Question

A micellization model. You have developed a model for the formation of micelles, based on expressions for the chemical potential \(\mu(\mathrm{mono})\) of the free monomeric molecules in solution and the chemical potential \(\mu(\mathrm{mic})\) of the aggregated state of the monomers in micelles as a function of the mole fraction \(x\) of monomers in solution: $$ \mu(\text { mono })=0.5+0.1 x^{2} $$ and $$ \mu(\mathrm{mic})=1.3-25 x^{2} $$ (a) What is the state of the system at low concentration? (b) What is the state of the system at high concentration? (c) What is the concentration at which the micelle transition occurs?

Short answer

a) Free monomers, b) Micelles, c) \( x \approx 0.178 \)

Step-by-step solution

- Interpret the Given Functions

First identify the given functions for the chemical potentials. For the free monomeric molecules, \( \mu(\text { mono }) = 0.5 + 0.1 x^{2} \), and for the aggregated state in micelles, \( \mu( \mathrm{mic} )= 1.3 - 25 x^{2} \). The variable \( x \) represents the mole fraction of monomers in solution.

- Low Concentration

At low concentration, \( x \) is very small (approaching 0). Plugging \( x \) into \( \mu(\text { mono }) \) : \[ \mu(\text { mono }) = 0.5 + 0.1(0)^{2} = 0.5 \]. For \( \, \mu( \mathrm{mic} )\) : \[ \mu( \mathrm{mic} ) = 1.3 - 25(0)^{2} = 1.3 \]. Since \( 0.5 < 1.3 \), at low concentration, the free monomers state is favored.

- High Concentration

At high concentration, \( x \) is large (approaching 1). Plugging \( x \) into \( \mu(\text { mono }) \) and \( \mu( \mathrm{mic} )\) : \[ \mu(\text { mono }) = 0.5 + 0.1(1)^{2} = 0.6 \], and \[ \mu( \mathrm{mic} ) = 1.3 - 25(1)^{2} = -23.7 \]. Since \( -23.7 < 0.6 \), at high concentration, the micelles state is favored.

- Identify Micelle Transition

The micelle transition occurs when the chemical potentials are equal: \( \mu(\text { mono }) = \mu( \mathrm{mic} ) \). Set the equations equal to each other and solve for \( x \): \[ 0.5 + 0.1 x^{2} = 1.3 - 25 x^{2} \]. Rearranging terms gives: \[ 0.1 x^{2} + 25 x^{2} = 1.3 - 0.5 \]. Combining terms: \[ 25.1 x^{2} = 0.8 \]. Solving for \( x \): \[ x^{2} = \frac{0.8}{25.1} \implies x = \sqrt{\frac{0.8}{25.1}} \approx 0.178 \]

Key concepts

chemical potential

Chemical potential (\ref{mu}) is a thermodynamic property that indicates the potential for a substance to change phases or react. It essentially tells us how the free energy of a system changes as the number of particles changes. In the context of the micellization model, the chemical potential of monomer molecules (\ref{mu(\text {mono })}) is given as: \[ \ref{mu(\text {mono })} = 0.5 + 0.1 x^{2} \], while the chemical potential of the aggregated state in micelles (\ref{mu( \text {mic } )}) is: \[ \ref{mu( \text {mic } )} = 1.3 - 25 x^{2} \] where \( x \) is the mole fraction of monomers in the solution. These expressions help us understand under which conditions monomers prefer to stay free, and under which conditions they aggregate into micelles.

In simple terms, the chemical potential serves as a guide to predict the behavior of substances within a solution. When the chemical potential of monomers is lower than that of micelles, monomers remain unaggregated. Conversely, when the chemical potential of micelles is lower than that of the free monomeric state, aggregation occurs.

mole fraction

The mole fraction (\ref{x}) is a way of expressing the concentration of one component in a mixture. It is the ratio of the number of moles of a particular component to the total number of moles in the solution. The mole fraction ranges from 0 (pure solvent) to 1 (pure solute).

In our micellization model, \(x\) represents the mole fraction of monomers in the solution. This fraction has a significant impact on the chemical potential of the monomers and micelles. At low mole fractions, there are only a few monomers in the solution. However, as the mole fraction increases, the number of monomers also increases, leading to a higher probability of micelle formation.

The mole fraction concept is critical because it directly connects to the chemical potentials. At very small mole fractions, the chemical potential of free monomers dominates. In contrast, at very high mole fractions, the chemical potential of micelles becomes more relevant. Understanding mole fraction helps us predict when monomers would start to form micelles.

micelles formation

Micelles formation is the process where monomers in a solution aggregate to form a spherical structure known as a micelle. This occurs at a certain concentration known as the critical micelle concentration (CMC).

In our model, micelle formation depends on the chemical potential of the monomers and micelles. Initially, at low concentrations (low mole fractions), monomers exist freely in the solution because their chemical potential is lower than the potential in the micelles. However, as the concentration increases (higher mole fractions), the chemical potential of the micelles drops below that of the free monomers, making micelle formation energetically favorable.

The transition to micelle formation happens when the chemical potential of the monomers equals the micelles' chemical potential. Solving the equation where the chemical potentials are equal gives us the mole fraction at which this transition occurs. In our model, this transition mole fraction was found to be approximately 0.178.

Simply put, micelles form when keeping monomers separate becomes less energetically favorable than aggregating them into micelles. This balance point, dictated by the chemical potentials and affected by the mole fraction, determines when and how micelles will form in a solution.