Problem Statement
Explain the concept of critical micelle concentration (CMC) and how it manifests in surface tension measurements.
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: Surfactant molecules have a hydrophilic head and hydrophobic tail. In water:
Step 2 — Apply the relevant physical law or equation:
- Below CMC: monomers adsorb to air-water interface, strongly reducing $\sigma$.
- At CMC: the interface is saturated; additional surfactant forms micelles — spherical aggregates with tails inward and heads outward.
- Above CMC: $\sigma$ remains approximately constant (interface already saturated; new molecules join micelles).
Step 3 — Solve algebraically for the unknown: On a $\sigma$ vs $\ln c$ plot, $\sigma$ decreases linearly until the CMC, then plateaus. The CMC is identified as the kink point.
Step 4 — Substitute numerical values with units: Typical CMC: SDS (sodium dodecyl sulfate) $\approx 8\ \text{mM}$; CTAB $\approx 1\ \text{mM}$. Below CMC, the Gibbs equation applies: $\Gamma = -(1/RT)(d\sigma/d\ln c)$.
Worked Calculation
$$\text{Numerical result} = \text{given expression substituted with values}$$
$$\boxed{\Gamma = -(1/RT)(d\sigma/d\ln c)}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
Surfactant molecules have a hydrophilic head and hydrophobic tail. In water:
- Below CMC: monomers adsorb to air-water interface, strongly reducing $\sigma$.
- At CMC: the interface is saturated; additional surfactant forms micelles — spherical aggregates with tails inward and heads outward.
- Above CMC: $\sigma$ remains approximately constant (interface already saturated; new molecules join micelles).
On a $\sigma$ vs $\ln c$ plot, $\sigma$ decreases linearly until the CMC, then plateaus. The CMC is identified as the kink point.
Typical CMC: SDS (sodium dodecyl sulfate) $\approx 8\ \text{mM}$; CTAB $\approx 1\ \text{mM}$. Below CMC, the Gibbs equation applies: $\Gamma = -(1/RT)(d\sigma/d\ln c)$.
Answer
$$\boxed{\Gamma = -(1/RT)(d\sigma/d\ln c)}$$
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.
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