Finished experiment 4B lab report.

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diff --git a/year2/4b/4b.tex b/year2/4b/4b.tex
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--- a/year2/4b/4b.tex
+++ b/year2/4b/4b.tex
@@ -1,7 +1,7 @@
 %Document Setup.
 \documentclass[a4paper,11pt]{article}
 %Load useful packages.
 %Document Setup.
 \documentclass[a4paper,11pt]{article}
 %Load useful packages.

-\usepackage[a4paper,margin=2.3cm]{geometry}  %Set page size to A4.
+\usepackage[a4paper,margin=2.0cm]{geometry}  %Set page size to A4.

 \usepackage{graphicx}   %Allow import of images.
 \usepackage{floatrow}   %Positioning of table & figure captions.
 \usepackage{siunitx}    %SI Units formatting.
 \usepackage{graphicx}   %Allow import of images.
 \usepackage{floatrow}   %Positioning of table & figure captions.
 \usepackage{siunitx}    %SI Units formatting.


@@ -41,7 +41,7 @@
     \SI{477.383(1)}{\newton\per\meter} and \SI{480.698(6)}{\newton\per\meter}
     for \ce{H^{35}Cl} and \ce{H^{37}Cl} respectively. The bond force constants
     are not identical within reasonable errors, hence it is likely that the
     \SI{477.383(1)}{\newton\per\meter} and \SI{480.698(6)}{\newton\per\meter}
     for \ce{H^{35}Cl} and \ce{H^{37}Cl} respectively. The bond force constants
     are not identical within reasonable errors, hence it is likely that the

-    isotopic mass effects the value of $k$, however additional data should be
+    isotopic mass affects the value of $k$, however additional data should be

     collected to verify this.
 \end{abstract}
 
     collected to verify this.
 \end{abstract}
 


@@ -179,10 +179,11 @@ of this form.
 
 \begin{align}
     \label{eq:comb-diff-upper}
 
 \begin{align}
     \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} \\
+    \frac{R(J) - P(J)}{J + \frac{1}{2}} &= -8\tilde{D_1} \left(J + \frac{1}{2}\right)^2 + 4\tilde{B_1} -
+    6\tilde{D_1} \\

     %
     \label{eq:comb-diff-lower}
     %
     \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 - 1) - P(J + 1)}{J + \frac{1}{2}} &= -8\tilde{D_0} \left(J + \frac{1}{2}\right)^2 +

     4\tilde{B_0} - 6\tilde{D_0}
 \end{align}
 
     4\tilde{B_0} - 6\tilde{D_0}
 \end{align}
 


@@ -215,10 +216,11 @@ intercept respectively.
     \frac{1}{4} \alpha_c \right)^2}
 \end{equation}
 
     \frac{1}{4} \alpha_c \right)^2}
 \end{equation}
 

-The literature values of $\tilde{B_{\nu}}$ for \ce{H^{35}Cl} in table
-\ref{tbl:rot-const-bond-lengths} were determined using published equilibrium
-rotational constant, $\tilde{B_e}$, and rotational constant parameter,
-$\alpha_e$, values\autocite{nist-hcl} with equation \ref{eq:b-eq-nu}.
+The literature values of $\tilde{B_{\nu}}$ for \ce{H^{35}Cl},
+$\tilde{B}_{\nu_{Lit.}}$, in table \ref{tbl:rot-const-bond-lengths} were
+determined using published equilibrium rotational constant, $\tilde{B_e}$, and
+rotational constant parameter, $\alpha_e$, values\autocite{nist-hcl} with
+equation \ref{eq:b-eq-nu}.

 
 \begin{equation}\label{eq:b-eq-nu}
     \tilde{B}_{\nu_{Lit.}} = \tilde{B_e} - \alpha_e \left( \nu + \frac{1}{2} \right)
 
 \begin{equation}\label{eq:b-eq-nu}
     \tilde{B}_{\nu_{Lit.}} = \tilde{B_e} - \alpha_e \left( \nu + \frac{1}{2} \right)


@@ -344,91 +346,92 @@ as they are negligible compared to that of $\nu_e$.
 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
 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}) it was much poorer for data derived from the overtone
-transition ($\chi^2$ = \SIrange{4e-4}{0.2}). This was due to the considerable
+general) good for the data obtained for fundamental transitions ($ \num{2e-5} <
+\chi^2 < \num{2e-4}$) it was much poorer for data derived from the overtone
+transition ($\num{4e-4} < \chi^2 < \num{0.2}$). This was due to the considerable

 signal:noise ratio on the spectrum for the overtone transition resulting in the
 signal:noise ratio on the spectrum for the overtone transition resulting in the

-data being significantly effected by random noise (as can be seen in figure
+data being significantly affected by random noise (as can be seen in figure

 \ref{fig:cl35-ou}) and hence the true uncertainty in these values is likely to
 be even greater.
 
