%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.
\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}
\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}
- \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}
\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)
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
-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.
-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}
-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}
-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
-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.
-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.
+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
-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 \\
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
-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
\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.}
\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
- \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}
projects / chemistry / university-chemistry-lab-reports.git / commitdiff