Water tools

Freundlich and Langmuir isotherm fit from batch data

A bottle-point test gives pairs of equilibrium concentration and loading. Both classical isotherms linearise, so a regression returns their parameters and tells you which fits.

Equilibrium concentration and loading, one pair per line (any consistent units)Loading q = V (C₀ − Ce) / M from each bottle.

Freundlich

Freundlich K

191.895

Freundlich 1/n

0.433

Typically 0.5 to 0.7 for compounds worth adsorbing; below 1 is favourable.

r²

0.9990

Langmuir

Langmuir capacity Qm

789.39

Langmuir b

0.3800

r²

0.9627

How it works

q_{e} = \frac{V (C_{0} - C_{e})}{M}

(1)

q = K C^{1 / n} \log q = \log K + \frac{1}{n} \log C

(2)

q = \frac{Q_{m} b C}{1 + b C} \frac{C}{q} = \frac{1}{b Q_{m}} + \frac{C}{Q_{m}}

(3)

where

$q_{e}, q$: adsorbent loading at equilibrium, mass per mass
$V$: bottle volume, L
$C_{0}, C_{e}$: initial and equilibrium concentration
$M$: adsorbent mass, g
$C$: equilibrium concentration
$K$: Freundlich capacity constant
$1 / n$: Freundlich intensity, dimensionless
$Q_{m}$: Langmuir monolayer capacity
$b$: Langmuir constant, litres per unit concentration

The log-linear and reciprocal fits weight the points unequally; weighted nonlinear regression is preferred for publication but the linear forms are what most design work uses. Single-solute data in organic-free water overstate the capacity in a real water, where natural organic matter competes. The Freundlich form is the one carried into bed-life and dose calculations.

Freundlich and Langmuir isotherm fit from batch data

How it works

Related reading