\end{tabular}
\end{table}
-\subsection{Determination of Rotational Constants}
-The values of $\tilde{B_0}$, $\tilde{B_1}$ and $\tilde{B_2}$ were calculated
+\subsection{Determination of Rotational Constants and Bond Lengths}
+The values of the rotational constants $\tilde{B_0}$, $\tilde{B_1}$ 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}.
\begin{align}
R(J) - P(J) &= (4\tilde{B_1} - 6\tilde{D_1})(J + \frac{1}{2}) -
8\tilde{D_1} (J + \frac{1}{2})^3 \nonumber \\
- %
\label{eq:comb-diff-upper}
- \frac{R(J) - P(J)}{J + \frac{1}{2}} &= -8\tilde{D_1} (J + \frac{1}{2})^2 + 4\tilde{B_1} - 6\tilde{D_1}
-\end{align}
-
-A similar rearrangement for the transitions with a common excited vibrational
-state yields equation \ref{eq:comb-diff-lower}.
-
-\begin{equation}\label{eq:comb-diff-lower}
- \frac{R(J - 1) - P(J + 1)}{J + \frac{1}{2}} = -8\tilde{D_0} (J + \frac{1}{2})^2 +
+ \frac{R(J) - P(J)}{J + \frac{1}{2}} &= -8\tilde{D_1} (J + \frac{1}{2})^2 + 4\tilde{B_1} - 6\tilde{D_1} \\
+ %
+ \label{eq:comb-diff-lower}
+ \frac{R(J - 1) - P(J + 1)}{J + \frac{1}{2}} &= -8\tilde{D_0} (J + \frac{1}{2})^2 +
4\tilde{B_0} - 6\tilde{D_0}
-\end{equation}
-
-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, $\tilde{D_{\nu}}$ (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.
+\end{align}
+Separate graphs were plotted of $\frac{R(J) - P(J)}{J + \frac{1}{2}}$ and $\frac{R(J - 1)
+- P(J + 1)}{J + \frac{1}{2}}$ against $(J + \frac{1}{2})^2$ then linear regressions were
+performed as shown in figures \ref{fig:cl35-fu} and \ref{fig:cl35-ou}. The values of the
+centrifugal distortion coefficients, $\tilde{D_{\nu}}$, were then determined (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}
+were calculated. 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 fit line gradient and
+intercept respectively.
+
+%TODO: y-axis label missing minus sign.
\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}
+ \caption{Graph showing the linear regression performed for the upper rotational (R) branch
+ of the fundamental transition in \ce{H^{35}Cl}.}\label{fig:cl35-fu}
\end{figure}
\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}
+ \caption{Graph showing the linear regression performed for the upper rotational (R)
+ branch for the overtone transition in \ce{H^{35}Cl}.}\label{fig:cl35-ou}
\end{figure}
\begin{equation}
\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.
+The bond lengths 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 propagate
+the uncertainties in $\tilde{B_{\nu}}$ for $r_{\nu}$ ($\alpha_r$) since the $\alpha_B$
+values are small and the equation used is given in equation \ref{eq:r-err-prop}. The
+uncertainties in the values of the constants and reduced mass used were insignificant
+compared to that of $\alpha_B$, and hence they were discarded.
+
+%TODO: Decide if this paragraph is relevant or needs to be deleted.
+%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{align} \label{eq:bond-length}
r_{\nu} &= \sqrt{\frac{h}{8 \pi^2 c \mu \tilde{B_{\nu}}}} \\
\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}
\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 values for the harmonic constant $\tilde{\nu_e}$ and the dimensionless anharmonicity
+constant $x_e$ were 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.
+included in section \ref{sec:vib-const-deriv} within the supplementary information.
\begin{align}
\label{eq:nu-e}
\end{align}
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.
+\ref{eq:xe-uncert} where the uncertainty in $R(0)$ was negligible compared to that in
+$\tilde{B_1}$ and $\tilde{B_2}$ since it is determined by reading the absorption
+wavenumber directly from the spectrum.
\begin{align}
\label{eq:nu-e-uncert}
\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
-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.
+The bond force constants, $k$, shown in table \ref{tbl:force-constants} were determined
+using equation \ref{eq:force-constant} where the error in $k$, $\alpha_k$, was determined
+using equation \ref{eq:force-constant-err} (which utilises the calculus-based
+approximation). The errors in $\mu$ and the constants were again negligible
+compared to that of $\nu_e$ and hence were discarded.
\begin{align}
\label{eq:force-constant}
however didn't combione since would introduce large random error to these values
for no reason.
+As can be clearly seen in figure \ref{fig:cl35-ou} there was a significant deviation of
+some of the results from the fitted line as shown since this fit had a $\chi^2$ value of
+%TODO: Add \chi^2
+. This patten of poor linear regression fit and high $\chi^2$ value can be seen in all of
+the graphs involving data recorded for the overtone transition. This is likely to partly
+due to the greater signal:noise ratio for this transition due to its lower probability of
+occurrence since it is a forbidden transition if the oscillator is completely harmonic.
+This also had the effect of reducing the number of data points which could be obtained
+since some will have been indistinguishable from the noise.
+
+This appears to be particularly bad around the low values for J
+%TODO: Investigate why.
+
+This could be improved by recording another spectrum with a greater concentration of
+\ce{HCl} gas to increase the intensity of the absorptions due to the overtone transition
+while ignoring the fundamental transition regions were the absorptions peaks will become
+too intense and will be cut. This could be done by reducing the range of the spectrometer
+to just include the overtone region.
+
+Repeated spectra could be obtained to increase the reproducibility of the results and
+verify the accuracy of them. Currently there are a limited number of data points.
+
+Uncertainties calculated (especially for the values derived from measurements in the
+overtone region) are likely to be large underestimates due to the limited number of points
+used in the linear regression. Repeat spectra and the isolated spectra for the overtone
+region would help with this.
+
+%TODO: Residual plots
+
+Analysis including effects of centrifugal distortion was completed since from the residual
+plots a systematic error could clearly be seen since there was a clear patten in the
+values of the residuals as opposed to them being randomly distributed.
+
+%TODO: k values and compare B values to literature.
+
\printbibliography
projects / chemistry / university-chemistry-lab-reports.git / commitdiff