{"id":62,"date":"2019-06-27T18:23:12","date_gmt":"2019-06-27T18:23:12","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/chapter\/sorption-and-chromatography\/"},"modified":"2026-03-16T01:13:01","modified_gmt":"2026-03-16T01:13:01","slug":"sorption-and-chromatography","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/chapter\/sorption-and-chromatography\/","title":{"raw":"Sorption and Chromatography","rendered":"Sorption and Chromatography"},"content":{"raw":"

Adsorption, Ion Exchange, and Chromatography<\/h2>\n$c_i$\u00a0= concentration of species i in the mobile phase (mass volume-1<\/sup>) or (mole volume-1<\/sup>)\n\n$k_i$ = empirical constant for species i for isotherms (units vary)\n\n$K_i$ = adsorption equilibrium constant for species i\n\n$n_i$= internal parameter for isotherms (units vary)\n\n$p_i$= partial pressure of species i (pressure)\n\n$q_i$= amount of species i adsorbed per unit mass of adsorbent at equilibrium (mass mass-1<\/sup>) or (mole mass-1<\/sup>)\n\n$q_{m_i}$ = amount of species i adsorbed per unit mass of adsorbent at maximum loading, where maximum loading corresponds to complete surface coverage (mass mass-1<\/sup>) or (mole mass-1<\/sup>)\n\n \n\nlinear isotherm:\n\n\\begin{equation}\n

\\tag{31.1}<\/p>\n

q_i=k_ip_i<\/p>\n\\end{equation}\n\nFreundlich isotherm:\n\n\\begin{equation}\n

\\tag{31.2}<\/p>\n

q_i=k_ip_i^{1\/n_i}<\/p>\n\\end{equation}\n\nLangmuir isotherm:\n\n\\begin{equation}\n

\\tag{31.3}<\/p>\n

q_i=\\frac{K_iq_{m_i}p_i}{1+K_ip_i}<\/p>\n\\end{equation}\n\nchromatography equilibrium:\n\n\\begin{equation}\n

\\tag{31.4}<\/p>\n

K_i=\\frac{q_i}{c_i}<\/p>\n\\end{equation}\n\nWatch a video from LearnChemE<\/a> for an explanation about the concept of adsorption: \u00a0Adsorption Introduction<\/a> (8:49)\n

Modeling Differential Chromatography<\/h2>\n$\\alpha_i$\u00a0= average partitioning of species i between the bulk fluid and sorbent (unitless)\n\n$\\epsilon_b$\u00a0= sorbent porosity, ranges from 0 to 1 (unitless)\n\n$\\epsilon^*_{p,i}$\u00a0= inclusion porosity, accounts for accessibility of sorbent pores to species i (unitless)\n\n$\\tau_f$\u00a0= sorbent tortuosity factor, usually approximately 1.4 (unitless)\n\n$\\omega_i$\u00a0= fraction of solute in the mobile phase, relative to sorbed solute, at equilibrium (unitless)\n\n \n\n$A$ = cross-sectional area of the column (area)\n\n$c_{f,i}$\u00a0= concentration of species i in the mobile phase (mass volume-1<\/sup>) or (mol volume-1<\/sup>)\n\n$D_{e,i}$\u00a0= effective diffusivity of species i within the sorbent pores (length2<\/sup> time-1<\/sup>)\n\n$E_i$\u00a0= coefficient that accounts for axial diffusion of species i and non-uniformities of flow (length2<\/sup> time-1<\/sup>)\n\n$H_i$ = height of theoretical chromatographic plate for species i (length)\n\n$k_{a,i}$\u00a0= kinetic rate constant of adsorption of species i to the sorbent (time-1<\/sup>)\n\n$k_{c,i}$\u00a0= mass transfer coefficient of species i in the mobile phase (length time-1<\/sup>)\n\n$k_{c,i,tot}$\u00a0= overall mass transfer coefficient of species i (length time-1<\/sup>)\n\n$K_{d,i}$\u00a0= equilibrium distribution coefficient of species i between the mobile phase and sorbent (unitless)\n\n$L$ = length of column (length)\n\n$m_{0_i}$\u00a0= amount of solute i fed to column (mass) or (mol)\n\n$R_{1,2}$\u00a0= resolution of species 1 and 2 in the proposed operating condition (unitless)\n\n$R_p$\u00a0= radius of sorbent particles (length)\n\n$s_i$\u00a0= variance of the Gaussian peak of the distribution of species i along the column length (time)\n\n$t$ = elapsed time since loading of the column (time)\n\n${\\overline t}_i$\u00a0= mean residence time of species i in the column (time)\n\n$u$ = actual fluid velocity through the bed (length time-1<\/sup>)\n\n$u_s$\u00a0= superficial fluid velocity through the bed (length time-1<\/sup>)\n\n$z$ = position along the length of the column, in the direction of flow (length)\n\n$z_{0,i}$\u00a0= mean position of species i along the length of the column as a function of time (length)\n\n \n\n\\begin{displaymath}\n

