From 8b59a67c8bfa26d97a9ae9d94dcef3fc7b2c3191 Mon Sep 17 00:00:00 2001 From: Sam W Date: Mon, 3 Dec 2018 23:16:50 +0000 Subject: [PATCH] Started discussion section of experiment 4B. --- year2/4b/4b.bcf | 1 - year2/4b/4b.pdf | Bin 269837 -> 271831 bytes year2/4b/4b.tex | 129 ++++++++++++++++++++++++++++++------------------ 3 files changed, 81 insertions(+), 49 deletions(-) diff --git a/year2/4b/4b.bcf b/year2/4b/4b.bcf index 44461dc..8633058 100644 --- a/year2/4b/4b.bcf +++ b/year2/4b/4b.bcf @@ -1983,7 +1983,6 @@ hughes-hase-uncertainties - crc-handbook diff --git a/year2/4b/4b.pdf b/year2/4b/4b.pdf index 49e095973db4a6932547ef8a99e0b10169287909..9df33317517b56bebe6cfd53220ea52019ecfad7 100644 GIT binary patch delta 74962 zcmV)pK%2jfyb#yR5U|e-0XCD+3@(4R5x(nJ%yGBuf$@H%$`|EXt{f6uH`|BxzPwdh zN!gX+_>$kA21tNt5ClmRlI^NoW~OGq#z&yP?nXC=p{M&3J^l3sU3h=_>ctm-Vd}&v zan!NXtGg4^jA&s_MmxfoIla0$eS7ihl2Lncb;;Gm5B}w~f4RLp3ra7z|NMW4%d=$c z;%oo;t$%s%U#`-phxEgn^uz1n`bp|@YV?%8yGqww4}FGq;ljJDdH5pq^kW9??WGbI 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-\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} @@ -203,13 +200,18 @@ the linear regression respectively. \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}}}} \\ @@ -217,10 +219,6 @@ used is insignificant compared to that of $\alpha_B$, hence they were discarded. \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} @@ -246,10 +244,10 @@ The reduced mass, $\mu$, was calculated to be $\SI{0.972937750}{\atomicmassunit \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} @@ -259,8 +257,9 @@ included within section \ref{sec:vib-const-deriv} of the supplementary informati \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} @@ -289,12 +288,11 @@ since it is determined by reading the 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 $\mu$ 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} @@ -327,6 +325,41 @@ 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. + \printbibliography -- 2.25.1