Graphs of $\frac{R(J) - P(J)}{J + \frac{1}{2}}$ and respectively $\frac{R(J - 1)
- P(J + 1)}{J + \frac{1}{2}}$ were plotted against $(J + \frac{1}{2})^2$ and
-linear regressions were performed in order to determine the values for
-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 then determined from the $\tilde{B_{\nu}}$ values using equation \ref{eq:bond-length}.
+linear regressions were performed in order to determine the values for the
+centrifugal distortion coefficients, $\tilde{D_{\nu}}$ (see table
+\ref{tbl:centrifugal-distortion-const} in the supplementary information) and
+hence the rotational constants, $\tilde{B_{\nu}}$ shown in table
+\ref{tbl:rot-const-bond-lengths}. The error propagation shown in equation
+\ref{eq:b-err-prop} was then completed to estimate the uncertainties in
+$\tilde{B_{\nu}}$, where $\alpha_B$, $\alpha_m$ and $\alpha_c$ are the
+uncertainties in $\tilde{B_{\nu}}$, the gradient and the intercept found in
+the linear regression respectively.
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=0.9\textwidth]{figures/cl35fu.jpg}
+ \caption{Graph showing the upper transition branch for the fundamental.}\label{fig:cl35-fu}
+\end{figure}
-\begin{equation}\label{eq:bond-length}
- r_{\nu} = \sqrt{4 \pi c \hbar \mu \tilde{B_{\nu}}}
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=0.9\textwidth]{figures/cl35ou.jpg}
+ \caption{Graph showing the upper transition branch for the overtone.}\label{fig:cl35-ou}
+\end{figure}
+
+\begin{equation}
+ \label{eq:b-err-prop}
+ \alpha_B = \sqrt{\left( \frac{3}{16} \alpha_m \right)^2 + \left(
+ \frac{1}{4} \alpha_c \right)^2}
\end{equation}
+The bond lengths shown in table \ref{tbl:rot-const-bond-lengths} were determined
+from the $\tilde{B_{\nu}}$ values using equation \ref{eq:bond-length}.
+Furthermore the calculus-based approximation\autocite{hughes-hase-uncertainties}
+was utilised to give the estimation in the uncertainty for $r_{\nu}$
+($\alpha_r$) given in equation \ref{eq:r-err-prop} since the $\alpha_B$ values
+are small. The uncertainties in the values of the constants and reduced mass
+used is insignificant compared to that of $\alpha_B$, hence they were discarded.
+
+\begin{align} \label{eq:bond-length}
+ r_{\nu} &= \sqrt{\frac{h}{8 \pi^2 c \mu \tilde{B_{\nu}}}} \\
+ \label{eq:r-err-prop}
+ \alpha_r &= \frac{1}{2} \sqrt{\frac{h}{8 \pi^2 c \mu \tilde{B_{\nu}}^3}} \alpha_B
+\end{align}
+
+The reduced mass, $\mu$, was calculated to be $\SI{0.972937750}{\atomicmassunit} =
+\SI{1.61560112e-27}{\kilo\gram}$ for
+\ce{H^{35}Cl} and $\SI{0.981077295}{\atomicmassunit} = \SI{1.62911715e-27}{\kilo\gram}$ for \ce{H^{37}Cl}.\autocite{crc-handbook}
+
\begin{table}[h]
\caption{Rotational constants and bond lengths.}
\label{tbl:rot-const-bond-lengths}
\hline
& $\nu$ & $\tilde{B_{\nu}}$ / \si{\per\centi\metre} & $r_{\nu}$ / \si{\pico\meter} \\
\hline
- \multirow{3}{*}{\ce{H^{35}Cl}} & 0 & & \\
+ \multirow{3}{*}{\ce{H^{35}Cl}} & 0 & \num{10.4408(1)} & \num{128.823(2)} \\
\cline{2-4}
- & 1 & & \\
+ & 1 & \num{10.1360(2)} & \num{130.745(1)} \\
\cline{2-4}
- & 2 & & \\
+ & 2 & \num{9.829(4)} & \num{132.77(2)} \\
\hline
- \multirow{3}{*}{\ce{H^{37}Cl}} & 0 & & \\
+ \multirow{3}{*}{\ce{H^{37}Cl}} & 0 & \num{10.4248(3)} & \num{128.385(2)} \\
\cline{2-4}
- & 1 & & \\
+ & 1 & \num{10.1214(1)} & \num{130.2953(9)} \\
\cline{2-4}
- & 2 & & \\
+ & 2 & \num{9.86(2)} & \num{132.0(1)} \\
\hline
\end{tabular}
\end{table}
-The error propagation shown in equation \ref{eq:b-err-prop} was completed to estimate the uncertainty in the
-values of $\tilde{B_{\nu}}$ where $\alpha_B$, $\alpha_D$, $\alpha_m$ are the uncertainties in
-$\tilde{B_{\nu}}$, $\tilde{D_{\nu}}$ and the gradient found in the linear
-regression respectively.
