Skip to content
Stan’s Legacy
Sign in

Surface tension with a surfactant

How far does this much surfactant lower the water's surface tension, and is it past the point of doing any more?

The formula
γ=γ0−a⁢ln1+K⁢c,a=R⁢T⁢Γ
γ
Surface tension, N/m
γ0
Clean surface tension, N/m
a
Thermal-adsorption scale, RTΓ, N/m
K
Adsorption constant, L/mol
c
Concentration, mol/L
Γ
Surface excess at saturation, mol/m²
LaTeX 
γ = γ0 - a \cdot \ln\left(1 + K \cdot c\right), \qquad a = R \cdot T \cdot Γ

Work it out

Fill from a part the library

mN/m

The carrier's surface tension with no surfactant. Water at 20 °C is 72.8 mN/m.

°C

Sets the thermal energy term. The clean surface tension is entered separately and is not adjusted for it.

µmol/m²

How densely the surfactant can pack the surface. Typically 2–4 µmol/m²; about 3.3 for SDS.

L/mol

How strongly the surfactant prefers the surface to the bulk. The default is fitted so SDS reaches 38 mN/m at its CMC.

mmol/L

How much surfactant is in the carrier.

mmol/L

Above this the surfactant forms micelles in the bulk and the surface tension stops falling. SDS: 8.2 mM.

or see a worked example with typical numbers

Method

  1. Convert the temperature to kelvin and the surface excess to moles per square metre. Multiply the gas constant R (8.314 J/mol·K) by both: this is the scale a, in newtons per metre, that sets how much surface tension one "e-fold" of surfactant removes. For SDS at 20 °C it is about 8 mN/m.
  2. Convert the concentration to moles per litre and multiply by the adsorption constant K; add one and take the natural log. This is the Langmuir isotherm's coverage term — how full the surface is at this concentration.
  3. Multiply the scale by that log and subtract from the clean surface tension. That is Szyszkowski's equation, and it holds up to the critical micelle concentration.
  4. Past the CMC, use the CMC's concentration instead: the surface is as full as it gets, further surfactant goes into micelles in the bulk, and the surface tension stays flat.

Assumptions

  • A single, pure surfactant in a clean carrier, adsorbing by a Langmuir isotherm. Mixed or impure surfactants, and any oil or salt in the water, change both Γ and K.
  • The defaults are SDS's, and K in particular is a fit, not a measurement: it is chosen so the curve passes through water's 72.8 mN/m and the roughly 38 mN/m every reference gives for SDS at its 8.2 mM CMC. Other surfactants need their own Γ, K and CMC — the shape of the curve is general, the numbers are not.
  • The sharp corner at the CMC is an idealisation. Real curves round off over a factor of two or so in concentration either side of it.
  • This is the surface tension against air. What keeps a particulate dispersed is the surfactant at the particle–liquid interface, a related but different adsorption; the air–water curve is the measurable proxy for how much surfactant is active, not the dispersing force itself.
  • Nothing here is Meyer's — the estate material does not name a surfactant or, so far as the archive holds, mention one. The calculation exists because a slurry of metal in water needs one to stay a slurry.