From 3b6df52042a7be77240918a99a7b1ff55e7bf742 Mon Sep 17 00:00:00 2001 From: Sam W Date: Sun, 2 Dec 2018 23:38:57 +0000 Subject: [PATCH] Continued with experiment 4B lab report and added figures. --- year2/4b/4b.bcf | 1 + year2/4b/4b.pdf | Bin 146061 -> 269837 bytes year2/4b/4b.tex | 138 ++++++++++++++++++++++++------------ year2/4b/figures/cl35fu.jpg | Bin 0 -> 59270 bytes year2/4b/figures/cl35ou.jpg | Bin 0 -> 60741 bytes 5 files changed, 93 insertions(+), 46 deletions(-) create mode 100644 year2/4b/figures/cl35fu.jpg create mode 100644 year2/4b/figures/cl35ou.jpg 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 54ab3970104ffd7babcc6118cf90b3974c4cb822..49e095973db4a6932547ef8a99e0b10169287909 100644 GIT binary patch literal 269837 zcmeEu1z43!(D0$VyFyE~=38wCUekuFhEML;Q~Q$T6{0}AR5uU@&| z^Z(EPz0Y~h?#|54%zl*!^1?F88d;e* zm=d!xbFdPhKd_AAW)==ccEpV07J3dwqDBVRhDNab{IK>8c1C)Zu+HP1=`!|{^(fsN z2k5+!H00PD5>~gmQ|UDjQ*WxhpI&W6;x47plDFWFvd)Br*#WqFi{!>VR-D7BIvfx` z1t72DY|mJO5uG@9j!bP>Rc3f>bP|KZb@j}AY$rC<9j|ZIYEK((Upt)HU3y=;Q|Q&* zK2>Qwy2Q|YfRFTPYjaCGaOrXP9W;4`LD7b|NNaa2q$g3p@bW}^BO<($9qj@JjrV(XMz(Z5EMqSvhnH(Ks#=Yk0As~fyl$N^1ff)9;-G8be9TYcXE&pxjW*9S0Jai zFFw{1uDID&Jq?PgM~VW8MsstY?_e8nj&X?pz-o= z^Y@Wa*$b$hf@qinnu)ulo+mJ!ykrR^pJ1ErYAo)9NlVRj7*}E_eTXzO5z^2dE|hP4 zj5&$*W>e9IYaTaTRXbcy#_Uuz*CvXHcNPk%G~F$5p8lbGhgEaxcm$;?*|yGbSFbM)|#Q(PTS{F*H$OlRj~rX%;=h4DaEb0&3`T-B$LHR}C>9|z>1*EOR* z;8}#n5vM7UMIqaM2cVmC!kPcdIRg{S|K*n1Be?!H8^n3Jo`ny2f`K_C;c|#; zVMdE&iL!oVv13x}dPDjv)5rCMyv6iVo2#c>&W|v73zLUKshkO)Z!UB8H}y&v;T2~M zMC_H&I-6l!Us<-s^V~+X9?EGlBpuwT`+$s2&ai5IjBtGOF1c)r@$0zS!uynK+<;D0 zJ1fL>HQkDRfdX-!)b@Hz8G6sV9@l(ncW%d%k29^&w^r%uikTWt@un2oGi8rcd@Qs_ zc-X|9Qts%kmuD}8Fb-1?p3HC@lrYu;o^_O!OlQS1*=v!<-ejv%u0+(7`i7)y+3I>X@6`L;-pw=*H@n}LBw;}TVe9Ca2WG$ zgrZQsPDT6>0&0?O91<7SDlvXdXNmCPkhC;L(ipy`oqVkGsE=6^I$YF3e7vJ{Bu)vg zixi2ZG`6U5YqB?utKm0;1o;B(VeE@ER*O}&x}9=V9QCmtcrzmU>i~3 zMayY>(afb;D<3KPt-okSt8M4dF6z8VW@NsN>lqVG(3VS{XF#6Xw!=R~O z1x@MU(I0`Ah1B4_eT{6smn zCAfdI2Z-t*_FU0_?q|d$kfzyPNxOySsoUOa`K)fP(2wvHD&HPK-B61{P`z4VLH)Aa zk*Pby5l!ZU#U`4@j^}d7A$j3b%CRChvooUKaCnl%vUnztDMdlcvBLEcM$X75-y#YzevVJf9qO1%DPOc{a=>{5!daM|)V_dc6Z zxf+t^v`oW2=5{TWYmNy za}P!}FFL#xR{bE?bNG9a1Mmk4d|h2E@uH|Kn$jtjj!9t)La0(by3|sYbm$K=s>vax zI*9$?n5$7nHh6bdo}PrXnu?f>v}bWvMYDUr1Lv!*BE<80YMjXmwkPDou0b%egNS&=Y+NZemS~Ont$k8mwb!T-?g_CnP7V$0N~qw<+L7-& z$#FmRk9MnVW#b=VQ`=9&fLeLeM@hC`m4_jE-NI1xx-2m%0f!CzU0;_sD)rd_5`qI* z&d3SV_r^!cK5B04ZEXDZR`>W#)Y?jJk+3!Ei<82b_yv$XQSKxEbWagI)-K(K+^k&L z*$pj9RN^iF?d52!