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zp;C-D%JaHpm8LmjQqZ)yr%kboNwD*hnUc|LN!at!>yo??Nyzh39g>`iJPQgwyT&LL z`fa-EOK`0kAOpe8rFuHZXWvSEe;=80*uC!YQqh~18nuC&r~3-Zu7x%_nCn|fb|{Kk z$Nx;T&svHxeQ%#iF#|JQmGhJm9J|+>xbH!~KOnaf8BNkw9H2?yx&8zGsi?#ifbjnV DnMxaA diff --git a/year2/4b/4b.tex b/year2/4b/4b.tex index c7b5f64..6d186e2 100644 --- a/year2/4b/4b.tex +++ b/year2/4b/4b.tex @@ -1,7 +1,7 @@ %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. @@ -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 - 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} @@ -179,10 +179,11 @@ of this form. \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} @@ -215,10 +216,11 @@ intercept respectively. \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) @@ -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 -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 \\ @@ -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 -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 @@ -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 -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.} @@ -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 - \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} -- 2.25.1