DOFPro Team

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The chemical reaction, \(\mathrm{A} \rightarrow \mathrm{B}\), has been studied and has been found to be a first-order reaction. In a constant-volume chemical reactor, if you start with a fixed initial concentration of \(\mathrm{A}\), \(C_\mathrm{A0}\), in grams per liter and no \(\mathrm{B}\), \(C_\mathrm{B}\), will evolve in time as \(C_\mathrm{B} = C_\mathrm{{A0}}(1-e^{- kt})\), where \(t\) is time in seconds and \(k\) is the rate constant or rate coefficient in units of per seconds. Your company hired four research labs to study the kinetics of the reaction at five different temperatures, 265 °C, 270 °C, 275 °C, 280 °C and 285 °C. The research labs have returned data tables that have \(T\), \(C_\mathrm{A0}\), and \(C_\mathrm{B}(t)\).
Start with
\[C_\mathrm{B} = C_\mathrm{{A0}}(1-e^{- kt})\]
Divide by \(C_\mathrm{{A0}}\)
Isolate \(e^{- kt}\)
\[1-\frac{C_\mathrm{B}}{C_\mathrm{{A0}}}=e^{- kt}\]
\[\ln \left(1-\frac{C_\mathrm{B}}{C_\mathrm{{A0}}}\right) = -kt\]
Plot \(\ln \left(1-\frac{C_\mathrm{B}}{C_\mathrm{{A0}}}\right)\) versus \(t\).
\(\mathrm{slope}=-k\)
\[k=A e^{-E_a/(RT)}\]
\(A\) is the pre-exponential or frequency factor.
\(E_a\) is the activation energy.
\[\ln{k}=\ln{A}-\frac{E_a}{RT}\]
Plot \(\ln k\) versus \(1/T\).
\(\mathrm{slope}=-\frac{E_a}{R}\)
\(\mathrm{intercept}=\ln{A}\)
\(E_a=-\mathrm{slope} \cdot R\)
\(A = \exp{(\mathrm{intercept})}\)
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The Full Story companion video is in the link in the upper left. The next video in the series, is in the upper right. To learn more about Chemical and Thermal Processes, visit the website linked in the description.

