From: Sam W Date: Tue, 4 Dec 2018 23:27:28 +0000 (+0000) Subject: Finished initial draft of experiment 4B results and analysis and discussion section. X-Git-Url: https://git.dalvak.com/public/?a=commitdiff_plain;h=aa7a090eb63055a3cfde3dec61cacb50c83a3905;p=chemistry%2Funiversity-chemistry-lab-reports.git Finished initial draft of experiment 4B results and analysis and discussion section. --- diff --git a/year2/4b/4b.bcf b/year2/4b/4b.bcf index 8633058..4af1ab5 100644 --- a/year2/4b/4b.bcf +++ b/year2/4b/4b.bcf @@ -1982,7 +1982,10 @@ specific.bib - hughes-hase-uncertainties + nist-hcl + hughes-hase-uncertainties + crc-handbook + biernacki-clerjaud diff --git a/year2/4b/4b.pdf b/year2/4b/4b.pdf index 9df3331..a571df4 100644 Binary files a/year2/4b/4b.pdf and b/year2/4b/4b.pdf differ diff --git a/year2/4b/4b.tex b/year2/4b/4b.tex index c2d268d..91cf5cc 100644 --- a/year2/4b/4b.tex +++ b/year2/4b/4b.tex @@ -33,6 +33,7 @@ \date{30/11/2018} \maketitle +%TODO: Write abstract. \begin{abstract} Experiment abstract. \end{abstract} @@ -200,12 +201,22 @@ intercept respectively. \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}. + +\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 +$\tilde{B_{\nu}}$ 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 +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. @@ -223,21 +234,22 @@ compared to that of $\alpha_B$, and hence they were discarded. \caption{Rotational constants and bond lengths.} \label{tbl:rot-const-bond-lengths} \centering - \begin{tabular}{|c|c|c|c|} + \begin{tabular}{|c|c|c|c|c|} \hline - & $\nu$ & $\tilde{B_{\nu}}$ / \si{\per\centi\metre} & $r_{\nu}$ / \si{\pico\meter} \\ + & $\nu$ & $\tilde{B_{\nu}}$ / \si{\per\centi\metre} & $r_{\nu}$ / \si{\pico\meter} & + $\tilde{B}_{\nu_{Lit.}}$ / \si{\per\centi\metre} \\ \hline - \multirow{3}{*}{\ce{H^{35}Cl}} & 0 & \num{10.4408(1)} & \num{128.823(2)} \\ - \cline{2-4} - & 1 & \num{10.1360(2)} & \num{130.745(1)} \\ - \cline{2-4} - & 2 & \num{9.829(4)} & \num{132.77(2)} \\ + \multirow{3}{*}{\ce{H^{35}Cl}} & 0 & \num{10.4408(4)} & \num{128.823(2)} & \num{10.43982} \\ + \cline{2-5} + & 1 & \num{10.1360(2)} & \num{130.745(1)} & \num{10.13264} \\ + \cline{2-5} + & 2 & \num{9.829(4)} & \num{132.77(2)} & \num{9.82546} \\ \hline - \multirow{3}{*}{\ce{H^{37}Cl}} & 0 & \num{10.4248(3)} & \num{128.385(2)} \\ - \cline{2-4} - & 1 & \num{10.1214(1)} & \num{130.2953(9)} \\ - \cline{2-4} - & 2 & \num{9.86(2)} & \num{132.0(1)} \\ + \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)} & \\ + \cline{2-5} + & 2 & \num{9.86(2)} & \num{132.0(1)} & \\ \hline \end{tabular} \end{table} @@ -316,49 +328,139 @@ compared to that of $\nu_e$ and hence were discarded. \end{table} \section{Discussion} -Two methods able to be used to determine the value of $\tilde{B_0}$: using -either the absorption data from the fundamental or the overtone transition. Much -larger signal:noise ratio for overtone due to lower probability of transition, -hence greater uncertainty in this value. For the values of $\tilde{B_0}$ only -the data from the fundamental transition was used due to the lower uncertainties -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. +\subsection{Errors and Justification of 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. + +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. + +\subsection{Discussion of Obtained 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}. + +\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} +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 +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. + +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. + +\begin{align} + E_{\nu} \psi_{\nu} &= \hat{H} \psi_{\nu} \nonumber \\ + \label{eq:s-eq} + &= \left( \frac{\hat{p}^2}{2 \mu} + V(x) \right) \psi_{\nu} +\end{align} + +\subsubsection{Bond Force Constants} +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. + +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? \printbibliography @@ -395,7 +497,10 @@ determined from the gradient found by completing linear regressions. \subsection{Derivation of Vibrational Constant Equations}\label{sec:vib-const-deriv} The energies associated with the discrete vibrational energy levels in a molecule are given by equation -\ref{eq:vib-energy}. +\ref{eq:vib-energy} which can be found through the application of perturbation +theory on the harmonic potential with the perturbation of the potential +including terms a of higher order than two from the Taylor expansion of the +potential energy. \begin{equation}\label{eq:vib-energy} \tilde{E_{\nu}} = \tilde{\nu_e} \left( \nu + \frac{1}{2} \right) - diff --git a/year2/4b/specific.bib b/year2/4b/specific.bib index ec46ffc..0a6b964 100644 --- a/year2/4b/specific.bib +++ b/year2/4b/specific.bib @@ -1,17 +1,21 @@ -@online{berkeley-pigments, - title = {Photosynthetic Pigments}, - author = {Brain R. Speer}, - date = {1997-07-09}, - url = {http://www.ucmp.berkeley.edu/glossary/gloss3/pigments.html}, - urldate = {2018-11-06} +@online{nist-hcl, + title = {Constants of Diatomic Molecules}, + author = {K.P. Huber and G. Herzberg}, + editor = {P.J. Linstrom and W.G. Mallard}, + url = {https://doi.org/10.18434/T4D303}, + urldate = {2018-12-04}, + organisation = {National Institute of Standards and Technology}, + note = {(data prepared by J.W. Gallagher and R.D. Johnson, III) in NIST Chemistry WebBook, NIST Standard Reference Database Number 69} } -@article{chlorophyll-uv-vis, - title = {Chlorophylls and Carotenoids: Measurement and Characterization by UV-VIS Spectroscopy}, - author = {Hartmut K. Lichtenthaler and Claus Buschmann}, +@article{biernacki-clerjaud, + title = {Force constant change upon isotopic substitution of hydrogen for deuterium}, + author = {S. W. Biernacki and B. Clerjaud}, year = {2001}, - journal = {Current Protocols in Food Analytical Chemistry}, - volume = {1}, - issue = {1}, - pages = {F4.3.1--F4.3.8} + journal = {Physical Review B}, + volume = {63}, + issue = {7}, + pages = {075201-1 -- 075201-6}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.63.075201} }