Engineering
Langmuir & Freundlich Adsorption Isotherm Capacity formulas
Compare equilibrium Langmuir and Freundlich uptake for a hypothetical solute/adsorbent system with explicit concentration and loading units.
Equilibrium uptake is not a filter sizing calculation
Ce is the equilibrium dissolved concentration in mg/L, not automatically the feed or initial concentration. qe is adsorbed mass per adsorbent mass in mg/g. Langmuir uses qmax in mg/g and KL in L/mg, making KL×Ce dimensionless.
Freundlich is an empirical power law. KF has units (mg/g)/(mg/L)^(1/n) under this convention; changing concentration units requires changing KF. Positive n is required. Values n>1 give a sublinear curve here, but do not by themselves establish contaminant removal performance.
The coefficients are hypothetical. Apply fitted parameters only for the relevant solute, adsorbent, temperature and chemistry over their measured concentration range. The high-concentration example illustrates Langmuir’s mathematical asymptote, not validated capacity at that concentration. Freundlich has no finite saturation plateau.
These equations alone do not size a PFAS filter, an ion-exchange bed or a treatment system. Flow, competing species, mass balance, kinetics and breakthrough are separate. A batch equilibrium calculation must solve Ce rather than simply substituting inlet concentration.
| Adsorption Isotherm Model | Formula | C_e = 12 mg/L (q_max=120, KL=0.25, KF=35, n=2.5) | High C_e = 4,000 mg/L |
|---|---|---|---|
| Langmuir Monolayer q_e | (B2 * B3 * B4) / (1 + B3 * B4) | 360 / 4 = 90.00 mg/g (75% sites) | 119.88 mg/g (-> q_max 120) |
| Surface Coverage Fraction theta | (B3 * B4) / (1 + B3 * B4) | 3 / 4 = 0.7500 | 0.9990 |
| Freundlich Heterogeneous q_e | B5 * (B4 ^ (1 / B6)) | 35 * (12 ^ 0.4) = 94.57 mg/g | Unsaturated power law |
Langmuir Monolayer Equilibrium Uptake q_e (mg / g)
Evaluates monolayer loading from maximum capacity q_max (B2), affinity K_L (B3), and equilibrium concentration C_e (B4).
=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<=0,B6<=0),"Invalid inputs",ROUND((B2 * B3 * B4) / (1 + B3 * B4), 2)))Freundlich Power-Law Uptake q_e = K_F * C_e^(1 / n)
Uses fractional exponent 1 / n (1 / B6) on equilibrium concentration B4.
=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<=0,B6<=0),"Invalid inputs",ROUND(B5 * (B4 ^ (1 / B6)), 2)))Langmuir fractional coverage
For this idealized model, theta=KL×Ce/(1+KL×Ce). This is a model site-coverage fraction, not a measured removal percentage from a flowing stream.
=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<=0,B6<=0),"Invalid inputs",ROUND(B3*B4/(1+B3*B4),4)))Try the example
Use the original values or change them to test your understanding.