# Appendix B: Derivation of k’ for Anionic Compounds

$k' = \dfrac{moles_{micelle}}{moles_{aqueous}} \tag{equation 3.1}$

$υ_{apparent} = υ_{eof} \dfrac{η_{aqueous}}{η_{micelle} + η_{aqueous}} + υ_{eph\_analyte} \dfrac{η_{aqueous}}{η_{micelle} + η_{aqueous}} + υ_{micelle} \dfrac{η_{micelle}}{η_{micelle} + η_{aqueous}} \tag{equation 3.15}$

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

$\dfrac{V μ_{eph\_analyte}}{L} = \dfrac{V μ_R}{L} - \dfrac{V μ_{eof}}{L} \tag{equation 3.29}$

$\dfrac{t_{eof}}{t_{eph\_analyte}} = \dfrac{μ_{eof}}{μ_{eph\_analyte}} \tag{equation 3.30}$

$k' = \dfrac{t_R (1 + \dfrac{μ_{eof}} {μ_{eph\_analyte}}) − t_{eof}}{t_{eof} (1 − \dfrac{t_R}{t_{micelle}})} \tag{equation 3.31}$