\date{30/11/2018}
\maketitle
+%TODO: Write abstract.
\begin{abstract}
Experiment abstract.
\end{abstract}
\frac{1}{4} \alpha_c \right)^2}
\end{equation}
+The equilibrium rational constant, $\tilde{B_e}$, was obtained using values from
+literature\autocite{nist-hcl} for the \ce{H^{35}Cl} isotopologue and the
+rotational constants at at the vibrational level $\nu$ according to this,
+$\tilde{B}_{\nu_{Lit.}}$ were calculated using equation \ref{eq:b-eq-nu} and
+tabulated in table \ref{tbl:rot-const-bond-lengths}.
+
+\begin{equation}\label{eq:b-eq-nu}
+ \tilde{B}_{\nu_{Lit.}} = \tilde{B_e} - \alpha_e \left( \nu + \frac{1}{2} \right)
+\end{equation}
+
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
+$\tilde{B_{\nu}}$ 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
+uncertainties in the values of the constants and reduced mass used\autocite{crc-handbook} 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.
\caption{Rotational constants and bond lengths.}
\label{tbl:rot-const-bond-lengths}
\centering
- \begin{tabular}{|c|c|c|c|}
+ \begin{tabular}{|c|c|c|c|c|}
\hline
- & $\nu$ & $\tilde{B_{\nu}}$ / \si{\per\centi\metre} & $r_{\nu}$ / \si{\pico\meter} \\
+ & $\nu$ & $\tilde{B_{\nu}}$ / \si{\per\centi\metre} & $r_{\nu}$ / \si{\pico\meter} &
+ $\tilde{B}_{\nu_{Lit.}}$ / \si{\per\centi\metre} \\
\hline
- \multirow{3}{*}{\ce{H^{35}Cl}} & 0 & \num{10.4408(1)} & \num{128.823(2)} \\
- \cline{2-4}
- & 1 & \num{10.1360(2)} & \num{130.745(1)} \\
- \cline{2-4}
- & 2 & \num{9.829(4)} & \num{132.77(2)} \\
+ \multirow{3}{*}{\ce{H^{35}Cl}} & 0 & \num{10.4408(4)} & \num{128.823(2)} & \num{10.43982} \\
+ \cline{2-5}
+ & 1 & \num{10.1360(2)} & \num{130.745(1)} & \num{10.13264} \\
+ \cline{2-5}
+ & 2 & \num{9.829(4)} & \num{132.77(2)} & \num{9.82546} \\
\hline
- \multirow{3}{*}{\ce{H^{37}Cl}} & 0 & \num{10.4248(3)} & \num{128.385(2)} \\
- \cline{2-4}
- & 1 & \num{10.1214(1)} & \num{130.2953(9)} \\
- \cline{2-4}
- & 2 & \num{9.86(2)} & \num{132.0(1)} \\
+ \multirow{3}{*}{\ce{H^{37}Cl}} & 0 & \num{10.4248(3)} & \num{128.385(2)} & \\
+ \cline{2-5}
+ & 1 & \num{10.1214(1)} & \num{130.2953(9)} & \\
+ \cline{2-5}
+ & 2 & \num{9.86(2)} & \num{132.0(1)} & \\
\hline
\end{tabular}
\end{table}
\end{table}
\section{Discussion}
-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. 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.
-
-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.
+\subsection{Errors and Justification of Analysis Method}\label{sec:errors}
+It is likely that the errors used for the data collected are underestimates
+since they are derived from linear regressions performed with a limited number
+of data points (less than 12).
+
+Furthermore while the linear regression was in general good for the data
+obtained for fundamental transitions ($\chi^2$ = \SIrange{2e-5}{2e-4}) in
+general it was much poorer for data derived from the overtone transition
+($\chi^2$ = \SIrange{4e-4}{0.2}). This was due to the considerable signal:noise
+ratio on the spectrum for the overtone transition resulting in the data
+being significantly effected by random noise. This can be seen clearly in figure
+\ref{fig:cl35-ou} and is likely to further increase the uncertainties in the
+quantities derived from this data over those stated.
+
+The high signal:noise ratio for the overtone absorption peaks occurs due to the
+lower probability of this transition occurring compared to the fundamental
+(since in a symmetric purely harmonic potential we have the selection rule
+$\Delta \nu = \pm 1$) and hence resulting in the related peaks having a reduced
+intensity. The signal:noise ratio for the overtone absorption peaks could be
+reduced by recording additional with a greater concentration of \ce{HCl} gas in the
+gas cell, hence increasing the intensity of the peaks. When completing this the
+spectrum range could be reduced to \SIrange{6000}{5000}{\per\centi\meter} since
+any fundamental transition peaks will be unlikely to yield any useful data as
+they will have a much greater intensity than previously thus are likely to be
+'cut' resulting in them not having a well defined wavenumber for their maxima.
+
+In the data analysis the centrifugal distortion was accounted for since when
+it was ignored the linear regression produced residuals which were clearly not
+randomly distributed hence suggesting the presence of a systematic error. When
+the centrifugal distortion was accounted for the residual plot showed a more
+random distribution.
+
+\subsection{Discussion of Obtained Data}
+
+\subsubsection{Rotational Constants}\label{sec:rot-const}
+The value of $\tilde{B_0}$ could have been obtained by considering either the
+fundamental or overtone transition. The value tabulated in table
+\ref{tbl:rot-const-bond-lengths} was derived from the fundamental transition
+since less random noise affected the values for the fundamental transition (as
+discussed in section \ref{sec:errors}, hence using only this transition reduced
+the random errors in the value determined.
