From ac0255b735085fe40ce110a59ba92ef5ba97e108 Mon Sep 17 00:00:00 2001 From: Sam W Date: Tue, 20 Mar 2018 22:27:33 +0000 Subject: [PATCH] Expanded and corrected copper exchange error propagation and completed zinc calculations up to the error propagation. --- project/project.pdf | Bin 289553 -> 310006 bytes project/project.tex | 259 +++++++++++++++++++++++++++++++++++++++----- 2 files changed, 232 insertions(+), 27 deletions(-) diff --git a/project/project.pdf b/project/project.pdf index 89118b17d93a77f9dc83c90efeb0a955d33ffed8..afc156f3015bd6603c460fcf5185d2f06d94734c 100644 GIT binary patch delta 97323 zcmZs>Q*@wR)U6rYwr$%^Dz4=Ahl2)PSE4O) zlv{v|j2NTLxL0-ZzxPic9~n@NN48XbmO3dVrM@z0D(*=6FBdAakDZZ?Y~n@<*~BS* zO6)b+H2B<$;R^yas*ZJvK8E*_S*-lCB?c|cw~^NO8sg2y-zt;z7JB#PeSm>&AI^d9 z8n;8ozJJSM>)vr^4Hq{hdAFl(p8d}w9Gf>IL*g{s2J-G-8aHi`0c#2d+=2bJo~Hz@myb5YYeWk zSz-${tzQJ&SoqiO7Jt?(2!Qzt>Tik7_uh?Bh1z(M&Qc;D<+aEoIt?F}1|#h?i-p~x z>SIMIVX$HP(vDNFdfK(AR9y`ItV?j0ys(BL-``7VTSc1=VBm2s)j!fi^$e0lWV}SC z?L>o?S(COLFkA87*}i5@m_W6Ot~L-oyH{uPthF8!?IW}%Q5C$Ad;stKUI(zanR=vm zsPqpfE-crlD>pkk zwuq4nM&mqL<0M6vAmiIyme3~kpe{3F*1@-lR~%=Wd?l38odl~Uym}9bg_7sjg;+PS zvSFZVjr>)?Db;S8*)K5VP9Tr9>9o zvL{kxl(4%Pm10ujLcozhA;Wi3ptWq6QnKYff4e^rOiN~clmR$id-bYl85lG_d8^<& z9vLjHLff91Q)0tc|DtbTcs$uyzsF5b9IN4HC=qHJobR~hN{Na`ziJsh6~Et=k05TV ztc{8O!~HT=Yeqfze8HZYX4q|{NrAw2|M~ao#*^I!WkzG6PPV8`;OSY<@r1He(l(Eg zHTxutx68qHwGdD+Z%8D^>XI*)kJtI9Ld0Ro5Ut*RS-|3E+8;QxEKxMMCz4+!M5Myb zND4O{xlNl$`_VieD^KkUNt%taQql3zUz4<*#EV2+!bYmy%yAa?cfdAQaYOTRSck)* ziBYFqSFdoOI#RjroUC6(KFp}AV6;B6$Rc=5AP85coCkP+It>#Eg;Wjp03X}DC+@as z=!z6JY$Fvhqcmp$wHf;1A-T^?!6#U!zalQHk3G!}U8-ZkppTsag={wB*hK9p@4fwgX6S!lvBn|Bj^|-Wlla zV%8u0ih6=%9lQF7Zc?%`7iUw6l)BHX9JZWSEK7QNzDzf=XdJ*R!aPt)^FvD5lE_q- z31P}yo_O2ni+46>Q>qR>@4pZXg4nCpj5(aj(FV3KWE&uB55z(_&6P5l2%qa^Z~EIrakysDyzkd_Flpj2Ssp-{sP@+4aEsKFgB zPDgtVy(zh&BG~5scjz0%m!D(CZ#b0Q^y>Bykj&MVx%N;Wv4-mn!QU)Rbzo~d?na*l za{v+O=VJt357`ZMZ`_gyc$YWSI#L`-t=wTLv_gZM>{>4C6*UUHjv zhD}}lU7*cjEajuzgmnA!L2O?$cY*U_-|%kUeck;;be|@L0}E1ktL6+81R?mKw|wDR zyT{HBSNlX-J4Wrx0viN;Cn;Iyuh@_D0-!%*IEXB?{S^n_H3Hn0>+mOqg##64`Ear&>-v*R-<@bZXHnyW6Z%^fEJVBt9WntF`@SF< zLgWbdmOa?On)nT9he%k=yAa+TimmENbfBjjfc@iVeHnIh%SblN{$5`5K$_75>$#mUj+^gO7R z7lAywa_8HTN^`=@vrnuxw<47L0XdKE==i>a)|{3?vPc zl$-qf?4NX6P*;i2I^Wgw!9ilZqr2`FAmpXK!Z(v3tHnm5Lop-)Ll(ypqQ!FN$IJBK z`kXOJ;(O-J6@kgzfo}`Xumb72em*%M)L2c*$MJO_@D4 zEaGQY^8D?zN`-z__z%@qDvJmcX(53q%`tJ)R;fj7$jMl}NEexLEfk~=ck@$ee7$u!GMJbOm(_6$wzojd*FeAKI4SzqP?uEj#Am5|8J zlG;^!yCoYn1`|{A!KT6!