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Chemistry LibreTexts

Appendix A: Derivation of k’ for Neutral Compounds

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

\[υ_{apparent} = υ_{eof} \dfrac{η_{aqueous}}{η_{micelle} + η_{aqueous}} + υ_{micelle} \dfrac{η_{micelle}}{η_{micelle} + η_{aqueous}} \tag{equation 3.2}\]

substitute equation 3.1

\[υ_{apparent} = υ_{eof}\dfrac{1}{1 + k'} + υ_{micelle}\dfrac{k'}{1 + k'}\tag{equation 3.3}\]

Given that velocity is the arrival time at the capillary window

\[υ_{apparent} = \dfrac{l}{t_R} \tag{equation 3.4a}\]

\[υ_{eof} = \dfrac{l}{t_{eof}} \tag{equation 3.4b}\]

\[υ_{micelle} = \dfrac{l}{t_{micelle}} \tag{equation 3.4c}\]

substitute equations 3.4a-c

\[\dfrac{l}{t_R} = \dfrac{l}{t_{eof}} ( \dfrac{1}{1 + k'} ) + \dfrac{l}{t_{micelle}} ( \dfrac{k'}{1 + k'} ) \tag{equation 3.5}\]

\[\dfrac{1}{t_R} = \dfrac{1}{t_{eof}} ( \dfrac{1}{1 + k'} ) + \dfrac{1}{t_{micelle}} ( \dfrac{k'}{1 + k'} ) \tag{equation 3.5a}\]

\[\dfrac{1 + k'}{k' t_R} = (\dfrac{1 + k'}{k'} )[\dfrac{1}{t_{eof}} ( \dfrac{1}{1 + k'} ) + \dfrac{1}{t_{micelle}}( \dfrac{k'}{1 + k'} )] \tag{equation 3.6}\]

\[\dfrac{1 + k'}{k' t_R} = \dfrac{1}{k' t_{eof}} + \dfrac{1}{t_{micelle}} \tag{equation 3.7}\]

\[\dfrac{1 + k'}{k' t_R} − \dfrac{1}{k' t_{eof}} = \dfrac{1}{t_{micelle}} \tag{equation 3.8}\]

\[\dfrac{1 + k'}{k' t_R} − \dfrac{(\dfrac{t_R}{t_{eof}})}{k' t_R} = \dfrac{1}{t_{micelle}} \tag{equation 3.9}\]

\[\dfrac{1}{k' t_R} (1 + k − \dfrac{t_R}{t_{eof}}) = \dfrac{1}{t_{micelle}}\tag{equation 3.10}\]

\[(1 + k' − \dfrac{t_R}{t_{eof}}) = \dfrac{k' t_R}{t_{micelle}} \tag{equation 3.11}\]

\[k' − \dfrac{k' t_R}{t_{micelle}} = \dfrac{t_R}{t_{eof}} − 1 \tag{equation 3.12}\]

\[k' (1 − \dfrac{t_R}{t_{micelle}}) = \dfrac{1}{t_{eof}}  (t_R − t_{eof}) \tag{equation 3.13}\]

\[k' = \dfrac{(t_R − t_{eof})}{t_{eof} (1 − \dfrac{t_R}{t_{micelle}})} \tag{equation 3.14}\]