+
+It was expected that the values of $\tilde{B_{\nu}}$ for \ce{H^{35}Cl} should be
+larger than the corresponding values for \ce{H^{37}Cl} due to the greater
+reduced mass of the \ce{H^{37}Cl} isotopologue making the $\tilde{B_{\nu}}$ value
+larger by equation \ref{eq:rot-const-b}.
+
+\begin{equation}\label{eq:rot-const-b}
+ \tilde{B_{\nu}} = \frac{h}{8 \pi^2 c \mu r^2}
+\end{equation}
+
+The obtained data (presented in table \ref{tbl:rot-const-bond-lengths}) supports
+this since all values of $\tilde{B_{\nu}}$ calculated for \ce{H^{35}Cl} are
+greater than the corresponding values for \ce{H^{37}Cl} except for the
+$\tilde{B_2}$ value where this condition can be satisfied with a probability of
+\SI{6.06}{\percent} (calculated by assuming a normal distribution around the
+data point and normalising). As discussed in section \ref{sec:errors} it is
+likely the errors used are underestimates hence the agreement of this datum with
+the expected result is likely to be stronger than this.
+
+The data for $\tilde{B_{\nu}}$ in table \ref{tbl:rot-const-bond-lengths}
+all agree with the literature values, $\tilde{B}_{\nu_{Lit.}}$, within
+\SI{\pm0.004}{\per\centi\metre}. While this is one order of magnitude greater
+than the estimated uncertainties in all values (except for $\tilde{B_{2}}$) it
+is likely that the obtained data does agree with these literature values,
+however (as discussed in section \ref{sec:errors}) the uncertainties stated are
+underestimates.
+
+In order to obtain better estimates of the uncertainties for the rotational
+constants additional spectra could be recorded hence allowing a better estimation
+of the uncertainty in the calculated values based any differences which arise
+between the data sets collected. This will also help suggest the reproducibility
+of the collected data.
+
+\subsubsection{Bond Lengths}
+It was also expected that the bond length ($r_{\nu}$) values for both
+isotopologues would increase as the vibrational energy level ($\nu$) increased
+since the anharmonicity of the bond potential results in the mean bond length
+increasing as the vibrational energy level increases. All of the values of
+$r_{\nu}$ obtained in table \ref{tbl:rot-const-bond-lengths} agree with this.
+
+In table \ref{tbl:rot-const-bond-lengths} it can also be seen that the bond
+lengths of the \ce{H^{37}Cl} isotopologue in each vibrational state are greater
+than for the \ce{H^{35}Cl} isotopologue. This is the expected result since
+from the one dimensional time-independent Schr\"{o}dinger equation (equation
+\ref{eq:s-eq}) it can be seen that for the same potential function $V(x)$
+(assumed since the chemical bonding should identical for both isotopologues) and
+molecular (stationary) states described by $\psi_{\nu}$ the energy of the state
+labelled by $\nu$ will decrease for an increased reduced mass. Hence from the
+asymmetry of the anharmonic bond potential the \ce{H^{37}Cl} isotopologue should
+have a lower mean bond length in each vibrational state.
+
+\begin{align}
+ E_{\nu} \psi_{\nu} &= \hat{H} \psi_{\nu} \nonumber \\
+ \label{eq:s-eq}
+ &= \left( \frac{\hat{p}^2}{2 \mu} + V(x) \right) \psi_{\nu}
+\end{align}
+
+\subsubsection{Bond Force Constants}
+It was expected that the bond force constants, $k$, displayed in table
+\ref{tbl:force-constants} should be equal for both isotopologues as it was
+assumed that $k$ depends only on the chemical bonding and hence should be the
+same for both isotopologues.
+
+Despite this the force constants differ by \SI{3.315}{\newton\per\meter} while
+both values have very small uncertainties of the order of \num{1e-3}. It is
+unlikely that an underestimation of the uncertainties in the $\tilde{B_0}$ and
+$\tilde{B_1}$ values can explain this difference since if the uncertainties in
+the $\tilde{B_0}$ and $\tilde{B_1}$ values for both isotopologues are increased
+by a factor of ten (this would hence give good agreement between the $\tilde{B_0}$ and
+$\tilde{B_1}$ values for \ce{H^{35}Cl} and the literature values) then when this
+is propagated the uncertainty in the bond constants also increases by
+slightly more than a factor of ten. This results in a difference between the
+bond constants of over 500 standard deviations.
+
+It is considered unlikely that a systematic error could result in the difference
+between the calculated bond force constants since both were obtained from taking
+differences between values on the spectrum (which should remove systematic
+errors) and both values were obtained from the same spectrum thus removing the
+influence of any calibration errors.
+
+It is possible that the mass of a molecule affects the bond force constant in an
+indirect way as concluded by Biernacki and Clerjaud for the \ce{SiH4} and
+\ce{SiD4} isotopologues.\autocite{biernacki-clerjaud} In order to confirm this
+with more confidence more spectra should be recorded of the overtone and
+fundamental transitions (as discussed in \ref{sec:errors} and
+\ref{sec:rot-const}). In addition to this further spectra could be obtained for
+deuterated hydrogen chloride gas, \ce{DCl}, since this will increase the reduced
+mass by almost a factor of two, hence should
+result in an even larger difference in the bond force constants.
+
+
+%TODO: Residual plots?
\printbibliography
\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}.
+\ref{eq:vib-energy} which can be found through the application of perturbation
+theory on the harmonic potential with the perturbation of the potential
+including terms a of higher order than two from the Taylor expansion of the
+potential energy.
\begin{equation}\label{eq:vib-energy}
\tilde{E_{\nu}} = \tilde{\nu_e} \left( \nu + \frac{1}{2} \right) -
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