			<td style="color: black; border: 1px solid black; text-align: center;">&#8211;3.231</td>
		</tr>
	</tbody>
</table>
 <img alt="Chromatography diagram illustrating ion exchange process; shows resin and ion separation method." src="/files/ftp_upload/11380/11380_Image%201.png" style="height: 350px;" data-altupdatedat="1754904545000" data-alt="Protein structure diagram; amino acid sequence; polypeptide chain illustration; biochemical analysis." />
On plotting the derived data points with ln k versus 1/T, a line-graph exhibiting a linear relationship between ln k and 1/T is generated, as shown.
The slope of the line, which corresponds to the activation energy, can be estimated using any two of the experimental data pairs. 
 <img alt="RNA secondary structure diagram, nucleotide base pairing and loop configuration, educational use." src="/files/ftp_upload/11380/11380_Equation_3.png" style="height: 350px;" data-altupdatedat="1754904545000" data-alt="Chemical reaction mechanism, electrophilic addition of HBr to ethylene diagram, illustrating process." />
An alternative approach in deriving activation energy involves the utilization of the rate constant at two different temperatures. In this approach, the Arrhenius equation is rearranged to a convenient two-point form:
 <img alt="Static equilibrium, &#931;Fx=0 balance equations; mechanical system diagram for physics study." src="/files/ftp_upload/11380/11380_Equation_4v2.png" style="height: 75px;" data-altupdatedat="1754904545000" data-alt="Electrical circuit diagram with resistors, voltage sources, Kirchhoff&#8217;s laws application." />
On rearranging the equation, an expression for the activation energy is generated.
 <img alt="Chromatography diagram; benzene structure; methyl orange separation; chemical analysis process." src="/files/ftp_upload/11380/11380_Equation_5.png" style="height: 75px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium diagram; &#931;Fx=0, MA=0; showing forces and moments on beam structure." />
By substituting any two data pairs and further calculation yields the value for the activation energy in joules per mole or kilojoules per mole.
 <img alt="Symbiotic nitrogen fixation process, N2 to NH3, chemical reaction diagram in nitrogen cycle study." src="/files/ftp_upload/11380/11380_Equation_6.png" style="height: 135px;" data-altupdatedat="1754904545000" data-alt="Thermodynamics: Gibbs free energy equations, &#916;G=&#916;H&#8722;T&#916;S, diagram explains spontaneity, equilibrium." />
This alternative two-point approach yields the same result as the graphical approach. However, in practice, the graphical approach typically provides more reliable results while working with actual experimental data.
This text is adapted from <a href="https://openstax.org/books/chemistry-2e/pages/12-5-collision-theory">Openstax, Chemistry 2e, Section 12.5: Collision Theory</a>.

The Arrhenius equation relates the activation energy and the rate constant, k, for chemical reactions. In the Arrhenius equation, k = Ae&#8722;Ea/RT, R is ...

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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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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.

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Determining Activation Energy from Temperature

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