Question

Does surface tension change with concentration?

Answer 1

Yes
`((dgamma)/(dc))_T= -(RTGamma_S)/c`
`gamma`=surface tension

Derivation
dG=`-SdT+VdP+gammadsigma+mudn_s`
At constant temperature and pressure,
dG=`gammadsigma+mudn_s` .............. (1)
`sigma`=surface area
n=no. of moles of the surfactant
`mu`=chemical potential= `((dG)/(dn))`

so,G=`gammasigma+mun_s`
`rArr dG=gammadsigma+sigmadgamma+mudn_s+n_sdmu` from (Gibbs-Duhem equation)...........(2)

by comparing the two equations (1) & (2) for dG, we get
`sigmadgamma+ndmu`=0 (Gibbs isotherm)

Now for an interface in which oil and water for example, are separated by a geometrically flat surface.The approximation implies that on the surfactant S accumulates at the surface and hence that `Gamma_(oil)` and `Gamma_(water)` are both zero.
The equation becomes
`dgamma= -Gamma_sdmu_s` where (`Gamma=n/sigma`)

for dilute solutions,
`dmu_s=RTlnc` WHERE c IS THE MOLAR CONCENTRATION OF THE SURFACTANT.

therefore at constant temperature, `((dgamma)/(dc))_T=-RTGamma_S/c`

I've tried to make the derivation look as simple as possible.If still not understood , leave a comment below.