#5+6rZoY^X9kz{N*UPmAJru%$eIx`$Y?J-bJiyG|c+oslt-NaT ziB!*^lBDn=X6%u=i*BxZlBjaCbun4_3)rxaezd5}`^`oo;cw|HgpMAe3d}%pq^-9p z2;B{mKSG=^ohgJT&i-`H4d@Z7we{|wip77KRlQhJwHoPRdqCbt(o%P7+X^0g$djJU zFaswEijL2|Aa~()0&3*`b3VUZ3hhOqbqKwW%l=X~fYzJ%pvX=~y_vl^PCOgBRfo#1 zw#;e94l+s*#ZH3;l{Tu}sX0YF8{)pstJ!6tVYaFjseMs`q^g2KLlIlPlW+wr=d&5% zC!_LmJpp=TuYWOt--NuaI>Ky15a&=a?7V^F$%Vlw`e_{hw8pr_fzaRb5a`8DG9I>$DpyFdy`LE|I>^`SClaPY#s77 z23mrP(cs!>8=%o_{-|~bziy~);QGA(R4LXMcFO_p*T-SEfR2AJA=Y@eO}~x`^Snf# zlcxcTS@(oOsTumzW?Xc)dA5K~Dj?Hmw+6v&7uaT2&-?+c z$5j)Rl|5gEwAxkAZL1kB1n5Fh@?UqsO(h?nxqCQfz4x}azwnC|@cA?RCFV$%UZl@lcy9jgHPJZ!abHs7!*#dT3@gEuu~R1S{MaXlS;c3OvGOuC+^EBYa}$hT+BagL#-O`>=$X7yAA#w)C9#}dU!(bFJUUZRjoX8oLM3Cb<$UiO2YKbRSJ0BA$aF3 zVdoQ|RlD`p>$c+Y24{67dUWmD48-FB1w>5f5Q!4>QBZ}E4XNe(B$#|Q#xMa<^*!E;qUJJ((;}Kw1j=v=nr={Wxy4aCDK`<8JDD1Kt7rT3bYA_54iO;K&60c+eg4isrqi z9;F?DQyu+Q{T@WWjSIO$T>9 zaay(g1z)zIGq-`+`Q!0xK7Go>wm5`S*O7879aak8pninNy)d6yM>ricB)d~VbtoGJ zlQNj~{DnGTt3hLflCdgZ=HP{y|+OH0EY zGS}D~a&tV8+d1AK3#*EkeA_z$T^cgll-pcOD~_~+`bf!M4U?5knSA@&BSYM9pS9Ad zaLacJpBh(K?J4}8^vVhr50@ZX3E4`8#|9Y>b7lHLORr_`-(bSVt$U@pwTLT+K34KD zoQWUO>hvZKu1sDPrBuhD=nZ@Od`t^m_cz?NN88A?&nQ1(EA2SLc1<%k>HGT*C5Q~v zpeQ*ksp2|=hViJK*w#K7b09fs3naR?u=FOHX5#ubj+TuU|<~_&p5;jIZr`s zsaB^%m^OG2Bcjl|pF&M>C`~%N4Q_s5=;dz&i!b&M)8#CT(Ww)pXBYl3u zlt}{JrpG@bPocb}42-n3<;erwoRHdUA&Fpth+KJj9fP0}*ytHHix3+z*mwI!*G!i~ z?;C{Y#xi+C+cg^1a7mCr`>2wJQ(N0q$P~snxD=u-`Eo#T*iI1NM3M|~WR;BbpgGSCLezvjbA`;JNG6MG4ddC7p&v`cZ5d5*8=ZrQA`PyBCY1~KNslYUu&_$7U*a` zX2D$@K~%MLfXBgl{i<*z#-nH?#6v%64*77Do5bX>^U<21IrpPb5nw+G#!+#J7|
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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 0000000000000000000000000000000000000000..fe3bdd09750731e0cfb2f9881b99eeee74e26a8a GIT binary patch literal 59270 zcmeFa2RxN;{5O7%ePpk)vS%o($lmkVJ4!Ys$0{kZcQVV~TS_!2m6erjB~-}Bs3`hM z)N`N1v5KhE@AZHF&+j;|b6?|gecqpSjr+QxJYRlZQiUt%!0IfFUV$-2V&^33(K`*RlrxR-=8C)YL1M0#lY@rLCm@ z8!TZ(H0T^1d9C6V=|2^Q3|xmjERed6P}qKo@#hEMU%8Yfg~2l4!mnQ z&ZhRkwQQ9Tl?gC!emY{X!cefb#*!NdYkSwTthBXY((8M){a&a*`EAdt;z5Qq{wBY0=5`(L2Pz3WHQ_N~b*t*ObZsV*H$ zrc5L!C8JGBCb~ufL>P7&2!uuX?)aUj+We9$0;BFifJd-Ll7$^xGKPvKDpiqYUqX~m z4XJOXfwImrh#NGejHP$12X&AuoUknFF|h*tvL(^^WokNoi{Zt~0S5wMjA@M=wRbM6 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