Freundlich Adsorption Isotherm: $\frac{x}{m} = k P^{1/n} \quad \text{where } 0 < \frac{1}{n} < 1$

Freundlich Isotherm (Logarithmic / Linear Form): $\log\left(\frac{x}{m}\right) = \log k + \frac{1}{n} \log P$

• Slope of log(x/m) vs log P plot = $\frac{1}{n}$
• Y-intercept = $\log k$

Langmuir Adsorption Isotherm: $\frac{x}{m} = \frac{aP}{1 + bP}$

• At low P: $\frac{x}{m} \approx aP$ (linear, Henry's law)
• At high P: $\frac{x}{m} \approx \frac{a}{b}$ (monolayer saturation)

Michaelis-Menten Enzyme Kinetics: $v = \frac{V_{max}[S]}{K_m + [S]}$

• When $[S] = K_m$: $v = \frac{V_{max}}{2}$

Thermodynamics of Adsorption: $\Delta G = \Delta H - T\Delta S$

• Adsorption is spontaneous: $\Delta G < 0$
• Entropy decreases: $\Delta S < 0$
• Therefore adsorption is always exothermic: $\Delta H < 0$

Hardy-Schulze Empirical Relationship: $\text{Coagulating Power} \propto z^6$ where $z$ = valency of the coagulating ion

Gold Number Definition: $\text{Gold number} = \frac{\text{mg of protective colloid}}{10 \text{ mL gold sol protected from } 1 \text{ mL of } 10\% \text{ NaCl}}$