Real Solutions - Ideal Dilute Solution Model

Deviation from Raoult's Law: In ideal solutions, the assumption is that all intermolecular interactions are identical; thus, the behavior of each component in the solution is purely dependent on its mole fraction. For such solutions, Raoult’s Law applies, which states that the partial vapor pressure of each component is proportional to its mole fraction and the vapor pressure of the pure component. However, real solutions often deviate from this ideal behavior because the interactions between molecules of different components are not always the same as the interactions between molecules of the same component.

In a real binary solution, as shown in the graph, deviations from Raoult's Law are observed due to these unequal interactions. For instance, if the A-B interactions are stronger than A-A or B-B, the components may have a tendency to stay together more, reducing the vapor pressure compared to the ideal prediction. Conversely, if A-B interactions are weaker, we observe an increase in vapor pressure as each component has a higher tendency to escape to the vapor phase.

Context and Implications: Deviations from Raoult's Law can be positive (resulting in higher vapor pressures) or negative (resulting in lower vapor pressures). Understanding these deviations is crucial in applications like distillation, where separating components in a mixture depends heavily on their individual vapor pressures.

 

Ideal vs Real Solutions

Ideal Solution Equations: In an ideal solution, where all interactions are identical, the thermodynamic properties of mixing can be represented by:

\[ \Delta_{\text{mix}} G = nRT \sum_{i} x_i \ln \chi_i \]
 
\[ \Delta_{\text{mix}} S = -nR \sum_{i} x_i \ln \chi_i \]
 
\[ \Delta_{\text{mix}} V = 0 \]

These equations imply that, for ideal solutions, mixing is an entropy-driven process with no enthalpic contribution, as

\[ \Delta_{\text{mix}} H = 0 \]

In other words, the change in Gibbs free energy upon mixing is entirely due to the increase in entropy.

 

Real Solution Equations: For real solutions, however, we find:

\[ \Delta_{\text{mix}} G < 0 \]

\[ \Delta_{\text{mix}} S > 0 \]

\[ \Delta_{\text{mix}} V \neq 0 \]
.

These deviations indicate that mixing in real solutions involves both entropic and enthalpic changes. In some cases, additional volume changes occur upon mixing due to intermolecular forces that lead to contraction or expansion of the solution.

Example: A common example of a real solution is the mixing of water and ethanol. The hydrogen-bonding interactions between water molecules and ethanol molecules are different from those in pure water or pure ethanol. As a result, the solution does not follow Raoult’s Law precisely and has volume changes upon mixing.

 

Partial Molar Volume

Partial molar volume represents the change in total volume of the solution when an infinitesimal amount of one component is added, while the amount of other components is kept constant. For an ideal solution, the total volume is given as:

\[ V_{\text{ideal}} = \chi_A V_{m,A}^* + (1 - \chi_A) V_{m,B}^* \]

In real solutions, the interactions between different molecules often lead to deviations from ideal behavior, resulting in non-additive volumes. For a real solution, we can express the partial molar volume as:

\[ V_{m,i,A}(p,T,n_A,n_B) = \left( \frac{\partial V}{\partial n_A} \right)_{p,T,n_B} \]

Using this definition, we can derive that the total volume is: 

\[ V = n_A V_{m,A} + n_B V_{m,B} \]

The relationship between the partial molar volumes of the components in a binary solution can be expressed as:

\[ \chi_A dV_{m,A} + \chi_B dV_{m,B} = 0 \]

Solving for one of the partial molar volume differentials, we get:

\[ dV_{m,A} = - \frac{\chi_B}{\chi_A} dV_{m,B} \]

This equation indicates that changes in the partial molar volume of one component are counteracted by opposite changes in the other component, maintaining the overall volume balance. This relationship is analogous to the Gibbs-Duhem equation used for chemical potentials in solutions, indicating the interdependence of the components' properties in a solution.