(06;#vW2~{?wOJwCa%qbF$JO3jRv_TstnGBE~d)S#A~rO zNbb?tzP~;O!!5&xfa)Lc0fn&aqk1Da=ui=s8Sk={IY~dx70&ine#+NyXDsd52!N~z zb=~}P2y+%}c%Rxa`-^w_mV!A)?K#P46T(Y|Wssy)4oM85;Qy-xSR%OvdHF;kl3IMd zJuGP5;%N%26bU{2(8V5bJUkcrsg||w;=T+w(z~K4l_-5RN0HGP-M(C%}SBau%nC_!Vaj?Q@Xu%&6bNsR4`i}Fy zG&vjqs+ujDW5M6Fozw$qp)pQ>8n&z@;pvxNNilRq;GXkHKd=yG61t{TY_@_2Xo zyt5=*L{>wuC7wE9Wqz=2&}o{eYZE5YsDAVF^nKb&?N6d6ON?Td?UWth(VDRLc$%sc z$UNMShqapQ`un5DBNe73a|!Ok!qx5RkCj+6i|i;nRV*%A&{Y)f)$bT_Gg*hsN4|*= zW4Tx3>g=5zpmlPssmOm|7M{jW3x*S}6|MhgZ#1ER7*RVC)c5x9kWiaTdiPTo2=iML z+%iIw9@8kRbY3wX%kGfsmwPs2gUX)A%t5nn61OgKvote@xYGi?@P zMX0iNYt*xn>+1uhFK)^WyUCK>NW7Sfaj>G6#Jns_Z(@n3`NXS2wV0^)Nw0xEyI$ae3T$Pj~24k!S|aatO`x$Db;p)rxHfx@0HV>bR|47NJjE z+r7-(KvjJa1n=AwZysZ74YT8Q@V4i7ZQ$io)e>FP`UB$7U>7QvuZG2IBa}~_PW|&`hSQAd!~KN zKE5dP#cKWCrbnElgfkSvaaf?6g2&WV&aGhxXb#j1C(jvgYb;cVHfbnvJ@#wT1@bDm zC-JEp>cWOT(oFtRaKXa#HU=!wh38Ne5)!5HyN<2ntp+%b{~by24%Qq)bBZ(s)hu6m*2?r3PZUPfedU zEB8lm>Ji!b>a8*Vg0PSFJG3v1V9E%Ob+`y!n{1Nej*h!6&Q|eLm`$`ch`;|L+Y!R$ zk1U1$gnbfpR}ozHTtLNaQZVqXuQjCt#vYWmZyiuy)bEC4O9sGETbHHnd^RH{Z(1R9 zxh~3U#33d3>Wve;0r;P9N)zua}0^D2zqoK|; 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zgpFN7RFLHV-}3J`Idca~H!Bi0W=^1R8x$>Irh_d+1Y>CT?yBb3cW1>tg#ijtJ#-PY zr@UDfbT;b=oNCcZrP(qiWviDT$?&w zOov%S5CJ%M4pl=Vj_03C&s76or4vv{P)8$K6#i7gwpRIR6By%~&5&rWqIec4DvFI% zl)#=q@-2bcnv$U?r7^){PJ~wm`;-!O4)UVt{*chWIH7US_z%NFsmx{L$)IG~Yo sXO2rJ$t ZnEDTA^2-} +\end{equation} + \begin{table} - \label{tbl:zn-standardisation} - \caption{Titration results from standardisation of EDTA solution with standard zinc sulphate solution.} + \caption{Titration results from standardisation of \ce{EDTA} solution with standard zinc sulphate solution.}\label{tbl:zn-standardisation} \centering \begin{tabular}{|c|c|c|c|} \hline @@ -163,11 +178,15 @@ After one week the zeolite had settled in the bottom of the solutions. The clear \hline \end{tabular} \end{table} -%TODO: Add note about incomplete standardisation. + +Due to time constraints the standardisation of the \ce{EDTA} solution was not fully completed, hence the accurate titre volume ($V_{\ce{EDTA}_\text{std.}}$) has been assumed to be the titre volume from the second titre (see table \ref{tbl:zn-standardisation}) in the absence of additional available titrations to confirm this. + +\begin{equation}\label{eq:edta-v-standardisation} + V_{\ce{EDTA}_{std.}} = \SI{31.95}{\centi\metre\cubed} +\end{equation} \begin{table} - \label{tbl:zn-analytical-titration} - \caption{Titration results between zinc solution after ion-exchange process and standardised EDTA solution.} + \caption{Titration results between zinc solution after ion-exchange process and standardised \ce{EDTA} solution.}\label{tbl:zn-analytical-titration} \centering \begin{tabular}{|c|c|c|c|} \hline @@ -180,9 +199,16 @@ After one week the zeolite had settled in the bottom of the solutions. The clear \hline \end{tabular} \end{table} + +The average titre volume for the titration with the post ion-exchange solution ($V_{\ce{EDTA}_\text{prod.