Appendix 1: Derivation of the Fundamental Resolution Equation
- Page ID
- 72012
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)\[\mathrm{R_S = \dfrac{2(t_2-t_1)}{W_1 + W_2}}\tag{1}\]
\[\mathrm{W_1 \cong W_2}\]
\[\mathrm{W = 4\sigma}\]
\[\mathrm{R_S = \dfrac{2(t_2-t_1)}{8\sigma_2} = \dfrac{t_2-t_1}{4\sigma_2}}\tag{2}\]
\[\mathrm{N = \left(\dfrac{t_2}{\sigma_2} \right )^2 \hspace{60px} \sigma_2 = \dfrac{t_2}{\sqrt{N}}}\]
\[\mathrm{R_S = \dfrac{t_2-t_1}{4\left( \dfrac{t_2}{\sqrt{N}}\right )}}\tag{3}\]
\[\mathrm{R_S = \left(\dfrac{\sqrt{N}}{4}\right )\left(\dfrac{t_2-t_1}{t_2}\right ) = \left(\dfrac{\sqrt{N}}{4} \right) \left(1 - \dfrac{t_1}{t_2} \right )}\tag{4}\]
Write an expression for the fraction of material in the mobile phase (φM):
\[\mathrm{\varphi_M = \dfrac{C_MV_M}{C_MV_M + C_SV_S}}\tag{5}\]
\[\mathrm{\varphi_M = \dfrac{\dfrac{C_MV_M}{C_MV_M}}{\dfrac{C_MV_M}{C_MV_M} + \dfrac{C_SV_S}{C_MV_M}} = \dfrac{1}{1 + k'}}\tag{6}\]
Express the average migration velocity of component 2:
\[\mathrm{v_{S_2} = \varphi_M v}\tag{7}\]
(where v is the mobile phase velocity)
\[\mathrm{v_{S_2} = \dfrac{L}{t_2}\hspace{20px}(8) \hspace{60px} v = \dfrac{L}{t_0} \hspace{20px} (9)}\]
\[\mathrm{t_2 = \dfrac{t_0}{\varphi_{M_2}} \hspace{60px} t_1 = \dfrac{t_0}{\varphi_{M_1}}}\tag{11}\]
\[\mathrm{t_2 = \dfrac{t_0}{\left(\dfrac{1}{1+k_2'}\right)} \hspace{60px} t_1 = \dfrac{t_0}{\left(\dfrac{1}{1+k_1'}\right)}}\tag{12}\]
\[\mathrm{t_2 = t_0(1+k_2') \hspace{60px} t_1=t_0(1+k_1')}\tag{13}\]
\[\mathrm{\dfrac{t_1}{t_2} = \dfrac{t_0(1+k_1')}{t_0(1+k_2')} =\dfrac{1+k_1'}{1+k_2'}}\tag{14}\]
Substitute (14) into (4):
\[\mathrm{R_S = \left(\dfrac{\sqrt{N}}{4}\right) = \left(1 - \dfrac{1+k_1'}{1+k_2'}\right) }\tag{15}\]
Consider the \(\mathrm{\left(1 - \dfrac{1+k_1'}{1+k_2'}\right) }\) term:
\[\mathrm{\left(1 - \dfrac{1+k_1'}{1+k_2'}\right) =\dfrac{1+k_2'}{1+k_2'} - \dfrac{1+k_1'}{1+k_2'}}\tag{15a}\]
\[\mathrm{=\dfrac{k_2' - k_1'}{1+k_2'}}\tag{15b}\]
\[\mathrm{=\dfrac{k_1'\left( \dfrac{k_2'}{k_1'} - 1\right )}{1+k_2'}}\tag{15c}\]
\[\mathrm{=\dfrac{\left(\dfrac{k_2'}{k_1'}\right)k_1'\left( \dfrac{k_2'}{k_1'} - 1\right )}{\left(\dfrac{k_2'}{k_1'}\right)(1+k_2')}
= \dfrac{k_2'\left( \dfrac{k_2'}{k_1'} - 1\right )}{\left(\dfrac{k_2'}{k_1'}\right)(1+k_2')}}\tag{15d}\]
\[\mathrm{=\left(\dfrac{k_2'}{1+k_2'}\right)\left(\dfrac{\dfrac{k_2'}{k_1'}-1}{\dfrac{k_2'}{k_1'}} \right )}\tag{15e}\]
\[\mathrm{\alpha=\dfrac{k_2'}{k_1'}}\]
\[\mathrm{=\left( \dfrac{k_2'}{1+k_2'}\right )\left(\dfrac{\alpha - 1}{\alpha}\right )}\tag{15f}\]
Substitute (15f) into (15):
\[\mathrm{R_S=\left( \dfrac{\sqrt{N}}{4}\right )\left(\dfrac{\alpha - 1}{\alpha}\right )\left(\dfrac{k_2'}{1+k_2'} \right )}\]
Relationship to retention time:
\[\mathrm{v_{S_2} = \dfrac{L}{t_2} \hspace{60px} t_2 = \dfrac{L}{v_{S_2}}}\]
\[\mathrm{v_{S_2} = \varphi_M v = \left(\dfrac{1}{1+k_2'} \right )v}\tag{16}\]
\[\mathrm{H = \dfrac{L}{N} \hspace{60px} L = HN}\]
\[\mathrm{t_2 = \dfrac{HN(1 + k_2')}{v}}\tag{17}\]
Rearrange the fundamental resolution equation to solve for N:
\[\mathrm{N = 16{R_S}^2 \left( \dfrac{\alpha}{\alpha - 1}\right )^2 \left( \dfrac{1 + k_2'}{k_2'}\right )^2}\]
\[\mathrm{t_2 = \left(\dfrac{16{R_S}^2H}{v}\right ) \left( \dfrac{\alpha}{\alpha - 1}\right )^2 \left( \dfrac{(1 + k_2')^3}{(k_2')^2}\right )}\tag{18}\]