Surface tension in terms of temperature and concentration of an added substance

[Prestohdus]

Hi! Here's a tricky thermodynamics problem, I hope you can help with it.

1. Homework Statement

The boundary between two different materials can be divided into two different kind of phases: bulk phases and surface phases. For example, let's consider a boundary between water and air. We can divide the boundary into two phases: a liquid phase (v) and a surface phase (σ). You can then write Euler's equation for each phase:

U^v(S, V, n_i) = TS – pV + ∑μ_in_i^v
and
U^σ(S, A, n_i) = TS + γA + ∑μ_in_i^σ

where γ is the surface tension, A is the area of the boundary surface, μ_i is the chemical potential of component i, and n_i^v and n_i^σ are the molar amounts of component i in the liquid phase and surface phase, respectively.

a) Write the Gibbs–Duhem equations for both the liquid and the surface phase.
b) Write the surface tension as a function of the temperature and concentration of an added substance A.
c) Analyze the previous result: what happens to the surface tension when a surfactant is added to the system in constant temperature?

Additionally, we can assume that the chemical potential of water stays almost constant when substance A is added. Also, the chemical potential of substance A is

μ_A = μ_A^° + RTln(x_A),

where μ_A^° is a constant and x_A is the mole fraction of substance A such that x_A = n_A / (n_A + n), where n_A is the molar amount of substance A added and n is the rest of the matter.

Homework Equations

The Attempt at a Solution

a) This part I think I understand, and confirmed from Wikipedia. For the liquid phase:

∑n_i^vμ_i = -SdT + Vdp

and for the surface phase:

∑n_i^σμ_i = -SdT – Adγ

b) Here I am stuck. How can I find the surface tension as a function of the concentration and temperature of the added substance? I assume the function is something like γ = γ₀ + [?], where γ₀ is the original surface tension before adding anything. Other than that, I don't know. Where do I even get concentration from?

Thanks very much for help!

 

[Chestermiller]

Hi! Here's a tricky thermodynamics problem, I hope you can help with it.

1. Homework Statement

The boundary between two different materials can be divided into two different kind of phases: bulk phases and surface phases. For example, let's consider a boundary between water and air. We can divide the boundary into two phases: a liquid phase (v) and a surface phase (σ). You can then write Euler's equation for each phase:

U^v(S, V, n_i) = TS – pV + ∑μ_in_i^v
and
U^σ(S, A, n_i) = TS + γA + ∑μ_in_i^σ

where γ is the surface tension, A is the area of the boundary surface, μ_i is the chemical potential of component i, and n_i^v and n_i^σ are the molar amounts of component i in the liquid phase and surface phase, respectively.

a) Write the Gibbs–Duhem equations for both the liquid and the surface phase.
b) Write the surface tension as a function of the temperature and concentration of an added substance A.
c) Analyze the previous result: what happens to the surface tension when a surfactant is added to the system in constant temperature?

Additionally, we can assume that the chemical potential of water stays almost constant when substance A is added. Also, the chemical potential of substance A is

μ_A = μ_A^° + RTln(x_A),

where μ_A^° is a constant and x_A is the mole fraction of substance A such that x_A = n_A / (n_A + n), where n_A is the molar amount of substance A added and n is the rest of the matter.

Homework Equations

The Attempt at a Solution

a) This part I think I understand, and confirmed from Wikipedia. For the liquid phase:

∑n_i^vμ_i = -SdT + Vdp

and for the surface phase:

∑n_i^σμ_i = -SdT – Adγ

These Gibbs Duhem equations should have $d\mu \ 's$, not $\mu$'s

 

[Prestohdus]

Thank you! I had those but wrote incorrectly here.

 

[Prestohdus]

Solved it. :)

http://rkt.chem.ox.ac.uk/lectures/liqsolns/liquid_surfaces.html this helped a lot