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

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 = $\frac{1}{n}$
• Y-intercept = $\log k$, so $k = 10^{\text{intercept}}$

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

• Low P limit (bP << 1): $\frac{x}{m} \approx aP$ (linear)
• High P limit (bP >> 1): $\frac{x}{m} \approx \frac{a}{b}$ (monolayer saturation)

Thermodynamic Criteria for Adsorption: $\Delta G = \Delta H - T\Delta S < 0 \text{ (spontaneous)}$ $\Delta S < 0 \text{ (entropy decreases, 3D → 2D)}$ $\Rightarrow \Delta H < 0 \text{ (adsorption is always exothermic)}$

Michaelis-Menten Enzyme Kinetics: $v = \frac{V_{max}[S]}{K_m + [S]}$ $\text{At } [S] = K_m: \quad v = \frac{V_{max}}{2}$ $\text{At } [S] \gg K_m: \quad v \to V_{max}$ $\text{At } [S] \ll K_m: \quad v \approx \frac{V_{max}}{K_m}[S] \text{ (first order)}$

Hardy-Schulze Empirical Coagulating Power: $\text{Coagulating Power} \propto z^6$ where $z$ = valency of the coagulating ion (opposite sign to colloid)

Coagulation value $\propto \frac{1}{\text{coagulating power}} \propto \frac{1}{z^6}$

Gold Number (Zsigmondy): $\text{Gold number} = \frac{\text{mg of protective colloid needed}}{10 \text{ mL standard gold sol} + 1 \text{ mL } 10\% \text{ NaCl}}$

Freundlich Numerical Examples:

Example 1: x/m = 0.5P^(1/3) at P = 27 atm: $\frac{x}{m} = 0.5 \times 27^{1/3} = 0.5 \times 3 = 1.5 \text{ units/g}$

Example 2: x/m = kP^(1/n); at P = 1: x/m = 4; at P = 4: x/m = 8. Find 1/n: $4 = k(1)^{1/n} \Rightarrow k = 4$ $8 = 4 \times 4^{1/n} \Rightarrow 4^{1/n} = 2 \Rightarrow \frac{1}{n} = \frac{\log 2}{\log 4} = \frac{0.301}{0.602} = 0.5$