}}$) was determined from the second and third runs (see table \ref{tbl:zn-analytical-titration}) since the first run was a rough titration and the fourth run can be clearly seen be be anomalous. + +\begin{equation}\label{eq:edta-v-analytical} + V_{\ce{EDTA}_\text{prod.}} = \frac{\SI{26.65}{\centi\metre\cubed} + \SI{26.60}{\centi\metre\cubed}}{2} = \SI{20.63}{\centi\metre\cubed} +\end{equation} + %TODO: Titre value too small due to ZSM-5 suspended in solution: displaces liquid so actual aliquot size is smaller than appears. %TODO: Add notes about titration results: final anomalous reading possibly due human error or water in pipette filler, more solid in aliquot than others since less solution in volumetric flask and solid started to settle on bottom hence aliquot smaller than others. - +%TODO: add note explaining when separate ZnSO4 solution prepared for standardisation of EDTA solution - not a primary analytical standard. \section{Calculations} \subsection{Calculation of Maximum Theoretical Number of Ion Exchanges} @@ -303,18 +329,18 @@ Using \ref{eq:cu-percent-exchanged} with: Mr_{\text{HZSM-5 unit cell}} &= \SI{6217.6134}{\gram\per\mole} \text{ from equation \ref{eq:hzsm-5-mr}} \\ m_{\ce{CuSO4.5H2O}} &= \SI{0.5014 \pm 0.00005}{\gram} \text{ from table \ref{tbl:cu-masses}} \\ A_{\ce{Cu}_\text{std.}} &= \num{0.484} \text{ from table \ref{tbl:cu-absorbance}} \\ - V_{\ce{Cu}_\text{react.}} &= \SI{20.00 \pm 0.06 e-3}{\deci\metre\cubed} \\ + V_{\ce{Cu}_\text{react.}} &= \SI{20.00 \pm 0.06 e-3}{\deci\metre\cubed} \text{ from method (section \ref{sec:experimental})} \\ A_{\ce{Cu}_\text{prod.}} &= \num{0.110} \text{ from table \ref{tbl:cu-absorbance}} \\ - V_{\ce{Cu}_\text{prod.}} &= \SI{100.00 \pm 0.20 e-3}{\deci\metre\cubed} \\ + V_{\ce{Cu}_\text{prod.}} &= \SI{100.00 \pm 0.20 e-3}{\deci\metre\cubed} \text{ from method (section \ref{sec:experimental})}\\ m_{\text{HZSM-5}} &= \SI{0.4810 \pm 0.00005}{\gram} \text{ from table \ref{tbl:cu-masses}} \\ - V_{\ce{Cu}_\text{std.}} &= \SI{50.00 \pm 0.06 e-3}{\deci\metre\cubed} \\ + V_{\ce{Cu}_\text{std.}} &= \SI{50.00 \pm 0.06 e-3}{\deci\metre\cubed} \text{ from method (section \ref{sec:experimental})} \\ Mr_{\ce{CuSO4.5H2O}} &= \SI{249.577}{\gram\per\mole} \text{ from equation \ref{eq:molar-extinction-calc}} \end{align*} \begin{align*} \begin{split} \si{\percent} \text{ \ce{Cu^{2+}} Exchanged} &= \frac{2 \times \SI{6217.6134}{\gram\per\mole} \times \SI{0.50140}{\gram}\left(0.484 \times 20.00 - 0.110 \times 100.00\right)\num{e-3} \text{ \si{\deci\metre\cubed}}}{7.68 \times \SI{0.4810}{\gram} \times 0.484 \times \SI{50.00e-3}{\deci\metre\cubed} \times \SI{249.577}{\gram\per\mole}} \\ - &\text{ } \times \SI{100}{\percent} + &\phantom{=} \times \SI{100}{\percent} \end{split} \\ &= \SI{-18}{\percent} \end{align*} @@ -323,10 +349,85 @@ Using \ref{eq:cu-percent-exchanged} with: Let the percentage of \ce{Cu^2+} exchanged be $v_{\ce{Cu}}$ in the error propagation below: \begin{equation} -\label{eq:cu-error-propagation} +\label{eq:cu-error-propagation-initial} \begin{split} \delta v_{\ce{Cu}} = &v_{\ce{Cu}} \vast( %Manually putting in large brackets for square root. + \left( \frac{\delta Mr_{\text{HZSM-5 unit cell}}}{Mr_{\text{HZSM-5 unit cell}}}\right)^2 + + \left( \frac{\delta m_{\ce{CuSO4.5H2O}}}{m_{\ce{CuSO4.5H2O}}}\right)^2 + + \left( + \frac{ + \delta \left( A_{\ce{Cu}_\text{std.}} V_{\ce{Cu}_\text{react.}} - A_{\ce{Cu}_\text{prod.}} V_{\ce{Cu}_\text{prod.}} \right) + } + { + A_{\ce{Cu}_\text{std.}} V_{\ce{Cu}_\text{react.}} - A_{\ce{Cu}_\text{prod.}} V_{\ce{Cu}_\text{prod.}} + } + \right)^2 \\ + %New Line + %New Line. + &+ \left( \frac{\delta m_{\text{HZSM-5}}}{m_{\text{HZSM-5}}} \right)^2 + + \left( \frac{\delta A_{\ce{Cu}_\text{std.}}}{A_{\ce{Cu}_\text{std.}}} \right)^2 + + \left( \frac{\delta V_{\ce{Cu}_\text{std.}}}{V_{\ce{Cu}_\text{std.}}} \right)^2 + + \left( \frac{\delta Mr_{\ce{CuSO4.5H2O}}}{Mr_{\ce{CuSO4.5H2O}}} \right)^2 + \vast)^{1/2} +\end{split} +\end{equation} + +Let $A_{\ce{Cu}_\text{std.}} V_{\ce{Cu}_\text{react.}} - A_{\ce{Cu}_\text{prod.}} V_{\ce{Cu}_\text{prod.}} = S$, thus: + +\begin{align} + \delta S = + &\left( + \left( \delta \left( A_{\ce{Cu}_\text{std.}} V_{\ce{Cu}_\text{react.}} \right) \right)^2 + + \left( \delta \left( A_{\ce{Cu}_\text{prod.}} V_{\ce{Cu}_\text{prod.}} \right) \right)^2 + \right)^{1/2} \nonumber \\ + %Next Line. + \begin{split} + = &\vastt( + \left( + A_{\ce{Cu}_\text{std.}} V_{\ce{Cu}_\text{react.}} + \left( + \left( \frac{\delta A_{\ce{Cu}_\text{std.}}}{A_{\ce{Cu}_\text{std.}}} \right)^2 + + \left( \frac{ \delta V_{\ce{Cu}_\text{react.}}}{V_{\ce{Cu}_\text{react.}}} \right)^2 + \right)^{1/2} + \right)^2 \\ + %Split Equation. + &+ + \left( + A_{\ce{Cu}_\text{prod.}} V_{\ce{Cu}_\text{prod.}} + \left( + \left( \frac{\delta A_{\ce{Cu}_\text{prod.}}}{A_{\ce{Cu}_\text{prod.}}} \right)^2 + + \left( \frac{ \delta V_{\ce{Cu}_\text{prod.}}}{V_{\ce{Cu}_\text{prod.}}} \right)^2 + \right)^{1/2} + \right)^2 + \vastt)^{1/2} + \end{split} \nonumber \\ + %Next Line. + \begin{split} \label{eq:delta-s} + = &\vast( + A_{\ce{Cu}_\text{std.}}^2 V_{\ce{Cu}_\text{react.}}^2 + \left( + \left( \frac{\delta A_{\ce{Cu}_\text{std.