Lecture 10 Temperature Effects on Reaction Rates

10.1 Introduction

The effect of temperature on reaction rates is readily observable in numerous chemical processes. Food preservation through refrigeration, the acceleration of cooking at higher temperatures, and the use of heat to speed up laboratory reactions all demonstrate the significant influence of temperature on chemical kinetics. These observations reflect a fundamental principle: most chemical reactions proceed more rapidly as temperature increases.

In previous chapters, we examined how reaction rates depend on reactant concentrations, establishing methods for determining rate laws and exploring the relationship between mechanisms and observed kinetics. This chapter addresses the second major factor affecting reaction rates: temperature. Understanding the temperature dependence of reaction rates is essential for predicting reaction behavior under various conditions and provides insight into the energetic requirements of chemical transformations.

10.2 The Arrhenius Equation

In the late 19th century, Swedish chemist Svante Arrhenius observed that for many reactions, the rate constant varies with temperature according to a simple mathematical relationship. This relationship, now known as the Arrhenius equation, expresses the rate constant as:

\[k = A \exp \left( \frac{-E_\mathrm{a}}{RT} \right)\]

where: - \(k\) is the rate constant - \(A\) is the pre-exponential factor (with the same units as \(k\)) - \(E_\mathrm{a}\) is the activation energy (typically in kJ mol\(^{-1}\)) - \(R\) is the gas constant (8.314 J K\(^{-1}\) mol\(^{-1}\)) - \(T\) is the absolute temperature (in Kelvin)

The Arrhenius equation has proven remarkably effective in describing the temperature dependence of rate constants across diverse chemical systems. It establishes that rate constants increase exponentially with temperature, which explains the substantial acceleration of reactions observed when temperature is raised.

10.3 Linearised Form of the Arrhenius Equation

For experimental analysis, the Arrhenius equation is often recast in logarithmic form. Taking the natural logarithm of both sides yields:

\[\ln k = \ln A - \frac{E_\mathrm{a}}{RT}\]

This transformation produces a linear relationship between \(\ln k\) and \(\frac{1}{T}\), with:

Slope = \(-\frac{E_\mathrm{a}}{R}\)
y-intercept = \(\ln A\)

This linear form facilitates the experimental determination of Arrhenius parameters from rate constant measurements at different temperatures.

10.4 General Expression for Activation Energy

The activation energy can be defined more generally through the temperature dependence of the rate constant:

\[E_\mathrm{a}(T) = -R \frac{d \ln k}{d(1/T)}\]

For reactions that strictly follow the Arrhenius equation, \(E_\mathrm{a}\) is independent of temperature. However, many real systems show some temperature variation in \(E_\mathrm{a}\), requiring more complex models.

10.5 Alternative Forms of the Arrhenius Equation

The Arrhenius equation can also be expressed in terms of the Boltzmann distribution factor:

\[k = A e^{-E_\mathrm{a}/RT}\]

This formulation emphasizes that the rate constant comprises two distinct components: the pre-exponential factor \(A\) and the exponential term that contains both temperature and activation energy.

10.6 Physical Meaning of Arrhenius Parameters

The Arrhenius equation incorporates two parameters that characterize how a reaction responds to temperature changes. While these parameters were initially derived empirically, they possess physical interpretations that enhance our understanding of reaction kinetics.

10.6.1 The Activation Energy (\(E_\mathrm{a}\))

The activation energy represents an energy threshold that must be exceeded for a chemical reaction to occur. This concept provides insight into the temperature dependence of reaction rates:

\(E_\mathrm{a}\) corresponds approximately to the minimum energy required for reactants to transform into products
At higher temperatures, a larger fraction of molecules possess sufficient energy to surpass this threshold
The exponential term in the Arrhenius equation (\(e^{-E_\mathrm{a}/RT}\)) quantifies the proportion of molecules with energy exceeding the required minimum

10.6.2 The Pre-exponential Factor (\(A\))

The pre-exponential factor \(A\) represents the theoretical maximum rate constant—the value that would be observed if either the activation energy barrier were absent (\(E_\mathrm{a} = 0\)) or the temperature were infinite.

