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Prestohdus
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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:
Uv(S, V, ni) = TS – pV + ∑μiniv
and
Uσ(S, A, ni) = TS + γA + ∑μiniσ
where γ is the surface tension, A is the area of the boundary surface, μi is the chemical potential of component i, and niv and niσ 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
μA = μA° + RTln(xA),
where μA° is a constant and xA is the mole fraction of substance A such that xA = nA / (nA + n), where nA is the molar amount of substance A added and n is the rest of the matter.
a) This part I think I understand, and confirmed from Wikipedia. For the liquid phase:
∑nivμi = -SdT + Vdp
and for the surface phase:
∑niσμi = -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!
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:
Uv(S, V, ni) = TS – pV + ∑μiniv
and
Uσ(S, A, ni) = TS + γA + ∑μiniσ
where γ is the surface tension, A is the area of the boundary surface, μi is the chemical potential of component i, and niv and niσ 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
μA = μA° + RTln(xA),
where μA° is a constant and xA is the mole fraction of substance A such that xA = nA / (nA + n), where nA 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:
∑nivμi = -SdT + Vdp
and for the surface phase:
∑niσμi = -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!