4.6: Temperature Dependence of Rate Constants
- Page ID
- 516696
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In general, increases in temperature increase the rates of chemical reactions. It is easy to see why, since most chemical reactions depend on molecular collisions. The frequency with which molecules collide increases with increased temperature. But also, the kinetic energy of the molecules increases, which should increase the probability that a collision event will lead to a reaction. An empirical model was proposed by Arrhenius to account for this phenomenon. The Arrhenius model (Arrhenius, 1889) can be expressed as
\[ k = A e^{-E_a/RT} \label{Arrh}\]
Although the model is empirical, some of the parameters can be interpreted in terms of the energy profile of the reaction. \(E_a\), for example, is the activation energy, which represents the energy barrier that must be overcome in a collision to lead to a reaction.
If the rate constant for a reaction is measured at two temperatures, the activation energy can be determined by taking the ratio. This leads to the following expression for the Arrhenius model:
\[ \ln \left( \dfrac{k_1}{k_2} \right) = - \dfrac{E_a}{R} \left( \dfrac{1}{T_2} - \dfrac{1}{T_1} \right) \label{Arrhenius} \]
For a given reaction, the rate constant doubles when the temperature is increased form 25 °C to 35 °C. What is the Arrhenius activation energy for this reaction?
Solution
The energy of activation can be calculated from the Arrhenius Equation (Equation \ref{Arrhenius}).
\[ \ln \left( \dfrac{2k_1}{k_1} \right) = - \dfrac{E_a}{8.314 \, \dfrac{J}{mol\, K}} \left( \dfrac{1}{308\,K} - \dfrac{1}{298\,K} \right) \nonumber \]
From this reaction:
\[E_a = 52.9\, kJ/mol \nonumber \]
Preferably, the rate constant is measured at several temperatures. Then the activation energy can be determined using all of the measurements by fitting them to a function. In this case Ea can be calculated from the derivative of the fitting function:
\[E_a = \frac {RT^2}{k}\frac{dk}{dT} = RT^2 \frac{d lnk}{dT}\label{Eadef}\]
\[ \frac{dk}{dT} = \left(\frac{d}{dT}\right)A e^{-E_a/RT} = \frac{E_{a} \color {blue}{A e^{- \frac{E_{a}}{R T}}}}{R T^{2}}\label{Ea#1}\]
The blue part in \(\ref{Ea#1}\) is just k. So this can be rewritten as:
\[\frac{dk}{dT} = \frac{k E_{a} }{R T^{2}}\label{Ea#2}\]
Solving for Ea:
\[E_a = \frac {RT^2}{k}\frac{dk}{dT} = RT^2 \frac{d lnk}{dT}\]
If Ea is temperature independent then a linear fit to the ln(k) versus 1/T can be used:
\[ \ln (k) = - \dfrac{E_a}{RT} + \ln (A) \nonumber \]
This can be done graphically by plotting the natural logarithm of the rate constant as a function of \(1/T\) (with the temperature measured in K). For a well-behaved reactions over limited temperature ranges the result will be a line with a slope of \(–E_a/R\).
There are some theoretical models (such as collision theory and transition state theory) which suggest the form of the Arrhenius model, but the model itself is purely empirical. A general feature, however, of the theoretical approaches is to interpret the activation energy as an energy barrier which a reaction must overcome in order to lead to a chemical reaction.
Contributors and Attributions
Patrick E. Fleming (Department of Chemistry and Biochemistry; California State University, East Bay)
- J. Gutow (UW Oshkosh)