}}}{A_{\ce{Cu}_\text{std.}}} \right)^2 + + \left( \frac{ \delta V_{\ce{Cu}_\text{react.}}}{V_{\ce{Cu}_\text{react.}}} \right)^2 + \right) \\ + %Split Equation. + &+ + A_{\ce{Cu}_\text{prod.}}^2 V_{\ce{Cu}_\text{prod.}}^2 + \left( + \left( \frac{\delta A_{\ce{Cu}_\text{prod.}}}{A_{\ce{Cu}_\text{prod.}}} \right)^2 + + \left( \frac{ \delta V_{\ce{Cu}_\text{prod.}}}{V_{\ce{Cu}_\text{prod.}}} \right)^2 + \right) + \vast)^{1/2} + \end{split} +\end{align} + +Hence substituting equation \ref{eq:delta-s} into \ref{eq:cu-error-propagation-initial} gives: + +\begin{equation} +\label{eq:cu-error-propagation} +\begin{split} + \delta v_{\ce{Cu}} = &v_{\ce{Cu}} + \Vast( %Manually putting in large brackets for square root. \left(\frac{\delta Mr_{\text{HZSM-5 unit cell}}}{Mr_{\text{HZSM-5 unit cell}}}\right)^2 + \left(\frac{\delta m_{\ce{CuSO4.5H2O}}}{m_{\ce{CuSO4.5H2O}}}\right)^2 \\ %New Line @@ -336,7 +437,7 @@ Let the percentage of \ce{Cu^2+} exchanged be $v_{\ce{Cu}}$ in the error propaga \left(\frac{\delta A_{\ce{Cu}_\text{std.}}}{A_{\ce{Cu}_\text{std.}}}\right)^2 + \left(\frac{\delta V_{\ce{Cu}_\text{react.}}}{V_{\ce{Cu}_\text{react.}}}\right)^2 \right) - + A_{\ce{Cu}_\text{std.}}^2 V_{\ce{Cu}_\text{react.}}^2 + + A_{\ce{Cu}_\text{prod.}}^2 V_{\ce{Cu}_\text{prod.}}^2 \left( \left(\frac{\delta A_{\ce{Cu}_\text{prod.}}}{A_{\ce{Cu}_\text{prod.}}}\right)^2 + \left(\frac{\delta V_{\ce{Cu}_\text{prod.}}}{V_{\ce{Cu}_\text{prod.}}}\right)^2 @@ -352,7 +453,7 @@ Let the percentage of \ce{Cu^2+} exchanged be $v_{\ce{Cu}}$ in the error propaga + \left(\frac{\delta A_{\ce{Cu}_\text{std.}}}{A_{\ce{Cu}_\text{std.}}}\right)^2 + \left(\frac{\delta V_{\ce{Cu}_\text{std.}}}{V_{\ce{Cu}_\text{std.}}}\right)^2 + \left(\frac{\delta Mr_{\ce{CuSO4.5H2O}}}{Mr_{\ce{CuSO4.5H2O}}}\right)^2 - \vast)^{1/2} + \Vast)^{1/2} \end{split} \end{equation} @@ -366,12 +467,116 @@ Substituting values into equation \ref{eq:cu-error-propagation} thus yields: So the percentage of \ce{Cu^2+} exchanged is \SI{-18 \pm 0.00}{\percent}. \subsection{Calculation of Ion-Exchange Efficiency for Zinc Solution} +\subsubsection{Standardisation of EDTA Solution (Batch A)} +Letting $V_{\ce{Zn}_\text{std.}}$ be the volume, $[\ce{ZnSO4}]_\text{std.}$ be the concentration and $m_{\ce{ZnSO4.7H2O}_\text{std}.}$ be the mass of \ce{ZnSO4.7H2O} used for the preparation of the \ce{ZnSO4} standard solution used to standardise the \ce{EDTA} solution. + +\begin{equation}\label{eq:[znso4]-std} + [\ce{ZnSO4}]_\text{std.} = \frac{m_{\ce{ZnSO4.7H2O}_\text{std}.}}{Mr_{\ce{ZnSO4.7H2O}} V_{\ce{Zn}_\text{std.