From: Sam W Date: Sun, 2 Dec 2018 13:26:52 +0000 (+0000) Subject: Continued work on Experiment 4B lab report. X-Git-Url: https://git.dalvak.com/public/?a=commitdiff_plain;h=8cfef308e8b052a57a905f459c967b588effdd38;p=chemistry%2Funiversity-chemistry-lab-reports.git Continued work on Experiment 4B lab report. --- diff --git a/year2/4b/4b.pdf b/year2/4b/4b.pdf index b12f13c..54ab397 100644 Binary files a/year2/4b/4b.pdf and b/year2/4b/4b.pdf differ diff --git a/year2/4b/4b.tex b/year2/4b/4b.tex index 6e6fe13..3afa54e 100644 --- a/year2/4b/4b.tex +++ b/year2/4b/4b.tex @@ -38,7 +38,8 @@ \end{abstract} \section{Results and Analysis} -\begin{table}[h] +\subsection{Collected Data} +\begin{table}[H] \caption{Rotational absorbances for the fundamental transition.} \label{tbl:fundamental-results} \centering @@ -103,7 +104,7 @@ \end{tabular} \end{table} -\begin{table}[h] +\begin{table}[H] \caption{Rotational absorbances for the overtone transition.} \label{tbl:overtone-results} \centering @@ -146,11 +147,12 @@ \cline{2-6} & 7 & 5786.65 & 5493.84 & 292.81 & 310.49 \\ \cline{2-6} - & 8 & 5464.63 & & & \\ + & 8 & & 5464.63 & & \\ \hline \end{tabular} \end{table} +\subsection{Determination of Rotational Constants} The values of $\tilde{B_0}$, $\tilde{B_1}$ and $\tilde{B_2}$ were calculated accounting for the centrifugal distortion of the molecules by using equations \ref{eq:comb-diff-upper} and \ref{eq:comb-diff-lower}. @@ -178,7 +180,7 @@ the centrifugal distortion coefficients, $\tilde{D_{\nu}}$, shown in table \ref{tbl:centrifugal-distortion-const} and hence the rotational constants, $\tilde{B_{\nu}}$ shown in table \ref{tbl:rot-const-bond-lengths}. -The bond lengths were calculated from the $\tilde{B_{\nu}}$ values using equation \ref{eq:bond-length}. +The bond lengths were then determined from the $\tilde{B_{\nu}}$ values using equation \ref{eq:bond-length}. \begin{equation}\label{eq:bond-length} r_{\nu} = \sqrt{4 \pi c \hbar \mu \tilde{B_{\nu}}} @@ -192,13 +194,13 @@ The bond lengths were calculated from the $\tilde{B_{\nu}}$ values using equatio \hline & $\nu$ & $\tilde{B_{\nu}}$ / \si{\per\centi\metre} & $r_{\nu}$ / \si{\pico\meter} \\ \hline - \multirow{3}{*}{\ce{^{35}Cl}} & 0 & & \\ + \multirow{3}{*}{\ce{H^{35}Cl}} & 0 & & \\ \cline{2-4} & 1 & & \\ \cline{2-4} & 2 & & \\ \hline - \multirow{3}{*}{\ce{^{37}Cl}} & 0 & & \\ + \multirow{3}{*}{\ce{H^{37}Cl}} & 0 & & \\ \cline{2-4} & 1 & & \\ \cline{2-4} @@ -227,60 +229,55 @@ Furthermore the calculus-based approximation\autocite{hughes-hase-uncertainties} \subsection{Determination of Vibrational Constants} The values for the harmonic constant $\tilde{\nu_e}$ and the dimensionless anharmonicity constant $x_e$ in equation -\ref{eq:vib-energy} (which gives the expected vibrational energy levels if the rotational spectrum is -ignored). - -\begin{equation}\label{eq:vib-energy} - \tilde{E_{\nu}} = \tilde{\nu_e} \left( \nu + \frac{1}{2} \right) - - \tilde{\nu_e} x_e \left( \nu + \frac{1}{2} \right)^2 -\end{equation} - -From equation \ref{eq:vib-energy} we can obtain the energy related to the pure vibrational transition -$\tilde{E}(\nu_f \leftarrow 0)$ (the $\nu = 0$ to $\nu = \nu_f$ transition) as equation \ref{eq:delta-e-coeff}. +\ref{eq:vib-energy} was determined using equations \ref{eq:nu-e} and \ref{eq:xe}. The derivation