From: Sam W Date: Sun, 2 Dec 2018 23:38:57 +0000 (+0000) Subject: Continued with experiment 4B lab report and added figures. X-Git-Url: https://git.dalvak.com/public/?a=commitdiff_plain;h=3b6df52042a7be77240918a99a7b1ff55e7bf742;p=chemistry%2Funiversity-chemistry-lab-reports.git Continued with experiment 4B lab report and added figures. --- diff --git a/year2/4b/4b.bcf b/year2/4b/4b.bcf index 8633058..44461dc 100644 --- a/year2/4b/4b.bcf +++ b/year2/4b/4b.bcf @@ -1983,6 +1983,7 @@ hughes-hase-uncertainties + crc-handbook diff --git a/year2/4b/4b.pdf b/year2/4b/4b.pdf index 54ab397..49e0959 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 3afa54e..4f695f9 100644 --- a/year2/4b/4b.tex +++ b/year2/4b/4b.tex @@ -175,17 +175,52 @@ state yields equation \ref{eq:comb-diff-lower}. 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, $\tilde{D_{\nu}}$, shown in table -\ref{tbl:centrifugal-distortion-const} and hence the rotational constants, -$\tilde{B_{\nu}}$ shown in table \ref{tbl:rot-const-bond-lengths}. - -The bond lengths were then determined from the $\tilde{B_{\nu}}$ values using equation \ref{eq:bond-length}. +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. + +\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} +\end{figure} -\begin{equation}\label{eq:bond-length} - r_{\nu} = \sqrt{4 \pi c \hbar \mu \tilde{B_{\nu}}} +\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} +\end{figure} + +\begin{equation} + \label{eq:b-err-prop} + \alpha_B = \sqrt{\left( \frac{3}{16} \alpha_m \right)^2 + \left( + \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. + +\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 +\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} @@ -194,42 +229,26 @@ The bond lengths were then determined from the $\tilde{B_{\nu}}$ values using eq \hline & $\nu$ & $\tilde{B_{\nu}}$ / \si{\per\centi\metre} & $r_{\nu}$ / \si{\pico\meter} \\ \hline - \multirow{3}{*}{\ce{H^{35}Cl}} & 0 & & \\ + \multirow{3}{*}{\ce{H^{35}Cl}} & 0 & \num{10.4408(1)} & \num{128.823(2)} \\ \cline{2-4} - & 1 & & \\ + & 1 & \num{10.1360(2)} & \num{130.745(1)} \\ \cline{2-4} - & 2 & & \\ + & 2 & \num{9.829(4)} & \num{132.77(2)} \\ \hline - \multirow{3}{*}{\ce{H^{37}Cl}} & 0 & & \\ + \multirow{3}{*}{\ce{H^{37}Cl}} & 0 & \num{10.4248(3)} & \num{128.385(2)} \\ \cline{2-4} - & 1 & & \\ + & 1 & \num{10.1214(1)} & \num{130.2953(9)} \\ \cline{2-4} - & 2 & & \\ + & 2 & \num{9.86(2)} & \num{132.0(1)} \\ \hline \end{tabular} \end{table} -The error propagation shown in equation \ref{eq:b-err-prop} was completed to estimate the uncertainty in the -values of $\tilde{B_{\nu}}$ where $\alpha_B$, $\alpha_D$, $\alpha_m$ are the uncertainties in -$\tilde{B_{\nu}}$, $\tilde{D_{\nu}}$ and the gradient found in the linear -regression respectively. - -\begin{equation}\label{eq:b-err-prop} - \alpha_B = \tilde{B_{\nu}} \sqrt{\left( \frac{\alpha_m}{m} \right)^2 + \left( - \frac{\alpha_D}{\tilde{D_{\nu}}} \right)^2} -\end{equation} - -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. - -\begin{equation}\label{eq:r-err-prop} - \alpha_r = \sqrt{\frac{\pi c \hbar \mu}{\tilde{B_{\nu}}}} \alpha_B -\end{equation} - \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 derivation of these equations is +\ref{eq:vib-energy} was 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. \begin{align} @@ -245,18 +264,37 @@ since it is determined by reading the wavenumber directly from the spectrum. \begin{align} \label{eq:nu-e-uncert} - \alpha_{\tilde{\nu_e}} &= \sqrt{ \left( 3 \alpha_{\tilde{B_1}} \right)^2 + \left( \alpha_{\tilde{B_1}} + \alpha_{\tilde{\nu_e}} &= \sqrt{ \left( 3 \alpha_{\tilde{B_1}} \right)^2 + \left( \alpha_{\tilde{B_2}} \right)^2} \\ \label{eq:xe-uncert} \alpha_{x_e} &= x_e \sqrt{ \frac{ \left( \alpha_{\tilde{B_1}} \right)^2 + \left( \alpha_{\tilde{B_2}} \right)^2 }{ \left( \tilde{B_2} - \tilde{B_1} \right)^2} + \left( \frac{\alpha_{\tilde{\nu_e}}}{\tilde{\nu_e}} \right)^2} \end{align} +\begin{table}[h] + \caption{Vibrational Coefficients.} + \label{tbl:vib-const} + \centering + \begin{tabular}{|c|c|c|} + \hline + & \ce{H^{35}Cl} & \ce{H^{37}Cl} \\ + \hline + $\tilde{\nu_e}$ / \si{\per\centi\metre} & \num{2885.821(4)} & + \num{2883.78(2)} \\ + \hline + $x_e$ / \num{e-5} & \num{-5.32(6)} & \num{-4.5(3)} \\ + \hline + \end{tabular} +\end{table} + + \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 the calculus approximation (equation \ref{eq:force-constant-err}) where the error in $\mu$ -was assumed negligible compared to that in $\nu_e$. +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. \begin{align} \label{eq:force-constant} @@ -273,7 +311,7 @@ was assumed negligible compared to that in $\nu_e$. \hline & \ce{H^{35}Cl} & \ce{H^{37}Cl} \\ \hline - k / \si{\newton\per\centi\metre} & & \\ + k / \si{\newton\per\metre} & \num{477.383(1)} & \num{480.698(6)} \\ \hline \end{tabular} @@ -283,14 +321,22 @@ was assumed negligible compared to that in $\nu_e$. 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. +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. \printbibliography \section{Supplementary Information} -\subsection{Centrifugal Distortion Coefficients} +\subsection{Centrifugal Distortion Coefficients} + +The values of the centrifugal distortion coefficients, $\tilde{D_{\nu}}$, were +determined from the gradient found by completing linear regressions. + \begin{table}[h] \caption{Centrifugal distortion coefficients.} \label{tbl:centrifugal-distortion-const} @@ -299,17 +345,17 @@ hence greater uncertainty in this value. \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)} and \num{5.4(3)} \\ \cline{2-3} - & 1 & \num{-5.11(2)} \\ + & 1 & \num{5.11(2)} \\ \cline{2-3} - & 2 & \num{-4.3(8)} \\ + & 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)} and \num{9(2)} \\ \cline{2-3} - & 1 & \num{-5.13(1)} \\ + & 1 & \num{5.13(1)} \\ \cline{2-3} - & 2 & \num{-11(4)}\\ + & 2 & \num{11(4)}\\ \hline \end{tabular} \end{table} diff --git a/year2/4b/figures/cl35fu.jpg b/year2/4b/figures/cl35fu.jpg new file mode 100644 index 0000000..fe3bdd0 Binary files /dev/null and b/year2/4b/figures/cl35fu.jpg differ diff --git a/year2/4b/figures/cl35ou.jpg b/year2/4b/figures/cl35ou.jpg new file mode 100644 index 0000000..ae1e9e1 Binary files /dev/null and b/year2/4b/figures/cl35ou.jpg differ