Appendix 1: Derivation of the Fundamental Resolution Equation

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Thomas Wenzel
Analytical Sciences Digital Library

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\[\mathrm{R_S = \dfrac{2(t_2-t_1)}{W_1 + W_2}}\tag{1}\]

\[\mathrm{W_1 \cong W_2}\]

\[\mathrm{W = 4\sigma}\]

\[\mathrm{R_S = \dfrac{2(t_2-t_1)}{8\sigma_2} = \dfrac{t_2-t_1}{4\sigma_2}}\tag{2}\]

\[\mathrm{N = \left(\dfrac{t_2}{\sigma_2} \right )^2 \hspace{60px} \sigma_2 = \dfrac{t_2}{\sqrt{N}}}\]

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

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

Write an expression for the fraction of material in the mobile phase (φ_M):

\[\mathrm{\varphi_M = \dfrac{C_MV_M}{C_MV_M + C_SV_S}}\tag{5}\]

\[\mathrm{\varphi_M = \dfrac{\dfrac{C_MV_M}{C_MV_M}}{\dfrac{C_MV_M}{C_MV_M} + \dfrac{C_SV_S}{C_MV_M}} = \dfrac{1}{1 + k'}}\tag{6}\]

Express the average migration velocity of component 2:

\[\mathrm{v_{S_2} = \varphi_M v}\tag{7}\]

(where v is the mobile phase velocity)

\[\mathrm{v_{S_2} = \dfrac{L}{t_2}\hspace{20px}(8) \hspace{60px} v = \dfrac{L}{t_0} \hspace{20px} (9)}\]

\[\mathrm{\dfrac{L}{t_2} = \dfrac{L}{t_0}\varphi_M}\tag{10}\]

\[\mathrm{t_2 = \dfrac{t_0}{\varphi_{M_2}} \hspace{60px} t_1 = \dfrac{t_0}{\varphi_{M_1}}}\tag{11}\]

\[\mathrm{t_2 = \dfrac{t_0}{\left(\dfrac{1}{1+k_2'}\right)} \hspace{60px} t_1 = \dfrac{t_0}{\left(\dfrac{1}{1+k_1'}\right)}}\tag{12}\]

\[\mathrm{t_2 = t_0(1+k_2') \hspace{60px} t_1=t_0(1+k_1')}\tag{13}\]

\[\mathrm{\dfrac{t_1}{t_2} = \dfrac{t_0(1+k_1')}{t_0(1+k_2')} =\dfrac{1+k_1'}{1+k_2'}}\tag{14}\]

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:

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

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