\\tag{32.1}<\/p>\n

z_{0,i}(t)=\\omega_iut<\/p>\n\\end{displaymath}\n\n\\begin{equation}\n

\\tag{32.2}<\/p>\n

\\omega_i=\\frac{1}{1+\\frac{1-\\epsilon_b}{\\epsilon_b\\alpha_i}}<\/p>\n\\end{equation}\n\n\\begin{equation}\n

\\tag{32.3}<\/p>\n

\\alpha_i=\\frac{1}{\\epsilon_{p,i}^*(1+K_{d,i})}<\/p>\n\\end{equation}\n\n\\begin{equation}\n

\\tag{32.4}<\/p>\n

u=u_s\/\\epsilon_b<\/p>\n\\end{equation}\n\n\\begin{equation}\n

\\tag{32.5}<\/p>\n

\\overline t_i=\\frac{L}{\\omega_iu}<\/p>\n\\end{equation}\n\n\\begin{equation}\n

\\tag{32.6}<\/p>\n

c_{f,i}(z,t)=\\frac{m_{0_i}\\omega_i}{A\\epsilon_b(2\\pi H_iz_0)^{0.5}}\\textrm {exp}\\left(\\frac{-(z-z_0)^2}{2H_iz_0}\\right)<\/p>\n\\end{equation}\n\n\\begin{equation}\n

\\tag{32.7}<\/p>\n

H_i=2\\left[\\frac{E_i}{u}+\\frac{\\omega_i(1-\\omega_i)R_pu}{3\\alpha_ik_{ci,tot}}\\right]<\/p>\n\\end{equation}\n\n\\begin{displaymath}\n

\\tag{32.8}<\/p>\n

N_{{\\rm Pe},i}=N_{\\rm Re}N_{{\\rm Sc},i}=\\frac{2R_pu\\epsilon_b}{D_i}<\/p>\n\\end{displaymath}\n\nif $N_{{\\rm Pe},i}<<1$\n\n\\begin{displaymath}\n

\\tag{32.9}<\/p>\n

E_i=\\frac{D_i}{\\tau_f}<\/p>\n\\end{displaymath}\n\nelse\n\n\\begin{displaymath}\n

\\tag{32.10}<\/p>\n

E_i=\\frac{2R_pu\\epsilon_b}{N_{{\\rm Pe},E,i}}<\/p>\n\\end{displaymath}\n\n$N_{{\\rm Pe},E,i}$ calculated by 15-61 or 15-62, Seader\n\n\\begin{equation}\n

\\tag{32.11}<\/p>\n

\\frac{1}{k_{ci,tot}}=\\frac{1}{k_{c,i}}+\\frac{R_p}{5\\epsilon_{p,i}^*D_{e,i}}+\\frac{3}{R_pk_{a,i}\\epsilon_{p,i}^*}\\left[\\frac{K_{d,i}}{1+K_{d,i}}\\right]^2<\/p>\n\\end{equation}\n\n\\begin{equation}\n

\\tag{32.12}<\/p>\n

s_i^2=\\frac{\\overline t_iH_i}{\\omega_iu}<\/p>\n\\end{equation}\n\n\\begin{equation}\n

\\tag{32.13}<\/p>\n

R_{1,2}=\\frac{\\textrm {abs}(\\overline t_1-\\overline t_2)}{2(s_1+s_2)}<\/p>\n\\end{equation}\n