}}} +\end{equation} +From equation \ref{eq:edta-zn-reaction} there is a 1:1 stoichiometric ratio between the \ce{Zn^2+} and \ce{EDTA^4-} ions hence letting $[\ce{EDTA^4-}]$ be the concentration of the \ce{EDTA} solution $n_{\ce{Zn}_\text{std. analyte}}$ be the amount and $V_{\ce{Zn}_\text{std. aliquot}}$ be the volume of \ce{Zn^2+} ions in the analyte. -%Over 100% exchange is possible e.g. due to formation of oxide species phyllosilicate outside zeolite e.t.c. influence on cobalt salt precursers on cobalt speciation and catalytic properties of H-ZSM-5 modified ... mhamdi +\begin{equation}\label{eq:[edta]-1} + [\ce{EDTA^4-}] = \frac{n_{\ce{Zn}_\text{std. analyte}}}{V_{\ce{EDTA}_\text{std.}}} = \frac{[\ce{ZnSO4}]_\text{std.} V_{\ce{Zn}_\text{std. aliquot}}}{V_{\ce{EDTA}_\text{std.}}} +\end{equation} + +Thus substituting equation \ref{eq:[znso4]-std} into equation \ref{eq:[edta]-1} gives: + +\begin{equation}\label{eq:[edta]-final} + [\ce{EDTA^4-}] = \frac{m_{\ce{ZnSO4.7H2O}_\text{std}.} V_{\ce{Zn}_\text{std. aliquot}}}{Mr_{\ce{ZnSO4.7H2O}} V_{\ce{Zn}_\text{std.}} V_{\ce{EDTA}_\text{std.}}} +\end{equation} + +\subsubsection{Determination of Percentage of \ce{Zn^2+} Exchanged Compared to the Theoretical Maximum} +Let $[\ce{ZnSO4}]_\text{std. orig.}$ be the concentration of, $m_{\ce{ZnSO4.7H2O}_\text{orig.}}$ be the mass of zinc sulphate used and $V_{\ce{Zn}_\text{std. orig.}}$ be the volume of the standard zinc sulphate solution created for the ion exchange process. + +\begin{equation} + \label{eq:[znso4]-std-orig} + [\ce{ZnSO4}]_\text{std. orig.} = \frac{m_{\ce{ZnSO4.7H2O}_\text{orig.}}}{Mr_{\ce{ZnSO4.7H2O}} V_{\ce{Zn}_\text{std. orig.}}} +\end{equation} + +Using equation \ref{eq:[znso4]-std-orig} with $V_{\ce{Zn}_\text{orig.}}$ as the volume of the stanadard solution used in the ion-exchange process. + +\begin{align} + n_{\ce{Zn}_\text{orig.}} &= [\ce{ZnSO4}]_\text{std. orig.} V_{\ce{Zn}_\text{orig.}} \nonumber \\ + \label{eq:zn-amount-orig} + &= \frac{m_{\ce{ZnSO4.7H2O}_\text{orig.}} V_{\ce{Zn}_\text{orig.}}}{Mr_{\ce{ZnSO4.7H2O}} V_{\ce{Zn}_\text{std. orig.}}} +\end{align} + +Using equation \ref{eq:[edta]-final} the amount of zinc remaining in solution after the ion-exchange ($n_{\ce{Zn}_\text{prod.}}$) can be calculated with $V_{\ce{Zn}_\text{prod.}}$ being the volume of this resultant solution and $V_{\ce{Zn}_\text{prod. aliquot}}$ being the volume of the aliquot titrated. + +\begin{align} + n_{\ce{Zn}_\text{prod.}} &= \frac{V_{\ce{EDTA}_\text{prod.}} [\ce{EDTA^4-}] V_{\ce{Zn}_\text{prod.}}}{V_{\ce{Zn}_\text{prod. aliquot}}} \nonumber \\ + \label{eq:zn-amount-product} + &= \frac{V_{\ce{EDTA}_\text{prod.}} m_{\ce{ZnSO4.7H2O}_\text{std.}} V_{\ce{Zn}_\text{std. aliquot}} V_{\ce{Zn}_\text{prod.