26.5: Summary of Important Relationships for Chromatography
- Page ID
- 349945
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this chapter we have introduced many chromatographic variables, some directly measured from the chromatogram, provided by the manufacturer, or from the operating conditions, and some derived from these variables. The following two tables summarize these variables.
| variable | name | source |
|---|---|---|
| \(t_r\) | retention time for solute | chromatogram |
| \(t_m\) | retention time for non-retained solute | chromatogram |
| \(w\) | peak width | chromatogram |
| \(u\) | mobile phase flow rate | operating conditions |
| \(L\) | length of column's stationary phase | manufacturer |
| \(d_c\) | diameter of column | manufacturer |
| \(d_p\) | diameter of packing material | manufacturer |
| \(d_f\) | thickness of stationary phase | manufacturer |
| \(V_s\) | volume of stationary phase | operating conditions |
| variable | name | equation to derive value |
|---|---|---|
| \(V_m\) | volume of mobile phase | \(V_m = t_m u\) |
| \(k\) | retention factor | \(k = \frac{t_r - t_m}{t_m}\) |
| \(t_r^{\prime}\) | adjusted retention time | \(t_r^{\prime} = t_r - t_m\) |
| \(D\) | distribution ratio | \(D = k \times \frac{V_s}{V_m}\) |
| \(\alpha\) | selectivity factor | \(\alpha = \frac{k_B}{k_A}\) |
| \(R_{AB}\) | resolution | \(R_{AB} = \frac{\sqrt{N_B}}{4} \times \frac{\alpha - 1}{\alpha} \times \frac{k_B}{1 + k_B}\) |
| \(N\) | number of theoretical plates | \(N = 16 \left(\frac{t_r}{w} \right)^2\) |
| \(H\) | height of theoretical plates | \(H = \frac{Lw^2}{16 t_r^2} = \frac{L}{N}\) |
| \(H_p\) | height due to multiple paths | \(H_p = 2 \lambda d_p\) |
| \(H_d\) | height due to longitudinal diffusion | \(H_d = \frac{2 \gamma D_m}{u}\) |
| \(H_s\) | height due to mass transfer in stationary phase | \(H_s = \frac{qkd_f^2}{(1 + k)^2 D_s}u\) |
| \(H_m\) | height due to mass transfer in mobile phase | \(H_m = \frac{fn(d_p^2, d_c^2)}{D_m}u\) |