\n

Example<\/p>\n\n<\/header>\n

\n\n1.0 g of species A is added to a chromatography column of cross-sectional area 1.0 m2<\/sup> and length 1.0 m. Mobile phase is added at a flowrate of $4.0 \\times 10^{-3}$\u00a0m3<\/sup>\/s. Species A has a mass transfer coefficient of $2.0 \\times 10^{-5}$ m\/s in this solvent. The selected sorbent has a porosity of 0.40 m and average particle radius of $5.0\\times 10^{-6}$ m. For species A in this sorbent, the inclusion porosity is 0.80, $K_d = 50$, $E = 2.0\\times 10^{-8}$\u00a0m2<\/sup>\/s, $k_a = 100$ s-1<\/sup> and the effective diffusivity is $3.5\\times 10^{-12}$\u00a0m2<\/sup>\/s.\n\n(a) When is mean expected elution time for species A?\n\n(b) Plot the concentration profile for species A at 0.05 m increments along the column length in 10-minute increments, until all of the solute has eluted.\n\n(c) Find the variance of the peak for species A in the proposed operating condition.\n\n(d) The column feed also contains 1.0 g of species B. Species B has a mass transfer coefficient of $1.0\\times 10^{-5}$ m\/s in the mobile phase, inclusion porosity of 0.50, $K_d = 60$, $E = 3.0\\times 10^{-8}$\u00a0m2<\/sup>\/s, effective diffusivity of $4\\times 10^{-12}$\u00a0m2<\/sup>\/s and $k_a = 200$ s-1<\/sup>. What is the resolution of these two species in the proposed operating condition?\n\n<\/div>\n<\/div>","rendered":"

Adsorption, Ion Exchange, and Chromatography<\/h2>\n

$c_i$\u00a0= concentration of species i in the mobile phase (mass volume-1<\/sup>) or (mole volume-1<\/sup>)<\/p>\n

$k_i$ = empirical constant for species i for isotherms (units vary)<\/p>\n

$K_i$ = adsorption equilibrium constant for species i<\/p>\n

$n_i$= internal parameter for isotherms (units vary)<\/p>\n

$p_i$= partial pressure of species i (pressure)<\/p>\n

$q_i$= amount of species i adsorbed per unit mass of adsorbent at equilibrium (mass mass-1<\/sup>) or (mole mass-1<\/sup>)<\/p>\n

$q_{m_i}$ = amount of species i adsorbed per unit mass of adsorbent at maximum loading, where maximum loading corresponds to complete surface coverage (mass mass-1<\/sup>) or (mole mass-1<\/sup>)<\/p>\n

 <\/p>\n

linear isotherm:<\/p>\n

\\begin{equation}<\/p>\n

\\tag{31.1}<\/p>\n

q_i=k_ip_i<\/p>\n

\\end{equation}<\/p>\n

Freundlich isotherm:<\/p>\n

\\begin{equation}<\/p>\n

\\tag{31.2}<\/p>\n

q_i=k_ip_i^{1\/n_i}<\/p>\n

\\end{equation}<\/p>\n

Langmuir isotherm:<\/p>\n

\\begin{equation}<\/p>\n

\\tag{31.3}<\/p>\n

q_i=\\frac{K_iq_{m_i}p_i}{1+K_ip_i}<\/p>\n

\\end{equation}<\/p>\n

chromatography equilibrium:<\/p>\n

\\begin{equation}<\/p>\n

\\tag{31.4}<\/p>\n

K_i=\\frac{q_i}{c_i}<\/p>\n

\\end{equation}<\/p>\n

Watch a video from LearnChemE<\/a> for an explanation about the concept of adsorption: \u00a0Adsorption Introduction<\/a> (8:49)<\/p>\n

Modeling Differential Chromatography<\/h2>\n

$\\alpha_i$\u00a0= average partitioning of species i between the bulk fluid and sorbent (unitless)<\/p>\n

$\\epsilon_b$\u00a0= sorbent porosity, ranges from 0 to 1 (unitless)<\/p>\n

$\\epsilon^*_{p,i}$\u00a0= inclusion porosity, accounts for accessibility of sorbent pores to species i (unitless)<\/p>\n

$\\tau_f$\u00a0= sorbent tortuosity factor, usually approximately 1.4 (unitless)<\/p>\n

$\\omega_i$\u00a0= fraction of solute in the mobile phase, relative to sorbed solute, at equilibrium (unitless)<\/p>\n

