ENV-30

Langmuir isotherm

qe = qmax b Ce /(1 + b Ce). Monolayer on uniform sites.

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AdsorptionLangmuir 1918

Governing equation

q_e=q_{\max}\dfrac{b C_e}{1+b C_e}

$C_e \in \left[0.05, 80\right]$

where

q_{\max}: Monolayer capacity (mg/g)
b: Affinity b (L/mg)
C_e: Equilibrium conc. (mg/L)
q_e: Solid loading (mg/g)
\theta: Coverage (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

q_e=q_{\max}\dfrac{b C_e}{1+b C_e}

$C_e \in \left[0.05, 80\right]$

$C_e = 0.05$ $q_e = 0.784$

Along the curve

②

Domain of the variable — blue

C_e_{\min} = 0.05,\quad C_e_{\max} = 80

$C_e \in \left[0.05, 80\right]$

$C_e = 0.05$ $C_e = 0.05$

Along the curve

③

Response curve — green

q_e=q_{\max}\dfrac{b C_e}{1+b C_e}

$C_e \in \left[0.05, 80\right]$

$C_e = 0.05$ $q_e = 0.784$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}q_e}{\mathrm{d}C_e}

$C_e \in \left[0.05, 80\right]$

$C_e = 0.553$ $q_e^{\prime} = 11.03$

Along the curve

Lecture brief

Historical brief

Streeter–Phelps (1925) oxygen sag, settling theory and Guldberg–Waage kinetics made water and air quality a rate problem. The lab computes sag, overflow and a snapshot of reactor mass balance. This sheet (ENV-30 — Langmuir isotherm) is the form associated with Langmuir 1918. Working symbols: $q_{\max}$ , $b$ , $C_e$ $\rightarrow$ $q_e$ , $\theta$ . Langmuir balanced adsorption and desorption rates on a finite number of identical sites, giving a hyperbolic isotherm.

Purpose

Purpose: compute

q_e

\theta

from

q_{\max}

b

C_e

in Environmental via

q_e=q_{\max}\dfrac{b C_e}{1+b C_e}

qe = qmax b Ce /(1 + b Ce). Monolayer on uniform sites. Use it when a real environmental question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

q_{\max} = 40.000\,\mathrm{mg/g}

b = 0.400\,\mathrm{L/mg}

C_e = 5.000\,\mathrm{mg/L}

, the governing relation

q_e=q_{\max}\dfrac{b C_e}{1+b C_e}

yields

q_e = 26.667\,\mathrm{mg/g}

\theta = 0.667\,\mathrm{—}

. A surface of sites, a filling fraction θ. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Monolayer capacity q_{\max}40 mg/gAffinity b b0.4 L/mgEquilibrium conc. C_e5 mg/L

Outputs

Solid loading q_e26.667 mg/g
Coverage \theta0.667 —

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Narration of this film

A surface of sites, a filling fraction θ.

Langmuir balanced adsorption and desorption rates on a finite number of identical sites, giving a hyperbolic isotherm.

Reading speed

Watch on YouTube