• WannabeNewton
In summary, the conversation discusses the behavior of surfactant molecules in dilute solutions, specifically their migration to surfaces and adsorption by other porous media. The conversation then presents a problem and solution regarding the chemical potential with particle density in a canonical ensemble, and raises a question about the concentration of floating surfactants at the surface as a function of solution concentration. After calculations and discussion, it is determined that the correct answer is n2 = n3λeβε0, with the correct limit being ε0 → -∞.
WannabeNewton

## Homework Statement

A dilute solution of surfactants can be regarded as an ideal three-dimensional gas. As surfactant molecules can reduce their energy by contact with air, a fraction of them migrate to the surface where they can be treated as a two-dimensional ideal gas. Surfactants are similarly adsorbed by other porous media such as polymers and gels with an affinity for them.

(a) Consider an ideal gas of classical particles of mass ##m## in ##d## dimensions, moving in a uniform attractive potential ##\epsilon_d## in equilibrium with a thermal bath of temperature ##T##. Show that the chemical potential with particle density ##n_d## is given by ##\mu_d = -\epsilon_d + k_B T \ln (n_d \lambda^d)## where ##\lambda = \frac{h}{\sqrt{2\pi m k_B T}}## is the thermal wavelength.

(b) If a surfactant lowers its energy by ##\epsilon_0## in moving from the solution it is into the surface of the solution, calculate the concentration of floating surfactants at the surface as a function of the solution concentration ##n = n_3## at temperature ##T##.

## The Attempt at a Solution

I calculated (a) using the canonical ensemble so that ##Z = \frac{V^N}{h^{3N}N!}e^{\beta \epsilon_d}\int \Pi _{i = 1}^N d^d p_i e^{-\beta \sum_j p_j^2/2m} = \frac{2\pi V^N}{h^{3N}N!}\frac{(2mE)^{(Nd -1)/2}}{(Nd/2 - 1)!}e^{\beta \epsilon_d} = \frac{V^N}{\lambda^{Nd}N!}\epsilon^{\beta \epsilon_d}## where ##V## is the volume in ##d## dimensions that a single particle moves in. Then ##F = -k_B T \ln Z = Nk_B T \ln n_d \lambda^d - N\epsilon_d - Nk_B T ## where ##n_d = \frac{N}{V}##. Hence ##\mu_d = \frac{\partial F}{\partial N}|_{T,V} = -\epsilon_d + k_B T \ln n_d \lambda^d##.

My problem is with (b). Let ##\epsilon_0 \equiv \epsilon_3 - \epsilon_2##. Now in the canonical ensemble, the system doesn't exchange particles with the thermal bath but subsystems of the system can exchange particles with one another. In our case the surfactants in the solution will flow to the surface, driven by the difference in potential energy ##\epsilon_0## between the solution and surface. They should keep flowing until ##dF_{\text{total}} = dF_2 - dF_3 = 0## since the total Helmholtz free energy of the subsystems combined has to be minimized. But this implies ##\frac{\partial F_2}{\partial N}|_{T,V} = \frac{\partial F_3}{\partial N}|_{T,V} \Rightarrow \mu_2 = \mu_3## hence ##n_2 = n_3 \lambda e^{-\beta\epsilon_0} ## which is certainly the wrong answer since this implies ##n_2 \ll n_3## for ##\epsilon_0 \rightarrow \infty## whereas it should be the other way around; in fact the correct answer should be ##n_2 = n_3 \lambda e^{\beta\epsilon_0} ##.

So where am I going wrong? Thanks in advance.

[strike]How did you go from ##F=Nk_BT\ln n_d\lambda^d-N\epsilon_d-Nk_BT## to ##\mu_d=k_BT\ln n_d\lambda^d-\epsilon_d##?

What happened to the last ##-Nk_B T## term?[/strike]

EDIT: REDACTED

Last edited:
Are ##\mu_d = -\epsilon_d + k_B T \ln (n_d \lambda^d)##, ##\epsilon_0 \equiv \epsilon_3 - \epsilon_2## and ##n_2 = n_3 \lambda e^{\beta\epsilon_0} ## given as the right answers?

Yes blobly. This is problem 4.6 in Kardar if you're interested.

Oops! I forgot that ##\epsilon_0 < 0## as per my sign convention so my answer agrees with the book's answer and the correct limit is actually ##\epsilon_0 \rightarrow -\infty## which gives ##n_2 \gg n_3## as desired.

And thank you to matterwave for setting my head straight on this :)

This is one deeply confusing sign error, got to admit.

## 1. What is surfactant adsorption chemical equilibrium?

Surfactant adsorption chemical equilibrium refers to the balance between the amount of surfactant molecules that are adsorbed onto a solid surface and the amount of free surfactant molecules in a solution. This equilibrium is governed by various factors such as the surface properties of the solid, the concentration of surfactant, and the temperature.

## 2. Why is surfactant adsorption chemical equilibrium important?

Surfactant adsorption chemical equilibrium is important in various industrial and biological processes. In industrial processes, surfactant adsorption can affect the efficiency of cleaning products, emulsification processes, and the stability of foams. In biological processes, surfactant adsorption plays a role in lung function and can impact the effectiveness of certain medications.

## 3. How is surfactant adsorption chemical equilibrium measured?

Surfactant adsorption chemical equilibrium can be measured using techniques such as surface tension measurements, X-ray photoelectron spectroscopy, and atomic force microscopy. These methods allow researchers to observe changes in the surface properties of a solid as surfactant molecules adsorb onto it.

## 4. What factors influence surfactant adsorption chemical equilibrium?

Several factors can influence surfactant adsorption chemical equilibrium, including the type of surfactant, the surface properties of the solid, the concentration and temperature of the solution, and the presence of other molecules or ions that can compete for adsorption sites on the surface.

## 5. How can surfactant adsorption chemical equilibrium be manipulated?

Surfactant adsorption chemical equilibrium can be manipulated by altering the concentration of surfactant in a solution, changing the temperature, or modifying the surface properties of the solid. Additionally, using mixtures of different types of surfactants or adding other molecules or ions to the solution can also affect the equilibrium and lead to desired outcomes in industrial or biological processes.

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