of these equations is +included within section \ref{sec:vib-const-deriv} of the supplementary information. \begin{align} - \tilde{E}\left( \nu_f \leftarrow 0 \right) &= \tilde{E}_{\nu_f} - \tilde{E}_0 \nonumber \\ - &= \tilde{\nu_e}\left( \nu_f + \frac{1}{2} - \frac{1}{2} \right) - \tilde{\nu_e} x_e \left( - \frac{1}{4} - \left( \nu_f + \frac{1}{2} \right)^2 \right) \nonumber \\ - \label{eq:delta-e-coeff} - &= \tilde{\nu_e} \left( \nu_f - \left( \nu_f^2 + \nu_f \right) x_e \right) + \label{eq:nu-e} + \tilde{\nu_e} &= R(0) - 3 \tilde{B_1} + \tilde{B_2} \\ + \label{eq:xe} + x_e &= \frac{1}{2} \frac{\tilde{B_2} - \tilde{B_1}}{R(0) - 3 \tilde{B_1} + \tilde{B_2}} \end{align} -The energy related to the R branch transitions can be determined to yield -equation \ref{eq:r-trans-energy} where $J$ is the rotational state adopted in -the lower vibrational state ($\nu = 0$). - -\begin{equation}\label{eq:r-trans-energy} - R(J) = \Delta \tilde{E}(\nu_f \leftarrow 0) + (\tilde{B_{\nu_f}} + \tilde{B_0}) (J + 1) + (\tilde{B_{\nu_f}} - - \tilde{B_0}) ( J + 1)^2 -\end{equation} - -Setting $J = 0$ hence gives equation \ref{eq:r-trans-energy-j0}. - -\begin{equation}\label{eq:r-trans-energy-j0} - R(J) - 2 \tilde{B_{\nu_f}} = \Delta \tilde{E}(\nu_f \leftarrow 0) -\end{equation} - -Subsituting equation \ref{eq:delta-e-coeff} into equation \ref{eq:r-trans-energy-j0} yields equation -\ref{eq:r-trans-energy-sub}. - -\begin{equation}\label{eq:r-trans-energy-sub} - R(0) - 2 \tilde{B_{\nu_f}} = \tilde{\nu_e} \left( \nu_f - \left( \nu_f^2 + \nu_f \right) x_e \right) -\end{equation} - -Now from equation \ref{eq:r-trans-energy-sub} a system of linear equations can be -obtained by setting $\nu_f = 1$ and $\nu_f = 2$ (equations xx and xx) for which we can solve to give equations xx and xx. - - +The uncertainties in these values was hence estimated using equations \ref{eq:nu-e-uncert} and +\ref{eq:xe-uncert} where the uncertainty in $R(0)$ assumed to be negligible +since it is determined by reading the wavenumber directly from the spectrum. +\begin{align} + \label{eq:nu-e-uncert} + \alpha_{\tilde{\nu_e}} &= \sqrt{ \left( 3 \alpha_{\tilde{B_1}} \right)^2 + \left( \alpha_{\tilde{B_1}} + \right)^2} \\ + \label{eq:xe-uncert} + \alpha_{x_e} &= x_e \sqrt{ \frac{ \left( \alpha_{\tilde{B_1}} \right)^2 + \left( + \alpha_{\tilde{B_2}} \right)^2 }{ \left( \tilde{B_2} - \tilde{B_1} \right)^2} + \left( \frac{\alpha_{\tilde{\nu_e}}}{\tilde{\nu_e}} \right)^2} +\end{align} +\subsection{Determination of Bond Force Constants} +The bond force constants, $k$, shown in table \ref{tbl:force-constants} were determined using equation +\ref{eq:force-constant} where $\mu$ is the reduced mass of the molecule and the error in $k$, $\alpha_k$, was +determined using the calculus approximation (equation \ref{eq:force-constant-err}) where the error in $\mu$ +was assumed negligible compared to that in $\nu_e$. +\begin{align} + \label{eq:force-constant} + k &= 4 \pi^2 c^2 \mu \tilde{\nu_e}^2 \\ + \label{eq:force-constant-err} + \alpha_k &= 8 \pi^2 c^2 \mu \tilde{\nu_e} \alpha_{\tilde{\nu_e}} +\end{align} -for which setting $J = 0$ and $\nu_f = 1$ and subsequently $\nu_f = 2$ gives the -system of linear equations (equations xx and xx) for which we can solve to give equations xx and xx. +\begin{table}[H] + \caption{Force Constants.