Appendix 1: Derivation of the Fundamental Resolution Equation
- Page ID
- 95089
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)\[\mathrm{R_S = \dfrac{2 (t_2 - t_1)}{W_1 + W_2}} \label{1}\]
\[\mathrm{W_1 ≅ W_2}\]
\[\mathrm{W = 4σ}\]
\[\mathrm{R_S = \dfrac{2(t_2 - t_1)}{8σ_2} = \dfrac{t_2 - t_1}{4σ_2}} \label{2}\]
\[\mathrm{N = \left(\dfrac{t_2}{σ_2}\right)^2 \hspace{80px} σ_2 = \dfrac{t_2}{\sqrt{N}}}\]
\[\mathrm{R_S = \dfrac{t_2 - t_1}{4\left(\dfrac{t_2}{\sqrt N}\right)}} \label{3}\]
\[\mathrm{R_S = \left(\dfrac{\sqrt{N}}{4}\right)\left(\dfrac{t_2 - t_1}{t_2} \right)=\left(\dfrac{\sqrt{N}}{4}\right)\left(1 - \dfrac{t_1}{t_2} \right)} \label{4}\]
Write an expression for the fraction of material in the mobile phase (φM):
\[\begin{align} \mathrm{φ_M} &= \mathrm{ \dfrac{C_M V_M}{C_M V_M + C_S V_S}} \label{5} \\[4pt] &= \mathrm{\dfrac{\dfrac{C_MV_M}{C_MV_M}}{\dfrac{C_MV_M}{C_MV_M}+\dfrac{C_SV_S}{C_MV_M}}} \\[4pt] &= \mathrm{\dfrac{1}{1+k}} \label{6} \end{align}\]
Express the average migration velocity of component 2:
\[\mathrm{v_{S_2}=φ_Mv} \label{7}\]
(where \(v\) is the mobile phase velocity)
\[\mathrm{v_{S_2}=\dfrac{L}{t_2}} \label{8}\]
\[\mathrm{v=\dfrac{L}{t_0}} \label{9}\]
\[\mathrm{\dfrac{L}{t_2}=\dfrac{L}{t_0}φ_M} \label{10}\]
\[\mathrm{t_2=\dfrac{t_0}{φ_{M_2}} \hspace{80px} t_1=\dfrac{t_0}{φ_{M_1}}} \label{11}\]
\[\mathrm{t_2=\dfrac{t_0}{\left(\dfrac{1}{1+k_2}\right)} \hspace{80px} t_1=\dfrac{t_0}{\left(\dfrac{1}{1+k_1}\right)}} \label{12}\]
\[\mathrm{t_2=t_0(1+k_2) \hspace{80px} t_1=t_0(1+k_1)} \label{13}\]
\[\mathrm{\dfrac{t_1}{t_2}=\dfrac{t_0(1+k_1)}{t_0(1+k_2)}=\dfrac{1+k_1}{1+k_2}} \label{14}\]
Substitute Equation \ref{14} into Equation \ref{4}:
\[\mathrm{R_S=\left(\dfrac{\sqrt{N}}{4}\right)\left(1-\dfrac{1+k_1}{1+k_2}\right)} \label{15}\]
Consider the \[\mathrm{\left(1-\dfrac{1+k_1}{1+k_2}\right)}\] term:
\[\begin{align} \mathrm{\left( 1-\dfrac{1+k_1}{1+k_2}\right )} &= \mathrm{\left( \dfrac{1+k_2}{1+k_2}-\dfrac{1+k_1}{1+k_2}\right )} \label{15a} \\[4pt] &= \mathrm{\dfrac{k_2-k_1}{1+k_2}} \label{15b} \\[4pt] &= \mathrm{\dfrac{k_1\left(\dfrac{k_2}{k_1}-1\right)}{1+k_2}} \label{15c} \\[4pt] &= \mathrm{\dfrac{\left(\dfrac{k_2}{k_1}\right)k_1\left(\dfrac{k_2}{k_1}-1\right)}{\left(\dfrac{k_2}{k_1}\right)(1+k_2)}} \\[4pt] &= \mathrm{\dfrac{k_2\left(\dfrac{k_2}{k_1}-1\right)}{\left(\dfrac{k_2}{k_1}\right)(1+k_2)}} \label{15d} \\[4pt] &= \mathrm{\left(\dfrac{k_2}{1+k_2}\right)\left(\dfrac{\dfrac{k_2}{k_1}-1}{\dfrac{k_2}{k_1}}\right)} \label{15e} \end{align}\]
Equation \ref{15e} can be simplified via the following substitution
\[ \mathrm{α=\dfrac{k_2}{k_1}}\]
So Equation \ref{15e} becomes
\[ \mathrm{\left( 1-\dfrac{1+k_1}{1+k_2}\right )} = \mathrm{\left(\dfrac{k_2}{1+k_2}\right)\left(\dfrac{α-1}{α}\right)} \label{15f} \]
Substitute Equation \ref{15f} into Equation \ref{15}:
\[\mathrm{R_S=\left(\dfrac{\sqrt{N}}{4}\right)\left(\dfrac{α-1}{α}\right)\left(\dfrac{k_2}{1+k_2}\right)}\]
Relationship to retention time:
\[\mathrm{v_{S_2}=\dfrac{L}{t_2} \hspace{80px} t_2 = \dfrac{L}{v_{S_2}}}\]
\[\mathrm{v_{S_2} = φ_Mv=\left(\dfrac{1}{1+k_2}\right)v} \label{16}\]
\[\mathrm{H=\dfrac{L}{N} \hspace{80px} L=HN}\]
\[\mathrm{t_2=\dfrac{HN(1+k_2)}{v}} \label{17}\]
Rearrange the fundamental resolution equation to solve for \(N\):
\[\mathrm{N=16{R_S}^2\left(\dfrac{α}{α-1}\right)^2\left(\dfrac{1+k_2}{k_2}\right)^2}\]
\[\mathrm{t_2=\left(\dfrac{16{R_S}^2H}{v}\right)\left(\dfrac{α}{α-1}\right)^2\left(\dfrac{(1+k_2)^3}{(k_2)^2}\right)} \label{18}\]