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# Appendix 1: Derivation of the Fundamental Resolution Equation

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

$\mathrm{W_1 \cong W_2}$

$\mathrm{W = 4\sigma}$

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

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

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

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

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)

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

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

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

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

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

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

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:

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

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