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Freundlich isotherm: x/m = kP^(1/n), where 1/n is between 0 and 1 (typically 0.1-0.5). Log form: log(x/m) = log k + (1/n) log P. Plot log(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 = KP/(1+KP) or x/m = 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: P/(x/m) = 1/a + (b/a)P. Plot 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^(1/n). Plot log(x/m) vs log C. Applications: activated charcoal in sugar decolourisation, water purification.