}}}{V_{\ce{Zn}_\text{prod. aliquot}} Mr_{\ce{ZnSO4.7H2O}} V_{\ce{Zn}_\text{std.}} V_{\ce{EDTA}_\text{std.}}} +\end{align} + +Using equations \ref{eq:zn-amount-orig} and \ref{eq:zn-amount-product} to calculate the amount of \ce{Zn^2+} ions exchanged with the HZSM-5 ($n_{\ce{Zn}_\text{ex.}}$): + +\begin{align} + n_{\ce{Zn}_\text{ex.}} &= n_{\ce{Zn}_\text{orig.}} - n_{\ce{Zn}_\text{prod.}} \nonumber \\ + &= \frac{m_{\ce{ZnSO4.7H2O}_\text{orig.}} V_{\ce{Zn}_\text{orig.}}}{Mr_{\ce{ZnSO4.7H2O}} V_{\ce{Zn}_\text{std. orig.}}} - \frac{V_{\ce{EDTA}_\text{prod.}} m_{\ce{ZnSO4.7H2O}_\text{std.}} V_{\ce{Zn}_\text{std. aliquot}} V_{\ce{Zn}_\text{prod.}}}{V_{\ce{Zn}_\text{prod. aliquot}} Mr_{\ce{ZnSO4.7H2O}} V_{\ce{Zn}_\text{std.}} V_{\ce{EDTA}_\text{std.}}} \nonumber \\ + \begin{split} + \label{eq:zn-amount-exchanged} + &= \frac{V_{\ce{Zn}_\text{prod. aliquot}} V_{\ce{Zn}_\text{std.}} V_{\ce{EDTA}_\text{std.}} m_{\ce{ZnSO4.7H2O}_\text{orig.}} V_{\ce{Zn}_\text{orig.}}}{V_{\ce{Zn}_\text{std. orig.}} V_{\ce{Zn}_\text{prod. aliquot}} Mr_{\ce{ZnSO4.7H2O}} V_{\ce{Zn}_\text{std.}} V_{\ce{EDTA}_\text{std.}}} \\ + &\phantom{=} - \frac{V_{\ce{Zn}_\text{std. orig.}} V_{\ce{EDTA}_\text{prod.}} m_{\ce{ZnSO4.7H2O}_\text{std.}} V_{\ce{Zn}_\text{std. aliquot}} V_{\ce{Zn}_\text{prod.}}}{V_{\ce{Zn}_\text{std. orig.}} V_{\ce{Zn}_\text{prod. aliquot}} Mr_{\ce{ZnSO4.7H2O}} V_{\ce{Zn}_\text{std.}} V_{\ce{EDTA}_\text{std.}}} + \end{split} +\end{align} -%TODO: Subsubsub section package? -%\subsubsubsection{Copper} +Hence substituting equation \ref{eq:zn-amount-exchanged} into \ref{eq:cation-percent-exchanged} and setting $q=2$ gives: + +\begin{equation}\label{eq:zn-percent-exchanged} +\begin{split} + \text{\si{\percent} \ce{Zn} Exchanged} &= \frac{2 Mr_{\text{HZSM-5 unit cell}}}{7.68 m_{\text{HZSM-5}} V_{\ce{Zn}_\text{std. orig.}} V_{\ce{Zn}_\text{prod. aliquot}} Mr_{\ce{ZnSO4.7H2O}} V_{\ce{Zn}_\text{std.}} V_{\ce{EDTA}_\text{std.}}} \\ + &\phantom{=} \times (V_{\ce{Zn}_\text{prod. aliquot}} V_{\ce{Zn}_\text{std.}} V_{\ce{EDTA}_\text{std.}} m_{\ce{ZnSO4.7H2O}_\text{orig.}} V_{\ce{Zn}_\text{orig.}} \\ + &\phantom{= \times (} - V_{\ce{Zn}_\text{std. orig.}} V_{\ce{EDTA}_\text{prod.}} m_{\ce{ZnSO4.7H2O}_\text{std.}} V_{\ce{Zn}_\text{std. aliquot}} V_{\ce{Zn}_\text{prod.}}) \times \SI{100}{\percent} +\end{split} +\end{equation} + +Using equation \ref{eq:zn-percent-exchanged} with: + +%TODO: Uncertainties in Mr. +\begin{align*} + Mr_\text{HZSM-5 unit cell} &= \SI{6217.6134}{\gram\per\mole} \text{ from equation \ref{eq:hzsm-5-mr}} \\ + V_{\ce{Zn}_\text{prod. aliquot}} &= \SI{20.00 \pm 0.06 e-3}{\deci\metre\cubed} \text{ from method (section \ref{sec:experimental})} \\ + V_{\ce{Zn}_\text{std.