 <\/p>\n

$A$ = cross-sectional area of the column (area)<\/p>\n

$c_{f,i}$\u00a0= concentration of species i in the mobile phase (mass volume-1<\/sup>) or (mol volume-1<\/sup>)<\/p>\n

$D_{e,i}$\u00a0= effective diffusivity of species i within the sorbent pores (length2<\/sup> time-1<\/sup>)<\/p>\n

$E_i$\u00a0= coefficient that accounts for axial diffusion of species i and non-uniformities of flow (length2<\/sup> time-1<\/sup>)<\/p>\n

$H_i$ = height of theoretical chromatographic plate for species i (length)<\/p>\n

$k_{a,i}$\u00a0= kinetic rate constant of adsorption of species i to the sorbent (time-1<\/sup>)<\/p>\n

$k_{c,i}$\u00a0= mass transfer coefficient of species i in the mobile phase (length time-1<\/sup>)<\/p>\n

$k_{c,i,tot}$\u00a0= overall mass transfer coefficient of species i (length time-1<\/sup>)<\/p>\n

$K_{d,i}$\u00a0= equilibrium distribution coefficient of species i between the mobile phase and sorbent (unitless)<\/p>\n

$L$ = length of column (length)<\/p>\n

$m_{0_i}$\u00a0= amount of solute i fed to column (mass) or (mol)<\/p>\n

$R_{1,2}$\u00a0= resolution of species 1 and 2 in the proposed operating condition (unitless)<\/p>\n

$R_p$\u00a0= radius of sorbent particles (length)<\/p>\n

$s_i$\u00a0= variance of the Gaussian peak of the distribution of species i along the column length (time)<\/p>\n

$t$ = elapsed time since loading of the column (time)<\/p>\n

${\\overline t}_i$\u00a0= mean residence time of species i in the column (time)<\/p>\n

$u$ = actual fluid velocity through the bed (length time-1<\/sup>)<\/p>\n

$u_s$\u00a0= superficial fluid velocity through the bed (length time-1<\/sup>)<\/p>\n

$z$ = position along the length of the column, in the direction of flow (length)<\/p>\n

$z_{0,i}$\u00a0= mean position of species i along the length of the column as a function of time (length)<\/p>\n

 <\/p>\n

\\begin{displaymath}<\/p>\n

\\tag{32.1}<\/p>\n

z_{0,i}(t)=\\omega_iut<\/p>\n

\\end{displaymath}<\/p>\n

\\begin{equation}<\/p>\n

\\tag{32.2}<\/p>\n

\\omega_i=\\frac{1}{1+\\frac{1-\\epsilon_b}{\\epsilon_b\\alpha_i}}<\/p>\n

\\end{equation}<\/p>\n

\\begin{equation}<\/p>\n

\\tag{32.3}<\/p>\n

\\alpha_i=\\frac{1}{\\epsilon_{p,i}^*(1+K_{d,i})}<\/p>\n

\\end{equation}<\/p>\n

\\begin{equation}<\/p>\n

\\tag{32.4}<\/p>\n

u=u_s\/\\epsilon_b<\/p>\n

\\end{equation}<\/p>\n

\\begin{equation}<\/p>\n

\\tag{32.5}<\/p>\n

\\overline t_i=\\frac{L}{\\omega_iu}<\/p>\n

\\end{equation}<\/p>\n

\\begin{equation}<\/p>\n

\\tag{32.6}<\/p>\n

c_{f,i}(z,t)=\\frac{m_{0_i}\\omega_i}{A\\epsilon_b(2\\pi H_iz_0)^{0.5}}\\textrm {exp}\\left(\\frac{-(z-z_0)^2}{2H_iz_0}\\right)<\/p>\n

\\end{equation}<\/p>\n

\\begin{equation}<\/p>\n

\\tag{32.7}<\/p>\n

H_i=2\\left[\\frac{E_i}{u}+\\frac{\\omega_i(1-\\omega_i)R_pu}{3\\alpha_ik_{ci,tot}}\\right]<\/p>\n

