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Appendix 1: Derivation of the Fundamental Resolution Equation

  • Page ID
    95089
  • \[\mathrm{R_S = \dfrac{2 (t_2 - t_1)}{W_1 + W_2}} \label{1}\]

    \[\mathrm{W_1 ≅ W_2}\]

    \[\mathrm{W = 4σ}\]

    \[\mathrm{R_S = \dfrac{2(t_2 - t_1)}{8σ_2} = \dfrac{t_2 - t_1}{4σ_2}} \label{2}\]

    \[\mathrm{N = \left(\dfrac{t_2}{σ_2}\right)^2 \hspace{80px} σ_2 = \dfrac{t_2}{\sqrt{N}}}\]

    \[\mathrm{R_S = \dfrac{t_2 - t_1}{4\left(\dfrac{t_2}{\sqrt N}\right)}} \label{3}\]

    \[\mathrm{R_S = \left(\dfrac{\sqrt{N}}{4}\right)\left(\dfrac{t_2 - t_1}{t_2} \right)=\left(\dfrac{\sqrt{N}}{4}\right)\left(1 - \dfrac{t_1}{t_2} \right)} \label{4}\]

    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 \(N\):

    \[\mathrm{N=16{R_S}^2\left(\dfrac{α}{α-1}\right)^2\left(\dfrac{1+k_2}{k_2}\right)^2}\]

    \[\mathrm{t_2=\left(\dfrac{16{R_S}^2H}{v}\right)\left(\dfrac{α}{α-1}\right)^2\left(\dfrac{(1+k_2)^3}{(k_2)^2}\right)} \label{18}\]

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