From a physical perspective, \(A\) encompasses several factors: - The frequency of molecular collisions - Geometric requirements for successful reaction (such as molecular orientation) - Other factors that influence reaction probability independent of energy considerations

The pre-exponential factor has the same units as the rate constant \(k\) and is specific to each reaction.

10.7 Reaction Energy Profiles

A valuable conceptual tool for understanding activation energy is the reaction energy profile, also known as a reaction coordinate diagram:

This diagram illustrates the energy changes that occur as reactants transform into products. The activation energy \(E_\mathrm{a}\) corresponds to the height of the energy barrier that separates reactants from products. This representation provides an intuitive framework for understanding why energy input (often in the form of heat) facilitates chemical reactions.

More sophisticated treatments of reaction kinetics will provide additional refinements to this conceptual model, but the essential framework remains valid—reactions require energy to overcome a barrier, and the Arrhenius equation quantifies how temperature affects this process.

10.8 Determining Arrhenius Parameters from Experimental Data

Several methodologies exist for extracting Arrhenius parameters from experimental measurements of rate constants at different temperatures.

10.8.1 Linear Regression Analysis

The most common approach involves measuring rate constants at multiple temperatures and plotting \(\ln k\) versus \(\frac{1}{T}\). Linear regression analysis of this plot provides:

Slope = \(-\frac{E_\mathrm{a}}{R}\), from which \(E_\mathrm{a}\) is calculated
y-intercept = \(\ln A\), from which \(A\) is determined

While conceptually straightforward, this approach has certain limitations. The logarithmic transformation can distort experimental errors, particularly at extreme temperatures. Additionally, extrapolation to determine the y-intercept (\(\frac{1}{T} = 0\), corresponding to infinite temperature) introduces substantial uncertainties in the estimation of \(A\).

10.8.2 Non-linear Regression

Contemporary computational methods permit direct non-linear fitting of the Arrhenius equation in its original exponential form:

\[k = A \exp \left( \frac{-E_\mathrm{a}}{RT} \right)\]

This approach avoids the error distortion that occurs with logarithmic transformation and typically yields more reliable parameter estimates, especially when data span a limited temperature range.

10.8.3 Two-Point Estimation

When data are available at only two temperatures, the activation energy can still be estimated. For temperatures \(T_1\) and \(T_2\) with corresponding rate constants \(k_1\) and \(k_2\):

\[\ln k_1 = \ln A - \frac{E_\mathrm{a}}{RT_1}\] \[\ln k_2 = \ln A - \frac{E_\mathrm{a}}{RT_2}\]

Subtracting the first equation from the second eliminates \(\ln A\):

\[\ln k_2 - \ln k_1 = \frac{E_\mathrm{a}}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)\]

Rearranging to solve for \(E_\mathrm{a}\):

\[E_\mathrm{a} = \frac{R \ln(k_2/k_1)}{\frac{1}{T_1} - \frac{1}{T_2}}\]

This approach provides a first approximation but cannot account for deviations from Arrhenius behavior across broader temperature ranges.

10.8.4 Estimating \(A\) from Graphical Data

When working with Arrhenius plots (\(\ln k\) versus \(\frac{1}{T}\)), the pre-exponential factor \(A\) corresponds to the y-intercept (\(\ln A\)). Direct extrapolation to \(\frac{1}{T} = 0\) introduces large uncertainties because this point lies far from the experimental data range.

A more reliable method uses the slope and a point on the fitted line:

Determine the slope \(m = -\frac{E_\mathrm{a}}{R}\) from the linear fit
Select a point \((x_0, y_0)\) on the line, preferably within the data range
Calculate the intercept \(c = y_0 - mx_0\)
Compute the pre-exponential factor as \(A = e^c\)

This method circumvents the uncertainties associated with extreme extrapolation while providing a reliable estimate of \(A\).

10.9 Special Cases and Deviations from Arrhenius Behavior

While the Arrhenius equation successfully describes many reactions, several important cases exhibit more complex temperature dependence.

10.9.1 Negative Activation Energies

For most reactions, the activation energy is positive, and rates increase with temperature. However, certain reactions display the opposite behavior—their rates decrease as temperature rises, corresponding to a negative activation energy.