Examples:

  • In an acetone-chloroform mixture, hydrogen bonding can occur between the hydrogen in chloroform and the carbonyl group in acetone, affecting the volume upon mixing. This leads to a contraction or expansion of the total volume compared to the ideal solution prediction.
  • When a small amount of salt (solute) is dissolved in water (solvent), the water molecules rearrange around the ions, often reducing the overall volume compared to the simple additive case. This reflects the impact of solute-solvent interactions on partial molar volume.

 

Ideal Dilute Solution Model

In very dilute solutions, we can use the Ideal Dilute Solution Model, which applies when the solute is present in a very low concentration. In this model, interactions among solute molecules are negligible, and we can treat the solution as if the solute molecules are isolated within the solvent.

The chemical potential of a component $i$ in an ideal solution is defined as:

\[\mu_i = \mu_i^* + RT \ln \frac{p_i}{p_i^*}\]

at equilibrium::

\[\mu_i^* (\text{vap}) = \mu_i^* (liquid) = \mu_i^*\]

In real solutions, however, the Raoult's law is not valid, and thus:

\[p_i \neq \chi_i p_i^* \Rightarrow \mu_i \neq \mu_i^* + RT \ln \chi_i\]

 

Activity and Activity Coefficient:

  • Activity: Activity is a property that accounts for the effective concentration of a solute in a real solution. It is defined in terms of the solution and the pure solvent:

\[a_{\text{solvent}} = \frac{p_{\text{solvent}}}{p_{\text{solvent}}^*}\]

This relationship indicates that $a_{solvent}$ compares the partial pressure of the solvent in solution to the vapor pressure of the pure solvent.

  • Activity Coefficient (γ): The activity coefficient is used to express deviations from ideal behavior, and is defined as:

\[\gamma_{\text{solvent}} = \frac{a_{\text{solvent}}}{\chi_{\text{solvent}}}\]

This definition shows how far a real solution deviates from ideality, where γ = 1 represents an ideal solution.

Summary: Ideal behavior is observed as the mole fraction of solute, $\chi_{solute}$, approaches zero. This is when the solute acts ideally within the solvent environment. In real solutions, activity and activity coefficients describe the effective concentration of components in the solution, adjusting for non-ideal interactions. Raoult’s law is no longer applicable to all components, and these concepts help quantify deviations from ideality.

The chemical potential of a component $i$ in a real solution can, therefore, be expressed as:

\[\mu_i = \mu_i^* + RT \ln a_i\]

In this expression, $a_i$ is the activity, which adjusts the mole fraction to account for non-ideal interactions.

 

The ideal dilute solution model applies under two limits, depending on whether a component behaves as a solvent or as a solute:

  • Solvent behavior (Raoult’s Law): When a component is the solvent, it obeys Raoult’s law as $\chi_{solvent}$ approaches 1.

\[p_{\text{solvent}} = x_{\text{solvent}} \cdot p_{\text{solvent}}^* \quad \text{for} \; x_{\text{solvent}} \rightarrow 1\]

In this case, the solvent behaves ideally as its concentration dominates the solution. Thus, at the limit for the solvent ($\chi_i \to 1$), the activity coefficient $\gamma_i \approx 1 \implies a_i \to \chi_i$.

  • Solute behavior (Henry’s Law): When a component is the solute, it obeys Henry’s law as $\chi_{solute}$ approaches 0. Thus, at the limit for the solute ($\chi_i \to 0$), the activity coefficient $\gamma_i \approx 1 \implies a_i \to \chi_i$. Henry’s law states:

\[p_{\text{solute}} \neq x_{\text{solute}} \cdot p_{\text{solute}}^* \]

Instead, the partial pressure of the solute is proportional to its mole fraction via Henry’s constant, $K_H$:

\[p_{\text{solute}} = x_{\text{solute}} \cdot K_{\text{solute}}^H \quad \text{for} \; x_{\text{solute}} \rightarrow 0\]

This relationship shows that, in the dilute limit, the solute’s partial pressure depends on Henry’s law constant rather than its vapor pressure.

Note: For ideal solutions, Henry's constant $K_H$ is approximately equal to the vapor pressure of the pure solute, $p_i^*$.