-
-\begin{equation}\label{eq:b-err-prop}
- \alpha_B = \tilde{B_{\nu}} \sqrt{\left( \frac{\alpha_m}{m} \right)^2 + \left(
- \frac{\alpha_D}{\tilde{D_{\nu}}} \right)^2}
-\end{equation}
-
-Furthermore the calculus-based approximation\autocite{hughes-hase-uncertainties} was utilised to give the estimation in the uncertainty for $r_{\nu}$
-($\alpha_r$) given in equation \ref{eq:r-err-prop} since the $\alpha_B$ values are small.
-
-\begin{equation}\label{eq:r-err-prop}
- \alpha_r = \sqrt{\frac{\pi c \hbar \mu}{\tilde{B_{\nu}}}} \alpha_B
-\end{equation}
-
\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} was determined using equations \ref{eq:nu-e} and \ref{eq:xe}. The derivation of these equations is
+\ref{eq:vib-energy} was determined using equations \ref{eq:nu-e} and \ref{eq:xe}
+and tabulated within table \ref{tbl:vib-const}. The derivation of these equations is
included within section \ref{sec:vib-const-deriv} of the supplementary information.
\begin{align}
\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}}
+ \alpha_{\tilde{\nu_e}} &= \sqrt{ \left( 3 \alpha_{\tilde{B_1}} \right)^2 + \left( \alpha_{\tilde{B_2}}
\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}
+\begin{table}[h]
+ \caption{Vibrational Coefficients.}
+ \label{tbl:vib-const}
+ \centering
+ \begin{tabular}{|c|c|c|}
+ \hline
+ & \ce{H^{35}Cl} & \ce{H^{37}Cl} \\
+ \hline
+ $\tilde{\nu_e}$ / \si{\per\centi\metre} & \num{2885.821(4)} &
+ \num{2883.78(2)} \\
+ \hline
+ $x_e$ / \num{e-5} & \num{-5.32(6)} & \num{-4.5(3)} \\
+ \hline
+ \end{tabular}
+\end{table}
+
+
\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 $\nu_e$.
+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
+equation \ref{eq:force-constant-err} (which utilises the calculus-based
+approximation) where the error in $\mu$ was assumed negligible compared to that
+in $\nu_e$ and is hence discarded.
\begin{align}
\label{eq:force-constant}
\hline
& \ce{H^{35}Cl} & \ce{H^{37}Cl} \\
\hline
- k / \si{\newton\per\centi\metre} & & \\
+ k / \si{\newton\per\metre} & \num{477.383(1)} & \num{480.698(6)} \\
\hline
\end{tabular}
Two methods able to be used to determine the value of $\tilde{B_0}$: using
either the absorption data from the fundamental or the overtone transition. Much
larger signal:noise ratio for overtone due to lower probability of transition,
-hence greater uncertainty in this value.
+hence greater uncertainty in this value. For the values of $\tilde{B_0}$ only
+the data from the fundamental transition was used due to the lower uncertainties
+in these values. Check for agreement between values and they agreeed (CHECK)
+however didn't combione since would introduce large random error to these values
+for no reason.
\printbibliography
\section{Supplementary Information}
-\subsection{Centrifugal Distortion Coefficients}
+\subsection{Centrifugal Distortion Coefficients}
+
+The values of the centrifugal distortion coefficients, $\tilde{D_{\nu}}$, were
+determined from the gradient found by completing linear regressions.
+
\begin{table}[h]
\caption{Centrifugal distortion coefficients.}
\label{tbl:centrifugal-distortion-const}
\hline
& $\nu$ & $\tilde{D_{\nu}}$ / \num{e-4} \si{\per\centi\metre}\\
\hline
- \multirow{3}{*}{\ce{^{35}Cl}} & 0 & \num{-5.25(4)} and \num{-5.4(3)} \\
+ \multirow{3}{*}{\ce{^{35}Cl}} & 0 & \num{5.25(4)} and \num{5.4(3)} \\
\cline{2-3}
- & 1 & \num{-5.11(2)} \\
+ & 1 & \num{5.11(2)} \\
\cline{2-3}
- & 2 & \num{-4.3(8)} \\
+ & 2 & \num{4.3(8)} \\
\hline
- \multirow{3}{*}{\ce{^{37}Cl}} & 0 & \num{-5.20(3)} and \num{-9(2)} \\
+ \multirow{3}{*}{\ce{^{37}Cl}} & 0 & \num{5.20(3)} and \num{9(2)} \\
\cline{2-3}
- & 1 & \num{-5.13(1)} \\
+ & 1 & \num{5.13(1)} \\
\cline{2-3}
- & 2 & \num{-11(4)}\\
+ & 2 & \num{11(4)}\\
\hline
\end{tabular}
\end{table}
projects / chemistry / university-chemistry-lab-reports.git / commitdiff