This counterintuitive behavior typically indicates a complex reaction mechanism rather than a true “downhill” elementary process. For example, the oxidation of nitric oxide:

\[2\mathrm{NO} + \mathrm{O}_2 \rightarrow 2\mathrm{NO}_2\]

exhibits a negative temperature dependence with empirical rate law:

\[\frac{d[\mathrm{NO}_2]}{dt} = k[\mathrm{NO}]^2[\mathrm{O}_2]\]

Such behavior typically arises in multi-step reaction mechanisms. In the case of NO oxidation, a reversible first step becomes less favorable at higher temperatures, leading to the observed decrease in overall reaction rate.

10.9.2 Non-Arrhenius Behavior

Several situations can lead to deviations from simple Arrhenius behavior:

Multiple reaction pathways: When parallel reaction paths with different activation energies contribute to the overall process, the apparent activation energy may vary with temperature. This occurs because the relative contribution of each pathway changes with temperature—lower-\(E_\mathrm{a}\) paths dominate at lower temperatures, while higher-\(E_\mathrm{a}\) paths become increasingly important at higher temperatures.

Change in rate-determining step: For reactions with sequential steps in a mechanism, changes in temperature can shift which step limits the overall rate. This phenomenon occurs because steps with different activation energies respond differently to temperature changes, potentially altering which step is rate-determining as temperature varies.
Diffusion-limited reactions: When reaction rates approach the limit set by diffusion processes, the temperature dependence weakens. This occurs because diffusion typically has a lower activation energy than chemical transformation steps.
Quantum tunneling effects: At very low temperatures, quantum tunneling can allow reactions to proceed despite insufficient thermal energy to overcome the classical barrier, causing deviations from Arrhenius behavior.

10.10 Predicting Rate Constants at Different Temperatures

A practical application of the Arrhenius equation involves predicting rate constants at temperatures where direct measurements may be impractical. If the activation energy and rate constant are known at one temperature, the rate constant at another temperature can be estimated.

Starting with the Arrhenius equation in logarithmic form:

\[\ln k_1 = \ln A - \frac{E_\mathrm{a}}{RT_1}\] \[\ln k_2 = \ln A - \frac{E_\mathrm{a}}{RT_2}\]

Subtracting the first equation from the second:

\[\ln k_2 - \ln k_1 = \frac{E_\mathrm{a}}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)\]

Rearranging:

\[\ln \frac{k_2}{k_1} = \frac{E_\mathrm{a}}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)\]

This equation enables estimation of how rate constants change between two temperatures without requiring knowledge of the pre-exponential factor \(A\). This approach assumes that \(E_\mathrm{a}\) remains constant across the temperature range of interest—a reasonable approximation for many reactions within moderate temperature intervals.

For reactions with positive activation energies, rate constants increase with temperature. A general guideline derived from typical activation energy values suggests that many reaction rates approximately double with every 10°C increase in temperature, though the exact factor depends on the specific activation energy of the reaction in question.

10.11 Summary

The Arrhenius equation provides a fundamental framework for understanding and quantifying how temperature affects reaction rates. The exponential relationship between temperature and rate constants explains the significant acceleration of most chemical reactions at elevated temperatures.

The two Arrhenius parameters—the pre-exponential factor \(A\) and the activation energy \(E_\mathrm{a}\)—relate empirical kinetic behavior to a conceptual model of reactions requiring an energy threshold to proceed. The activation energy represents the energy barrier that reactant molecules must overcome, while the pre-exponential factor encompasses various factors affecting reaction probability independent of energy considerations.

Experimental determination of these parameters through regression analysis allows for characterization of reactions and prediction of their behavior across different temperature ranges. The Arrhenius equation thus serves as a powerful tool linking macroscopic kinetic measurements to fundamental aspects of chemical reactivity.

While the Arrhenius equation applies to a wide range of chemical processes, cases such as negative activation energies and non-Arrhenius behavior reveal the complexity of reaction mechanisms. These exceptions provide valuable insights into the intricate nature of chemical transformations and the limitations of simple kinetic models.