}} &= \SI{100.00 \pm 0.20 e-3}{\deci\metre\cubed} \text{ from method (section \ref{sec:experimental})} \\ + V_{\ce{EDTA}_\text{std.}} &= \SI{31.95 \pm 0.2 e-3}{\deci\metre\cubed} \text{ from equation \ref{eq:edta-v-standardisation}} \\ + m_{\ce{ZnSO4.7H2O}_\text{orig.}} &= \SI{0.6331 \pm 0.00005 e-3}{\gram} \text{ from table \ref{tbl:zn-zsm-5-masses}} \\ + V_{\ce{Zn}_\text{orig.}} &= \SI{20.00 \pm 0.06 e-3}{\deci\metre\cubed} \text{ from method (section \ref{sec:experimental})} \\ + V_{\ce{Zn}_\text{std. orig.}} &= \SI{50.00 \pm 0.06 e-3}{\deci\metre\cubed} \text{ from method (section \ref{sec:experimental})} \\ + V_{\ce{EDTA}_\text{prod.}} &= \SI{26.63 \pm 0.2 e-3}{\deci\metre\cubed} \text{ from equation \ref{eq:edta-v-analytical}} \\ + m_{\ce{ZnSO4.7H2O}_\text{std.}} &= \SI{0.4587 \pm 0.00005}{\gram} \text{ from table \ref{tbl:zn-std-masses}} \\ + V_{\ce{Zn}_\text{std. aliquot}} &= \SI{10.00 \pm 0.04 e-3}{\deci\metre\cubed} \text{ from method (section \ref{sec:experimental})} \\ + V_{\ce{Zn}_\text{prod.}} &= \SI{100.00 \pm 0.20 e-3}{\deci\metre\cubed} \text{ from method (section \ref{sec:experimental})} \\ + m_{\text{HZSM-5}} &= \SI{0.5274 \pm 0.00005}{\gram} \text{ from table \ref{tbl:zn-zsm-5-masses}} \\ + Mr_{\ce{ZnSO4.7H2O}} &= (65.38 + 32.066 + 4(15.999) + 7(2(1.008) + 15.999)) \text{ \si{\gram\per\mole}} \\ + &= \SI{287.547}{\gram\per\mole} +\end{align*} + +\begin{displaymath} + \text{\si{\percent} \ce{Zn} Exchanged} = \SI{66}{\percent} +\end{displaymath} + +\subsubsection{Error Propagation} +Let the percentage of \ce{Zn^2+} exchanged be $v_{\ce{Zn}}$ in the following error propagation: + +\begin{equation} +\label{eq:zn-error-propagation} +\begin{split} + \delta v_{\ce{Zn}} = &v_{\ce{Zn}} + \vast( + \left( \frac{\delta Mr_\text{HZSM-5 unit cell}}{Mr_\text{HZSM-5 unit cell}} \right)^2 + + \left( \frac{\delta m_{\text{HZSM-5}}}{m_{\text{HZSM-5}}} \right)^2 + + \left( \frac{\delta V_{\ce{Zn}_\text{std. orig.}}}{V_{\ce{Zn}_\text{std. orig.}}} \right)^2 + + \left( \frac{\delta V_{\ce{Zn}_\text{prod. aliquot}}}{V_{\ce{Zn}_\text{prod. aliquot}}} \right)^2 \\ + %New Line + &+ \left( \frac{\delta Mr_{\ce{ZnSO4.7H2O}}}{Mr_{\ce{ZnSO4.7H2O}}} \right)^2 + + \left( \frac{\delta V_{\ce{Zn}_\text{std.}}}{V_{\ce{Zn}_\text{std.}}} \right)^2 + + \left( \frac{\delta V_{\ce{EDTA}_\text{std.}}}{V_{\ce{EDTA}_\text{std.}}} \right)^2 + \dots + \vast)^{1/2} +\end{split} +\end{equation} + +%Over 100% exchange is possible e.g. due to formation of oxide species phyllosilicate outside zeolite e.t.c. influence on cobalt salt precursers on cobalt speciation and catalytic properties of H-ZSM-5 modified ... mhamdi \section{Analysis} -- 2.25.1