\date{30/11/2018}
\maketitle
-%TODO: Write abstract.
\begin{abstract}
- Experiment abstract.
+ The rotational constants, $\tilde{B_{\nu}}$, were determined for both common
+ isotopologues of \ce{HCl} gas and hence the bond force constants, $k$. The
+ rotational constants for \ce{H^{35}Cl} were in agreement with published
+ values and the bond force constants found were
+ \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
+ collected to verify this.
\end{abstract}
\section{Results and Analysis}
\subsection{Collected Data}
\begin{table}[H]
- \caption{Rotational absorbances for the fundamental transition.}
+ \caption{Rotational-vibrational absorbance wavelengths for the fundamental
+ transition bands.}
\label{tbl:fundamental-results}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\end{table}
\begin{table}[H]
- \caption{Rotational absorbances for the overtone transition.}
+ \caption{Rotational-vibrational absorbance wavelengths for the overtone
+ transition bands.}
\label{tbl:overtone-results}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\end{table}
\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}.
+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 plotting graphs of $\frac{R(J) - P(J)}{J + \frac{1}{2}}$ and
+$\frac{R(J - 1)- P(J + 1)}{J + \frac{1}{2}}$ against $(J + \frac{1}{2})^2$ and
+performing linear regressions (as shown in figures \ref{fig:cl35-fu} and
+\ref{fig:cl35-ou}). This was completed since if combination differences between
+the $R(J)$ and $P(J)$ and then the $R(J-1)$ and $P(J+1)$ bands are taken and the
+centrifugal distortion accounted for the resultant equations can be re-arranged
+to give equations \ref{eq:comb-diff-upper} and \ref{eq:comb-diff-lower}
+respectively, where $\tilde{D_{\nu}}$ are the centrifugal distortion
+coefficients. These equations are in the form of the general equation of a
+straight line, $y = m x + c$, hence the plotted data can be fitted by a function
+of this form.
\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} \\
%
4\tilde{B_0} - 6\tilde{D_0}
\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
+The values of $\tilde{D_{\nu}}$ 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 $\alpha_{\tilde{B_{\nu}}}$, $\alpha_m$ and $\alpha_c$ are the
+uncertainties in $\tilde{B_{\nu}}$, the gradient of the fitted line and the
intercept respectively.
-%TODO: y-axis label missing minus sign.
\begin{figure}[h]
\centering
- \includegraphics[width=0.9\textwidth]{figures/cl35fu.jpg}
+ \includegraphics[width=0.9\textwidth]{figures/cl35fu.png}
\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}
+ \includegraphics[width=0.9\textwidth]{figures/cl35ou.png}
\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}
\label{eq:b-err-prop}
- \alpha_B = \sqrt{\left( \frac{3}{16} \alpha_m \right)^2 + \left(
+ \alpha_{\tilde{B_{\nu}}} = \sqrt{\left( \frac{3}{16} \alpha_m \right)^2 + \left(
\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}.
+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}.
\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
-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\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.
-%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}
+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 $r_{\nu}$
+($\alpha_{r_{\nu}}$) as the $\alpha_{\tilde{B_{\nu}}}$ values are small hence producing equation
+\ref{eq:r-err-prop}. The uncertainties in the values of the constants and
+reduced mass used\autocite{crc-handbook} are insignificant compared to that of
+$\alpha_{\tilde{B_{\nu}}}$, and hence they were excluded from the error propagation.
