$word_{count}$: 200

Freundlich isotherm: x/m = kP^$\frac{1}{n}$, where 1/n is between 0 and 1 (typically 0.1-0.5). Log form: log$\frac{x}{m}$ = log k + $\frac{1}{n}$ log P. Plot log$\frac{x}{m}$ vs log P: straight line, slope = 1/n, intercept = log k. At low P: x/m proportional to $P^1$ (linear). At high P: x/m approaches constant (saturation). Limitation: empirical, fails at very high P. Langmuir isotherm: theta = $\frac{KP}{1+KP}$ or x/m = $\frac{aP}{1+bP}$. Assumptions: monolayer, equivalent sites, no lateral interaction. At low P: theta proportional to P (linear). At high P: theta = 1 (saturated). Linear form: $\frac{P}{x/m}$ = 1/a + $\frac{b}{a}$P. Plot $\frac{P}{x/m}$ vs P: straight line. BET isotherm: extends Langmuir to multilayer adsorption, used for surface area measurement. For solutions: Freundlich modified: x/m = kC^$\frac{1}{n}$. Plot log$\frac{x}{m}$ vs log C. Applications: activated charcoal in sugar decolourisation, water purification.