\\end{equation}<\/p>\n

\\begin{displaymath}<\/p>\n

\\tag{32.8}<\/p>\n

N_{{\\rm Pe},i}=N_{\\rm Re}N_{{\\rm Sc},i}=\\frac{2R_pu\\epsilon_b}{D_i}<\/p>\n

\\end{displaymath}<\/p>\n

if $N_{{\\rm Pe},i}<<1$<\/p>\n

\\begin{displaymath}<\/p>\n

\\tag{32.9}<\/p>\n

E_i=\\frac{D_i}{\\tau_f}<\/p>\n

\\end{displaymath}<\/p>\n

else<\/p>\n

\\begin{displaymath}<\/p>\n

\\tag{32.10}<\/p>\n

E_i=\\frac{2R_pu\\epsilon_b}{N_{{\\rm Pe},E,i}}<\/p>\n

\\end{displaymath}<\/p>\n

$N_{{\\rm Pe},E,i}$ calculated by 15-61 or 15-62, Seader<\/p>\n

\\begin{equation}<\/p>\n

\\tag{32.11}<\/p>\n

\\frac{1}{k_{ci,tot}}=\\frac{1}{k_{c,i}}+\\frac{R_p}{5\\epsilon_{p,i}^*D_{e,i}}+\\frac{3}{R_pk_{a,i}\\epsilon_{p,i}^*}\\left[\\frac{K_{d,i}}{1+K_{d,i}}\\right]^2<\/p>\n

\\end{equation}<\/p>\n

\\begin{equation}<\/p>\n

\\tag{32.12}<\/p>\n

s_i^2=\\frac{\\overline t_iH_i}{\\omega_iu}<\/p>\n

\\end{equation}<\/p>\n

\\begin{equation}<\/p>\n

\\tag{32.13}<\/p>\n

R_{1,2}=\\frac{\\textrm {abs}(\\overline t_1-\\overline t_2)}{2(s_1+s_2)}<\/p>\n

\\end{equation}<\/p>\n

\n
\n

Example<\/p>\n<\/header>\n

\n

1.0 g of species A is added to a chromatography column of cross-sectional area 1.0 m2<\/sup> and length 1.0 m. Mobile phase is added at a flowrate of $4.0 \\times 10^{-3}$\u00a0m3<\/sup>\/s. Species A has a mass transfer coefficient of $2.0 \\times 10^{-5}$ m\/s in this solvent. The selected sorbent has a porosity of 0.40 m and average particle radius of $5.0\\times 10^{-6}$ m. For species A in this sorbent, the inclusion porosity is 0.80, $K_d = 50$, $E = 2.0\\times 10^{-8}$\u00a0m2<\/sup>\/s, $k_a = 100$ s-1<\/sup> and the effective diffusivity is $3.5\\times 10^{-12}$\u00a0m2<\/sup>\/s.<\/p>\n

(a) When is mean expected elution time for species A?<\/p>\n

(b) Plot the concentration profile for species A at 0.05 m increments along the column length in 10-minute increments, until all of the solute has eluted.<\/p>\n

(c) Find the variance of the peak for species A in the proposed operating condition.<\/p>\n

(d) The column feed also contains 1.0 g of species B. Species B has a mass transfer coefficient of $1.0\\times 10^{-5}$ m\/s in the mobile phase, inclusion porosity of 0.50, $K_d = 60$, $E = 3.0\\times 10^{-8}$\u00a0m2<\/sup>\/s, effective diffusivity of $4\\times 10^{-12}$\u00a0m2<\/sup>\/s and $k_a = 200$ s-1<\/sup>. What is the resolution of these two species in the proposed operating condition?<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":1,"menu_order":8,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-62","chapter","type-chapter","status-publish","hentry"],"part":20,"_links":{"self":[{"href":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/wp-json\/pressbooks\/v2\/chapters\/62","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":1,"href":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/wp-json\/pressbooks\/v2\/chapters\/62\/revisions"}],"predecessor-version":[{"id":63,"href":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/wp-json\/pressbooks\/v2\/chapters\/62\/revisions\/63"}],"part":[{"href":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/wp-json\/pressbooks\/v2\/parts\/20"}],"metadata":[{"href":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/wp-json\/pressbooks\/v2\/chapters\/62\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/wp-json\/wp\/v2\/media?parent=62"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/wp-json\/pressbooks\/v2\/chapter-type?post=62"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/wp-json\/wp\/v2\/contributor?post=62"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/chemicalengineeringseparations\/wp-json\/wp\/v2\/license?post=62"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}