The chemical potential of a rod-like micelle which consists of \(m\) surfactant molecules is written as \(\mu_{m}=\mu_{m}^{0}+k_{B} T \ln n_{m}, \quad\) with \(\quad \mu_{m}^{0}=a+m b\) where \(a\) and \(b\) are constants independent of \(m\). Answer the following questions. (a) Show that at equilibrium the number density of micelles of size \(m\) can be written as $$ n_{m}=n_{c}\left(\frac{n_{1}}{n_{c}}\right)^{m} $$ Obtain an expression for \(n_{c^{-}}\) (b) \(n_{m}\) satisfies the equation $$ \sum_{m=1}^{\infty} m n_{m}=n $$ where \(n\) is the number density of surfactants in the solution. Express \(n_{1}\) as a function \(n\). (c) Express the surface tension as a function of surfactant density \(n\), and discuss the meaning of \(n_{c}\).

Chemical potential is a measure of the chemical energy of a species in a system, represented by the symbol \( \mu \) . For a rod-like micelle, its chemical potential \( \mu_{m} \) can be described as a combination of a standard chemical potential \( \mu_{m}^{0} \) and a term dependent on temperature \( T \) and the natural logarithm of the number density of micelles of size \( m \, n_{m} \). The standard chemical potential \( \mu_{m}^{0} \) can further be expressed as \( a + mb \), where \( a \) and \( b \) are constants. Understanding chemical potential is crucial as it dictates the conditions at equilibrium for micelle formation through the equation:

• \( \mu_{m} = \mu_{m}^{0} + k_{B}T \, \ln n_{m} \)

Here, \( k_{B} \) is the Boltzmann constant and \( T \) is the temperature. At equilibrium, the chemical potential of any micelle size is equal, leading to expressions that help determine the relationships between different micelle sizes and their densities.

The number density of micelles \( n_{m} \) refers to the concentration of micelles of a certain size \( m \) in a solution. At equilibrium, this can be expressed by simplifying the chemical potential equations. The general expression for the number density of micelles of size \( m \) is given by:

• \( n_{m} = n_{c} \left ( \frac{n_{1}}{n_{c}} \right )^{m} \)

Here, \( n_{c} \) is a constant representing the critical number density and \( n_{1} \) is the number density of individual surfactant molecules. This equation highlights how the density of larger micelles depends exponentially on their size \( m \). The number density also plays a crucial role in balancing the total surfactant concentration in the solution, described by the equation:

• \( \sum_{m=1}^{\infty} m n_{m} = n \)

where \( n \) is the overall number density of surfactants. Solving this series helps us understand the distribution of micelles of different sizes in equilibrium.

Surface tension \( \gamma \) is the energy required to increase the surface area of a liquid by a unit area. For solutions containing surfactants, surface tension changes as the concentration of surfactants \( n \) varies. Surfactants reduce surface tension up to a point known as the critical micelle concentration (CMC). The surface tension can be generally represented as a function of surfactant density:

• \( \gamma = f(n) \)

The surface tension profoundly decreases until the surfactant concentration reaches the CMC. Beyond this concentration, additional surfactants form micelles instead of further lowering the surface tension. This relationship emphasizes the role of surfactant concentration in phenomena like wetting, emulsification, and detergency.

The critical micelle concentration (CMC), symbolized as \( n_{c} \), is a key parameter in colloidal chemistry. It is the concentration of surfactants in a solution above which micelles start forming. Below the CMC, surfactants exist primarily as individual molecules, while above the CMC, any additional surfactant molecules aggregate to form micelles. The CMC is characterized by:

• \( n_{c} = e^{\frac{-b}{k_{B}T}} \)

where \( b \) is a constant related to micelle formation energy, \( k_{B} \) is the Boltzmann constant, and \( T \) is temperature. The significance of the CMC lies in its implications for surface tension and solubilization properties. It marks a distinct threshold affecting how surfactants modify interfacial properties, critical in processes like cleaning, drug delivery, and enhanced oil recovery.