Problem Statement
Derive the Gibbs adsorption isotherm relating surface tension change to solute concentration.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: For a liquid-vapour interface, the Gibbs surface excess $\Gamma_2$ of solute is related to surface tension by:
Step 2 — Apply the relevant physical law or equation: $$d\sigma = -\Gamma_2\,d\mu_2 = -\Gamma_2 RT\,d\ln a_2$$
$$\Gamma_2 = -\frac{1}{RT}\frac{d\sigma}{d\ln c_2} = -\frac{c_2}{RT}\frac{d\sigma}{dc_2}$$
Step 3 — Solve algebraically for the unknown: For a surface-active solute (surfactant): $d\sigma/dc < 0$ (surface tension decreases) → $\Gamma_2 > 0$ (solute enriched at surface).
Step 4 — Substitute numerical values with units: For a surface-inactive solute (e.g., salt): $d\sigma/dc > 0$ (surface tension increases) → $\Gamma_2 < 0$ (solute depleted from surface — negative adsorption).
Step 5 — Compute and check the result: This equation is fundamental to understanding surfactant behaviour and is experimentally verified by measuring $\sigma(c)$.
Worked Calculation
$$d\sigma = -\Gamma_2\,d\mu_2 = -\Gamma_2 RT\,d\ln a_2$$
$$\Gamma_2 = -\frac{1}{RT}\frac{d\sigma}{d\ln c_2} = -\frac{c_2}{RT}\frac{d\sigma}{dc_2}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
For a liquid-vapour interface, the Gibbs surface excess $\Gamma_2$ of solute is related to surface tension by:
$$d\sigma = -\Gamma_2\,d\mu_2 = -\Gamma_2 RT\,d\ln a_2$$
$$\Gamma_2 = -\frac{1}{RT}\frac{d\sigma}{d\ln c_2} = -\frac{c_2}{RT}\frac{d\sigma}{dc_2}$$
For a surface-active solute (surfactant): $d\sigma/dc < 0$ (surface tension decreases) → $\Gamma_2 > 0$ (solute enriched at surface).
For a surface-inactive solute (e.g., salt): $d\sigma/dc > 0$ (surface tension increases) → $\Gamma_2 < 0$ (solute depleted from surface — negative adsorption).
This equation is fundamental to understanding surfactant behaviour and is experimentally verified by measuring $\sigma(c)$.
Answer
$$\boxed{\Gamma_2 = -\frac{1}{RT}\frac{d\sigma}{d\ln c_2} = -\frac{c_2}{RT}\frac{d\sigma}{dc_2}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
Leave a Reply