%Allow manual adding of very large brackets.
\makeatletter
-\newcommand{\vast}{\bBigg@{5}}
+\newcommand{\vast}{\bBigg@{3}}
+\newcommand{\vastt}{\bBigg@{4}}
+\newcommand{\Vast}{\bBigg@{5}}
+\newcommand{\Vastt}{\bBigg@{6}}
\makeatother
%Document Headings.
\section{Introduction}
Zeolites are crystalline, microporous solids used for a large number of purposes such as for catalytic cracking, air purification, water softening and desiccants.~\autocite{hardsoftwater,petrov12} This project was completed using the ZSM-5 (Zeolite Socony Mobil-5)\autocite{zhang15} zeolite which has important uses in the petrochemical industry such as for the conversion of methanol to gasoline, dewaxing of distillates, separation of organic products (such as separating para-xylene from its isomers), the interconversion of hydrocarbons.\autocite{olson81,rasouli12,sarkany99}
-%TODO: Switch phrasing from aluminium atoms/silon atoms to ions? Ensure constancy throughout at least.
\subsection{Structure}
Each zeolite is comprised of a finite or infinite number of unique unit cells each of which is made from a constant, integral number of the same type of secondary building unit (SBU) with each vertex in the SBU being a tetrahedron of either \ce{[SiO4]} or \ce{[AlO4]^{-}} (which are themselves the primary building units).\autocite{zeolite-atlas,petrov12,ic-zeolite-structure,han09,danaher17} Each aluminium tetrahedron in a SBU introduces a negative charge -- since aluminium has a 3+ oxidation state compared the 4+ oxidation state of silicon -- which is balanced by the presence of cationic counterions.\autocite{gomez16,petrov12,han09,danaher17} The ZSM-5 zeolite used is a pentasil\autocite{danaher17,olson81} zeolite (constructed of eight five-membered rings) with an SBU containing twelve \ce{AO4} tetrahedra which form a pair of five-one units\autocite{zeolite-atlas,olson81,wu79} as shown in figure \ref{fig:zsm-5-sbu} (A-O-A bridges are shown as straight lines to increase the clarity of the images and since the A-O-A bond angle is around $\text{\SIrange{140}{150}{\degree}} \approx \SI{180}{\degree}$ for silicas and aluminosilicates and the A atoms are represented by the vertices).\autocite{zeolite-atlas}
A ZSM-5 zeolite with a \ce{SiO2}/\ce{AlO3} ratio of 23 was used since this maximised the number of sites which were available for ion-exchange due to the higher aluminium content. In addition this increased the efficiency of the ion-exchange process since zeolites with a high Si/Al ratio are hydrophobic\autocite{chen76,han09,olson99} hence the cation solution does not spontaneously enter the zeolite nanopores so ion-exchange happens only at sites close to the pore entrance.\autocite{han09,olson99} This will thus reduce the percentage uncertainties in the values recorded.
-
-\section{Experimental}
+\section{Experimental}\label{sec:experimental}
Standard solutions of \ce{Cu^{2+}} and \ce{Zn^{2+}} (\SI{50.00}{\centi\metre\cubed}) were made using \ce{CuSO4.5H2O} and \ce{ZnSO4.7H2O} with concentration \SI{2.008e-3}{\mole\per\deci\metre\cubed} and \SI{2.02e-3}{\mole\per\deci\metre\cubed} respectively. The absorbance of the standard copper sulphate solution was taken at \SI{806}{\nano\metre} (\num{0.484}) then \SI{20.00}{\centi\metre\cubed} of the standard solutions were added to \SI{0.4810}{\gram} (for the copper solution) and \SI{0.5274}{\gram} (for the zinc solution) of HZSM-5 zeolite with an \ce{AlO3}:\ce{SiO2} ratio of 23 -- forming an opaque white suspension -- before heating both solutions (with stirring) at \SI{70}{\celsius} for one hour. Centrifugation was completed on part of the resultant copper mixture, however time constraints prevented the completion of this process. The two mixtures were thus stored in a fridge for one week until the following laboratory session.
-After one week the zeolite had settled in the bottom of the solutions. The clear solution was decanted and the remainder was centrifuged for 30 minutes before the supernatant was reintroduced to the initially decanted solution producing a slightly cloudy copper solution and a moderately cloudy zinc solution. The solutions were made up to \SI{100.00}{\centi\metre\cubed} before the absorbance of the copper solution at \SI{806}{\nano\metre} was determined (\num{0.110}) and the zinc solution was titrated against a standard EDTA solution (\SI{0.4993}{\mole\per\deci\metre\cubed}) with \SI{2}{\centi\metre\cubed} of a pH 10 buffer solution and eriochrome black T as the indicator.
+After one week the zeolite had settled in the bottom of the solutions. The clear solution was decanted and the remainder was centrifuged for 30 minutes before the supernatant was reintroduced to the initially decanted solution producing a slightly cloudy copper solution and a moderately cloudy zinc solution. The solutions were made up to \SI{100.00}{\centi\metre\cubed} before the absorbance of the copper solution at \SI{806}{\nano\metre} was determined (\num{0.110}) and \SI{20.00}{\centi\metre\cubed} aliquots of the zinc solution was titrated against a standard EDTA solution (batch A: \SI{0.4993}{\mole\per\deci\metre\cubed}) with \SI{2}{\centi\metre\cubed} of a pH 10 buffer solution and eriochrome black T as the indicator (colour change from red to light blue).
%TODO: Need to create new standard solutn to standardise the EDTA. Might as well set to heat for 1hr while sorting out Cu? (Use 10.00 cm aliquot)
\subsection{Copper-Exchanged Zeolite}
\begin{table}[h]
- \label{tbl:cu-masses}
- \caption{Masses used in CuZSM-5 preparation.}
+ \caption{Masses used in CuZSM-5 preparation.}\label{tbl:cu-masses}
\centering
\begin{tabular}{|c|c|}
\hline
\end{table}
\begin{table}[h]
- \label{tbl:cu-absorbance}
- \caption{Spectrophotometric results.}
+ \caption{Spectrophotometric results.}\label{tbl:cu-absorbance}
\centering
\begin{tabular}{|c|c|}
\hline
\subsection{Zinc-Exchanged Zeolite}
\begin{table}[h]
- \label{tbl:zn-masses}
- \caption{Masses used in ZnZSM-5 preparation.}
+ \caption{Masses used in preparation of \ce{ZnSO4} standard solution.}\label{tbl:zn-std-masses}
+ \centering
+ \begin{tabular}{|c|c|}
+ \hline
+ Substance & Mass / \si{\gram} \\
+ \hline
+ \ce{ZnSO4.7H2O} & 0.4587 \\
+ \hline
+ \end{tabular}
+\end{table}
+
+\begin{table}[h]
+ \caption{Masses used in ZnZSM-5 preparation.}\label{tbl:zn-zsm-5-masses}
\centering
\begin{tabular}{|c|c|}
\hline
\end{tabular}
\end{table}
+The reaction which occurred during the titrations between the \ce{EDTA} and \ce{Zn^2+} ions in given in equation \ref{eq:edta-zn-reaction}.
+
+\begin{equation}\label{eq:edta-zn-reaction}
+ \ce{Zn^2+ + EDTA^4- -> 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
\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
\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}
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*}
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
\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
+ \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}
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}
projects / chemistry / university-chemistry-lab-reports.git / commitdiff