Appendix B: Derivation of k’ for Anionic Compounds
- Page ID
- 63542
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)\[k' = \dfrac{moles_{micelle}}{moles_{aqueous}} \tag{equation 3.1}\]
\[υ_{apparent} = υ_{eof} \dfrac{η_{aqueous}}{η_{micelle} + η_{aqueous}} + υ_{eph\_analyte} \dfrac{η_{aqueous}}{η_{micelle} + η_{aqueous}} + υ_{micelle} \dfrac{η_{micelle}}{η_{micelle} + η_{aqueous}} \tag{equation 3.15}\]
substitute equation 3.1
\[υ_{apparent} = υ_{eof}\dfrac{1}{1 + k'} + υ_{eph\_analyte}\dfrac{1}{1 + k'} + υ_{micelle}\dfrac{k'}{1 + k'}\tag{equation 3.3}\]
Given that velocity is the arrival time at the capillary window (l = length to detection window)
\[υ_{apparent} = \dfrac{l}{t_R} \tag{equation 3.4a}\]
\[υ_{eof} = \dfrac{l}{t_{eof}} \tag{equation 3.4b}\]
\[υ_{micelle} = \dfrac{l}{t_{micelle}} \tag{equation 3.4c}\]
\[υ_{eph\_analyte} = \dfrac{l}{t_{eph\_analyte}} \tag{equation 3.4d}\]
substitute equations 3.4a-d
\[\dfrac{l}{t_R} = \dfrac{l}{t_{eof}} ( \dfrac{1}{1 + k'} ) + \dfrac{l}{t_{eph\_analyte}}( \dfrac{1}{1 + k'} ) + \dfrac{l}{t_{micelle}} ( \dfrac{k'}{1 + k'} ) \tag{equation 3.16}\]
\[\dfrac{1}{t_R} = \dfrac{1}{t_{eof}} ( \dfrac{1}{1 + k'} ) + \dfrac{1}{t_{eph\_analyte}}( \dfrac{1}{1 + k'} ) + \dfrac{1}{t_{micelle}} ( \dfrac{k'}{1 + k'} ) \tag{equation 3.16a}\]
\[\dfrac{1 + k'}{k't_R} = ( \dfrac{1 + k'}{k'} )[\dfrac{1}{t_{eof}} ( \dfrac{1}{1 + k'} ) + \dfrac{1}{t_{eph\_analyte}}( \dfrac{1}{1 + k'} )+ \dfrac{1}{t_{micelle}}( \dfrac{k'}{1 + k'} )] \tag{equation 3.17}\]
\[\dfrac{1 + k'}{k't_R} = \dfrac{1}{k't_{eof}} + \dfrac{1}{k't_{eph\_analyte}}+ \dfrac{1}{t_{micelle}} \tag{equation 3.18}\]
\[\dfrac{1 + k'}{k't_R} − \dfrac{1}{k't_{eof}} - \dfrac{1}{k't_{eph\_analyte}} = \dfrac{1}{t_{micelle}} \tag{equation 3.19}\]
\[\dfrac{1 + k'}{k't_R} − \dfrac{(\dfrac{t_R}{t_{eof}})}{k't_R} −\dfrac{(\dfrac{t_R}{t_{eph\_analyte}})}{k't_R} = \dfrac{1}{t_{micelle}} \tag{equation 3.20}\]
\[\dfrac{1}{k't_R} (1 + k - \dfrac{t_R}{t_{eof}} - \dfrac{t_R}{t_{eph\_analyte}}) = \dfrac{1}{t_{micelle}} \tag{equation 3.21}\]
\[(1 + k' - \dfrac{t_R}{t_{eof}} - \dfrac{t_R}{t_{eph\_analyte}}) = \dfrac{k't_R}{t_{micelle}} \tag{equation 3.22}\]
\[k' - \dfrac{k't_R}{t_{micelle}} = \dfrac{t_R}{t_{eof}} + \dfrac{t_R}{t_{eph\_analyte}} - 1 \tag{equation 3.23}\]
\[k' (1 - \dfrac{t_R}{t_{micelle}} ) = \dfrac{1}{t_{eof}} ( t_R + \dfrac{t_Rt_{eof}}{t_{eph\_analyte}} - t_{eof}) \tag{equation 3.24}\]
\[k' = \dfrac{t_R + \dfrac{t_Rt_{eof}}{t_{eph\_analyte}} - t_{eof}}{t_{eof} (1 - \dfrac{t_R}{t_{micelle}} )} \tag{equation 3.25}\]
\[k' = \dfrac{t_R (1 + \dfrac{t_{eof}}{t_{eph\_analyte}}) - t_{eof}}{t_{eof} (1 - \dfrac{t_R}{t_{micelle}} )} \tag{equation 3.26}\]
The term teph_analyte is difficult to measure. In free zone capillary electrophoresis the apparent velocity of anionic analyte is the sum of two components: the velocity of the electroosmotic flow and the electrophoretic velocity of the anionic analyte.
\[υ_R = υ_{eof} + υ_{eph\_analyte} \tag{equation 3.27a}\]
\[υ_{eph\_analyte} = υ_R − υ_{eof} \tag{equation 3.27b}\]
To simplify the representation of the final equation we will substitute mobility, μ, with velocity, υ, using the in equation 3.29a (L=capillary length).
\[μ_R = \dfrac{υ_R L}{\mathrm V} = \dfrac{lL}{\mathrm{t_R V}} \tag{equation 3.28a}\]
\[μ_{eof} = \dfrac{υ_{eof} L}{\mathrm V} = \dfrac{lL}{t_{eof}\mathrm V} \tag{equation 3.28b}\]
\[μ_{eph\_analyte} = \dfrac{υ_{eph\_analyte} L}{\mathrm V} = \dfrac{lL}{t_{eph\_analyte}\mathrm V} \tag{equation 3.28c}\]
substitute equations 3.29 a,b,c into equation3.28b
\[\dfrac{V μ_{eph\_analyte}}{L} = \dfrac{V μ_R}{L} - \dfrac{V μ_{eof}}{L} \tag{equation 3.29}\]
\[\dfrac{t_{eof}}{t_{eph\_analyte}} = \dfrac{μ_{eof}}{μ_{eph\_analyte}} \tag{equation 3.30}\]
\[k' = \dfrac{t_R (1 + \dfrac{μ_{eof}} {μ_{eph\_analyte}}) − t_{eof}}{t_{eof} (1 − \dfrac{t_R}{t_{micelle}})} \tag{equation 3.31}\]