\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
+ \alpha_{r_{\nu}} &= \frac{1}{2} \sqrt{\frac{h}{8 \pi^2 c \mu \tilde{B_{\nu}}^3}} \alpha_{\tilde{B_{\nu}}}
\end{align}
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
- & $\nu$ & $\tilde{B_{\nu}}$ / \si{\per\centi\metre} & $r_{\nu}$ / \si{\pico\meter} &
- $\tilde{B}_{\nu_{Lit.}}$ / \si{\per\centi\metre} \\
+ & $\nu$ & $\tilde{B_{\nu}}$ / \si{\per\centi\metre} &
+ $\tilde{B}_{\nu_{Lit.}}$ / \si{\per\centi\metre} & $r_{\nu}$ / \si{\pico\meter} \\
\hline
- \multirow{3}{*}{\ce{H^{35}Cl}} & 0 & \num{10.4408(4)} & \num{128.823(2)} & \num{10.43982} \\
+ \multirow{3}{*}{\ce{H^{35}Cl}} & 0 & \num{10.4408(4)} & \num{10.43982} & \num{128.823(2)} \\
\cline{2-5}
- & 1 & \num{10.1360(2)} & \num{130.745(1)} & \num{10.13264} \\
+ & 1 & \num{10.1360(2)} & \num{10.13264} & \num{130.745(1)} \\
\cline{2-5}
- & 2 & \num{9.829(4)} & \num{132.77(2)} & \num{9.82546} \\
+ & 2 & \num{9.829(4)} & \num{9.82546} & \num{132.77(2)} \\
\hline
- \multirow{3}{*}{\ce{H^{37}Cl}} & 0 & \num{10.4248(3)} & \num{128.385(2)} & \\
+ \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)} & \\
+ & 1 & \num{10.1214(1)} & & \num{130.2953(9)} \\
\cline{2-5}
- & 2 & \num{9.86(2)} & \num{132.0(1)} & \\
+ & 2 & \num{9.86(2)} & & \num{132.0(1)} \\
\hline
\end{tabular}
\end{table}
\subsection{Determination of Vibrational Constants}
-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 in section \ref{sec:vib-const-deriv} within the supplementary information.
+The harmonic constant, $\tilde{\nu_e}$, and the anharmonicity constant, $x_e$,
+in table \ref{tbl:vib-const} were determined using equations \ref{eq:nu-e} and
+\ref{eq:xe}. The derivation method of these equations is included in section
+\ref{sec:vib-const-deriv} within the supplementary information.
\begin{align}
\label{eq:nu-e}
x_e &= \frac{1}{2} \frac{\tilde{B_2} - \tilde{B_1}}{R(0) - 3 \tilde{B_1} + \tilde{B_2}}
\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 $R(0)$ was negligible compared to that in
+The uncertainties in these values was estimated using equations
+\ref{eq:nu-e-uncert} and \ref{eq:xe-uncert} with the uncertainty in $R(0)$
+excluded from the propagation as it is 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.
\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 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.
+The bond force constants, $k$, in table \ref{tbl:force-constants} were determined
+using equation \ref{eq:force-constant} and the error in $k$, $\alpha_k$, was determined
+using equation \ref{eq:force-constant-err} (derived using the calculus-based
+approximation). The errors in $\mu$ and the constants were again not considered
+as they are negligible compared to that of $\nu_e$.
\begin{align}
\label{eq:force-constant}
k &= 4 \pi^2 c^2 \mu \tilde{\nu_e}^2 \\
+ %
\label{eq:force-constant-err}
- \alpha_k &= 8 \pi^2 c^2 \mu \tilde{\nu_e} \alpha_{\tilde{\nu_e}}
+ \alpha_k &= 8 \pi^2 c^2 \mu \tilde{\nu_e} \alpha_{\tilde{\nu_e}} = 8 \pi^2 c^2 \mu \tilde{\nu_e} \sqrt{ \left( 3 \alpha_{\tilde{B_1}} \right)^2 + \left( \alpha_{\tilde{B_2}}
+ \right)^2}
\end{align}
\begin{table}[H]
\end{table}
\section{Discussion}
-\subsection{Errors and Justification of Analysis Method}\label{sec:errors}
+\subsection{Errors and 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.
+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
+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
+\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
the centrifugal distortion was accounted for the residual plot showed a more
random distribution.
-\subsection{Discussion of Obtained Data}
+\subsection{Data}
\subsubsection{Rotational Constants}\label{sec:rot-const}
The value of $\tilde{B_0}$ could have been obtained by considering either the
projects / chemistry / university-chemistry-lab-reports.git / commitdiff
