11: Determination of an Activation Energy
- Page ID
- 516597
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. Prerequisite Math & Theory
- The Modified Arrhenius Equation: Because we measure reaction time (\(t\)) instead of rate constant (\(k\)), the Arrhenius slope equation inverts sign:
\[ \ln t = \frac{E_\text{a}}{R}\left(\frac{1}{T}\right) + \text{constant} \]
- Slope: \(m = +\frac{E_\text{a}}{R}\). A positive slope reflects longer reaction times at lower absolute temperatures.
- Calculation: \(E_\text{a} = \text{Slope} \times 8.314\text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}\).
- Catalyst Effect: A catalyst lowers the activation energy of a reaction. Consequently, the calculated \(E_\text{a}\) for Part B (Catalyzed) must be lower than that for Part A (Uncatalyzed).
2. Required Technical Skills
- Accurately monitoring solution temperatures using a temperature probe.
- Linearizing kinetic data (\(\ln t\) vs. \(\frac{1}{T}\)) using spreadsheet software.
3. Critical Safety
- Oxidizer Hazard: Ammonium peroxydisulfate (\(\mathrm{(NH_4)_2S_2O_8}\)) is a strong oxidizing agent. Avoid contact with skin and eyes.
- Heavy Metal Waste: Reaction mixtures containing copper catalyst and iodine products must be disposed of in the designated Heavy Metal Waste container.
- To investigate the effects of temperature and a homogeneous catalyst (\(\mathrm{Cu^{2+}}\)) on the rate of the iodide-peroxydisulfate clock reaction.
- To calculate the activation energy (\(E_\text{a}\)) for both uncatalyzed and catalyzed pathways by measuring reaction times across a range of absolute temperatures.
- To apply the logarithmic Arrhenius equation (\(\ln t = \frac{E_\text{a}}{R}\left(\frac{1}{T}\right) + \text{constant}\)) via graphical analysis of \(\ln t\) versus \(\frac{1}{T}\).
INTRODUCTION
Reaction rates depend strongly on temperature and the presence of a catalyst. From kinetic theory, only molecular collisions exceeding a minimum threshold energy—known as the activation energy (\(E_\text{a}\))—result in reaction, forming an activated complex. The Maxwell-Boltzmann distribution dictates that the fraction of molecules possessing this minimum energy increases exponentially with absolute temperature (\(T\)).
A catalyst accelerates a reaction by providing an alternative reaction pathway with a lower activation energy, allowing a larger fraction of molecules to react at any given temperature. The temperature dependence of the rate constant \(k\) is described by the Arrhenius equation:
\[ k = A\exp\left(-\frac{E_\text{a}}{RT}\right) \label{EARR} \]
where \(R\) is the gas constant (\(8.314\text{ J/mol}\cdot\text{K}\)) and \(A\) is the frequency factor. Taking the natural logarithm yields the linear form:
\[ \ln k = -\frac{E_\text{a}}{R}\left(\frac{1}{T}\right) + \ln A \label{LINARR} \]
Because the reaction rate constant \(k\) is inversely proportional to reaction time (\(t\) required to consume a fixed amount of reactant), substituting \(k \propto 1/t\) transforms the linear equation into:
\[ \ln t = \frac{E_\text{a}}{R}\left(\frac{1}{T}\right) + \text{constant} \]
Thus, graphing \(\ln t\) on the y-axis against \(\frac{1}{T}\) (in \(\text{K}^{-1}\)) on the x-axis yields a straight line with a positive slope \(m = \frac{E_\text{a}}{R}\).
The Iodine Clock Reaction
This experiment examines the oxidation of iodide (\(\mathrm{I^-}\)) by peroxydisulfate (\(\mathrm{S_2O_8^{2-}}\)):
\[ \mathrm{I^-}(aq) + \mathrm{S_2O_8^{2-}(aq) \rightarrow I_2(aq) + 2\,SO_4^{2-}(aq)} \label{I-toI2} \]
To measure the reaction time conveniently, a small, precisely measured amount of sodium thiosulfate (\(\mathrm{Na_2S_2O_3}\)) and starch indicator are added. Thiosulfate instantly consumes iodine as it is formed:
\[ \mathrm{I_2(aq) + 2\,S_2O_3^{2-}(aq) \rightarrow 2\,I^-(aq) + S_4O_6^{2-}(aq)} \label{I2toI-} \]
As long as thiosulfate remains in solution, no free iodine accumulates. The instant thiosulfate is completely exhausted, the next drop of generated iodine reacts with starch, producing an abrupt, deep blue-black color change that marks the exact reaction time \(t\).
Because reaction time \(t\) is inversely proportional to rate (\(k \propto 1/t\)), substituting into logarithmic form gives \(\ln(1/t) = -\ln t\).
- Slope Sign: Consequently, plotting \(\ln t\) vs. \(\frac{1}{T}\) produces a positive slope (\(+\frac{E_\text{a}}{R}\)), unlike a standard Arrhenius plot of \(\ln k\) vs. \(\frac{1}{T}\) which has a negative slope.
Arrhenius Activation Energy:
\[ E_\text{a} = \text{Slope} \times R \]
- 11.1: Determination of an Activation Energy - Experiment
- This page provides safety guidelines for handling chemical hazards, highlighting the need for personal protective equipment (PPE). It specifies essential equipment and chemicals for an experiment to analyze temperature and catalyst effects on reaction rates. The procedure involves controlled mixtures and measurements for reaction timing, along with a reality check for data accuracy. It concludes with instructions for the proper disposal of chemicals.
- 11.2: Determination of an Activation Energy - Pre-lab
- This page discusses activation energy (\(E_a\)) and the Arrhenius equation, detailing how to calculate \(E_a\) using an Arrhenius plot and the gas constant (\(R\)). It explores the involvement of thiosulfate and starch in a measurable chemical reaction, indicating a visual signal for the reaction's completion. Moreover, it encourages comparing slopes of uncatalyzed and catalyzed reactions through graphing activities.
- 11.3: Determination of an Activation Energy - Data and Report
- This page outlines a framework for analyzing reaction rates in both uncatalyzed and catalyzed mixtures. It includes data collection tables for temperatures and times, post-lab questions on temperature conversion and logarithmic calculations, and emphasizes the importance of unit management. Key concepts include the rule that reaction rates double with every 10°C increase in temperature and the analysis of catalysts' effects on reaction rates.