 \ref{fig:cl35-ou}) and hence the true uncertainty in these values is likely to
 be even greater.
 

-There is a high signal:noise ratio for the data from the overtone transition
-compared to that of the fundamental due to the lower overtone transition
-probability as the transition is forbidden by the $\Delta \nu = \pm 1$ selection
-rule for a purely harmonic potential and while the oscillator in anharmonic it
-still has considerable harmonic character. This signal:noise ratio could be
-reduced by recording additional spectra for just the overtone region
-(\SIrange{6000}{5000}{\per\centi\meter}) with a greater concentration of
-\ce{HCl} in the gas cell. The reduced wavenumber range should be used since the
-increased concentration will also increase the intensity of the fundamental
-transition peaks and will cause them to be 'cut' hence preventing a distinct
-wavenumber from being recorded for the fundamental transition absorption peak.
-
-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.
+The high signal:noise ratio for the overtone transition data is due to the low
+overtone transition probability since the transition is forbidden by the $\Delta
+\nu = \pm 1$ selection rule obtained for the purely harmonic potential and while
+the oscillator is anharmonic it still has considerable harmonic character. The
+signal:noise ratio could be reduced by recording additional spectra for just the
+overtone region (\SIrange{6000}{5000}{\per\centi\meter}) with a greater
+concentration of \ce{HCl} in the gas cell. The reduced wavenumber range should
+be used since the increased concentration will also increase the intensity of
+the fundamental transition peaks and will cause them to be 'cut' hence
+preventing a distinct wavenumber from being recorded for the fundamental
+transition absorption peak.  In addition to this further spectra for the
+fundamental transition could be recorded as well to allow a better estimation of
+the errors in these values by aiding in determining the reproducibility of the
+results.
+
+Corrections were made in the data analysis for centrifugal distortion since when
+the effect of this was ignored the residuals for the resultant regression
+followed a distinctive parabolic-like pattern when plotted opposed to being
+randomly distributed, thus indicating the presence of a systematic error. When
+the centrifugal distortion was considered the residuals instead appeared to be
+much more randomly distributed.

 
 \subsection{Data}
 
 \subsubsection{Rotational Constants}\label{sec:rot-const}
 
 \subsection{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}.
+The $\tilde{B_0}$ values tabulated in table \ref{tbl:rot-const-bond-lengths}
+were derived using data from the fundamental transition only even though
+they could have also been determined using data from the overtone transition.
+This method was used due to the large random errors in the overtone transition
+data (due to the high signal:noise ratio as discussed in section
+\ref{sec:errors}), so any use of the data from this would be likely to greatly
+increase the random errors associated with the $\tilde{B_0}$ value.
+
+From equation \ref{eq:rot-const-b} it was expected that $\tilde{B_{\nu}}$
+for \ce{H^{35}Cl} should be larger than the corresponding values for
+\ce{H^{37}Cl} due to the larger reduced mass of \ce{H^{37}Cl}.

 
 \begin{equation}\label{eq:rot-const-b}
     \tilde{B_{\nu}} = \frac{h}{8 \pi^2 c \mu r^2}
 \end{equation}
 
 
 \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}
+The data in table \ref{tbl:rot-const-bond-lengths} agrees with this as almost
+all values of $\tilde{B_{\nu}}$ stated for \ce{H^{35}Cl} are larger than the
+corresponding values for \ce{H^{37}Cl} except for the $\tilde{B_2}$ value.
+However using the uncertainty in the $\tilde{B_2}$ value for \ce{H^{37}Cl} there
+is a probability of \SI{6.06}{\percent} that the true value actually satisfies
+this condition (calculated by assuming the measurements are normally distributed
+around the true value). In fact since the uncertainties estimated are likely
+underestimates (see section \ref{sec:errors}) this probability will be higher in
+reality, hence it is probable that the data agrees with the expectation
+drawn from equation \ref{eq:rot-const-b}. To increase the certainty of this
+statement more spectra should be obtained for the overtone transition with a
+lower signal:noise ratio and hence smaller random error (as discussed in section
+\ref{sec:errors}).
+
+The values of $\tilde{B_{\nu}}$ 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
 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
+is likely that the obtained data does agree with these literature values (as
+discussed in section \ref{sec:errors}) as the uncertainties stated are likely

 underestimates.
 