} + \label{tbl:force-constants} + \centering + \begin{tabular}{|c|c|c|} + \hline + & \ce{H^{35}Cl} & \ce{H^{37}Cl} \\ + \hline + k / \si{\newton\per\centi\metre} & & \\ + \hline + \end{tabular} -%TODO: Total energy equation, determination of \nu_e x_e, calculation of values, determination of k with error -%propagation and table. +\end{table} \section{Discussion} Two methods able to be used to determine the value of $\tilde{B_0}$: using @@ -293,6 +290,7 @@ hence greater uncertainty in this value. \section{Supplementary Information} +\subsection{Centrifugal Distortion Coefficients} \begin{table}[h] \caption{Centrifugal distortion coefficients.} \label{tbl:centrifugal-distortion-const} @@ -315,6 +313,63 @@ hence greater uncertainty in this value. \hline \end{tabular} \end{table} + +\subsection{Derivation of Vibrational Constant Equations}\label{sec:vib-const-deriv} +The energies associated with the discrete vibrational energy levels in a molecule are given by equation +\ref{eq:vib-energy}. + +\begin{equation}\label{eq:vib-energy} + \tilde{E_{\nu}} = \tilde{\nu_e} \left( \nu + \frac{1}{2} \right) - + \tilde{\nu_e} x_e \left( \nu + \frac{1}{2} \right)^2 +\end{equation} + +From equation \ref{eq:vib-energy} we can obtain the energy related to the pure vibrational transition +$\tilde{E}(\nu_f \leftarrow 0)$ (the $\nu = 0$ to $\nu = \nu_f$ transition) as equation \ref{eq:delta-e-coeff}. + +\begin{align} + \tilde{E}\left( \nu_f \leftarrow 0 \right) &= \tilde{E}_{\nu_f} - \tilde{E}_0 \nonumber \\ + &= \tilde{\nu_e}\left( \nu_f + \frac{1}{2} - \frac{1}{2} \right) - \tilde{\nu_e} x_e \left( + \frac{1}{4} - \left( \nu_f + \frac{1}{2} \right)^2 \right) \nonumber \\ + \label{eq:delta-e-coeff} + &= \tilde{\nu_e} \left( \nu_f - \left( \nu_f^2 + \nu_f \right) x_e \right) +\end{align} + +The energy related to the R branch transitions can be determined to yield +equation \ref{eq:r-trans-energy} where $J$ is the rotational state adopted in +the lower vibrational state ($\nu = 0$). + +\begin{equation}\label{eq:r-trans-energy} + R(J) = \Delta \tilde{E}(\nu_f \leftarrow 0) + (\tilde{B_{\nu_f}} + \tilde{B_0}) (J + 1) + (\tilde{B_{\nu_f}} - + \tilde{B_0}) ( J + 1)^2 +\end{equation} + +Setting $J = 0$ hence gives equation \ref{eq:r-trans-energy-j0}. + +\begin{equation}\label{eq:r-trans-energy-j0} + R(J) - 2 \tilde{B_{\nu_f}} = \Delta \tilde{E}(\nu_f \leftarrow 0) +\end{equation} + +Substituting equation \ref{eq:delta-e-coeff} into equation \ref{eq:r-trans-energy-j0} yields equation +\ref{eq:r-trans-energy-sub}. + +\begin{equation}\label{eq:r-trans-energy-sub} + R(0) - 2 \tilde{B_{\nu_f}} = \tilde{\nu_e} \left( \nu_f - \left( \nu_f^2 + \nu_f \right) x_e \right) +\end{equation} + +Now from equation \ref{eq:r-trans-energy-sub} a system of linear equations can be +obtained by setting $\nu_f = 1$ and $\nu_f = 2$ (equations \ref{eq:coeff-1} and +\ref{eq:coeff-2}). + +\begin{align} + \label{eq:coeff-1} + R(0) - 2 \tilde{B_1} &= \tilde{\nu_e} \left( 1 - 2 x_e \right) \\ + \label{eq:coeff-2} + R(0) - 2 \tilde{B_2} &= \tilde{\nu_e} \left( 1 - 6 x_e \right) +\end{align} + +Solving equations \ref{eq:coeff-1} and \ref{eq:coeff-2} for $\tilde{\nu_e}$ and +$x_e$ then gives equations \ref{eq:nu-e} and \ref{eq:xe}. + \end{document} %Note: gq means : reformat the text included in the motion % vim:tw=80