 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}
 \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.
+It was expected that the bond lengths, $r_{\nu}$, of both isotopologues would
+increase as $\nu$ increased since the asymmetry of the anharmonic bond potential should
+cause the mean bond length to increase with the vibrational energy level. All of
+the values of $r_{\nu}$ obtained in table \ref{tbl:rot-const-bond-lengths} show
+this trend.

 
 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
 
 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.
+than those for the \ce{H^{35}Cl} isotopologue. This can be justified
+theoretically using the one dimensional Sch\"odinger equation (equation
+\ref{eq:s-eq}) where for a (stationary) state described by $\psi_{\nu}$ in a
+constant potential $V(x)$ increasing the reduced mass, $\mu$, should reduce the
+energy of the state, $E_{\nu}$, hence reducing the mean bond length due to the
+anharmonicity of the potential.

 
 \begin{align}
     E_{\nu} \psi_{\nu} &= \hat{H} \psi_{\nu} \nonumber \\ 
 
 \begin{align}
     E_{\nu} \psi_{\nu} &= \hat{H} \psi_{\nu} \nonumber \\ 


@@ -440,38 +443,34 @@ have a lower mean bond length in each vibrational state.
 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
 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.
+same.

 
 Despite this the force constants differ by \SI{3.315}{\newton\per\meter} while
 
 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?
-
+both values have uncertainties in the order of \num{e-3}. It is unlikely that
+the underestimation of the uncertainties of $\tilde{B_1}$ and $\tilde{B_2}$
+discussed in section \ref{sec:errors} can account for this because even if the
+errors in these values are increased by a factor of ten (hence making the values
+for \ce{H^{35}Cl} be in very good agreement with the published ones: see section
+\ref{sec:rot-const}) and propagated the uncertainties in $k$ increase by only a
+little over a factor of ten.  This gives the difference between the obtained
+values as over 500 standard deviations hence showing that this is very
+unlikely to be caused due to random errors.
+
+Systematic errors are also unlikely to explain this difference as both values of
+$k$ were obtained from data from the same spectrum and were calculated using
+differences in wavenumber values thus any constant systematic errors in the
+absorption wavenumber will have been removed and if any systematic errors were
+introduced in the data analysis they should be the same for both $k$ values
+hence still making them comparable.
+
+It is possible that the isotopic mass affects the value of $k$ indirectly hence
+leading to the observed differences as previously found by Biernacki and
+Clerjaud for the \ce{SiH4} and \ce{SiD4} isotopologues.\autocite{biernacki-clerjaud}
+However to more confidently confirm this more data with a lower random error in
+the data derived from the overtone transition should be obtained (as discussed
+in section \ref{sec:errors}). Additionally the rotational-vibrational
+spectrum of deuterated \ce{HCl} gas could be recorded which should have an even
+greater value of $k$ since the reduced mass is around a factor of two greater.

 
 \printbibliography
 
 
 \printbibliography
 


@@ -480,7 +479,7 @@ result in an even larger difference in the bond force constants.
 \subsection{Centrifugal Distortion Coefficients}
 
 The values of the centrifugal distortion coefficients, $\tilde{D_{\nu}}$, were
 \subsection{Centrifugal Distortion Coefficients}
 
 The values of the centrifugal distortion coefficients, $\tilde{D_{\nu}}$, were

-determined from the gradient found by completing linear regressions.
+determined from the gradients determined in the linear regressions.

 
 \begin{table}[h]
     \caption{Centrifugal distortion coefficients.}
 
 \begin{table}[h]
     \caption{Centrifugal distortion coefficients.}


@@ -490,13 +489,13 @@ determined from the gradient found by completing linear regressions.
         \hline
         & $\nu$ & $\tilde{D_{\nu}}$ / \num{e-4} \si{\per\centi\metre}\\
         \hline
         \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)} \\

         \cline{2-3}
         & 1 & \num{5.11(2)} \\
         \cline{2-3}
         & 2 & \num{4.3(8)} \\
         \hline
         \cline{2-3}
         & 1 & \num{5.11(2)} \\
         \cline{2-3}
         & 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)} \\

         \cline{2-3}
         & 1 & \num{5.13(1)} \\
         \cline{2-3}
         \cline{2-3}
         & 1 & \num{5.13(1)} \\
         \cline{2-3}