Therefore, an ideal dilute solution is one in which the solvent obeys Raoult’s law and the solute obeys Henry’s law. This is illustrated in the graph where the at high mole fractions, the solvent behaves ideally (follows Raoult's law), while at low mole fractions, the solute follows Henry's law. The red line shows the real solution's behavior, diverging from Raoult’s law as the solution becomes more dilute.

 

Defining the chemical potentials for the solvent and the solute

The chemical potential of a component in a solution is expressed based on the choice of standard state:

  • For the solvent (Raoult’s Law Standard State): When the solvent behaves ideally, we describe its chemical potential using Raoult’s law:

\[ \mu_i = \mu_i^* + RT \ln \frac{p_i}{p_i^*} \]

Since $a_i = \frac{p_i}{p_i^*}$ for the solvent, this becomes:

\[ \mu_i = \mu_i^* + RT \ln a_i \]

  • For the solute (Henry’s Law Standard State): When the solute is very dilute, we use Henry’s law to define its chemical potential:

Substituting Henry's law into the chemical potential formular we get:

\[ p_i = \chi_i \cdot K_i^H \implies \mu_i = \mu_i^* + RT \ln \frac{p_i}{p_i^*} \implies \]

\[ \mu_i = \mu_i^* + RT \ln \frac{K_i^H \cdot \chi_i}{p_i^*} \]

This leads to:

\[ \mu_i = \mu_i^* + RT \ln \frac{K_i^H}{p_i^*} + RT \ln \chi_i = \mu_i^{*H} + RT \ln \chi_i \]

where $μ_i^{*H}$ denotes the standard chemical potential based on Henry's law. Finally, we generalize the chemical potential in terms of activity as:

\[ \mu_i = \mu_i^{*H} + RT \ln a_i \]

Explanation: This approach allows us to define the chemical potential of a solute in terms of activity, making it applicable to non-ideal solutions. The chemical potential is thus adjusted for real behavior, allowing for calculations even when Raoult's law does not hold strictly.

 

Henry’s Law and Gas Solubility

Henry’s Law is used to describe the solubility of gases in liquids. For instance, the solubility of oxygen in water is proportional to the partial pressure of oxygen above the water. The concentration of dissolved oxygen can be expressed as:

\[ \chi_{\text{O}_2} = \frac{n_{\text{O}_2}}{n_{\text{O}_2} + n_{\text{H}_2O}} \]

As the concentration of $O_2$ approaches zero, Henry's Law provides a linear relationship between the concentration in solution and the partial pressure above it:

\[ \mu_{\text{O}_2} = \mu_{\text{O}_2}^* + RT \ln a_{\text{O}_2} \]
.

\[ \chi_A = \frac{n_{O_2}}{n_{O_2} + n_{H_2O}} \approx \frac{n_A}{n_{H_2O}} \Rightarrow n_A{O_2}= n_{H_2O} \cdot \chi_A = n_{H_2O} \cdot \frac{p_A}{K_A^H} \]

Application: Henry's Law is essential in fields like environmental science, where the solubility of gases like CO₂ and O₂ in water bodies affects aquatic life. It also applies in beverage carbonation, where CO₂ is dissolved in water at high pressure to make soda.

 

Chemical equilibrium in solutions:

We consider the chemical equilibrium between solutes in solution. For such cases, we often use Henry's law as the standard state for solutes. In this case, following the derivations above, the equilibrium condition is given in terms of the chemical potentials and a concentration-dependent:

\[ \sum_i \nu_i \mu_i (\text{solution}) = \sum_i \nu_i \mu_i^{*H} (\text{solution}) + RT \sum_i \nu_i \ln (a_i^{eq})^{\nu_i} = 0 \]

Here, $\nu_i$ represents the stoichiometric coefficient of species $i$, and $\mu_i^{*H}$ denotes the Henry's standard chemical potential. $a_i$ is the activity of species $i$.

We can relate this to the Gibbs free energy change, $\Delta_{rxn} G$ and to the equilibrium constant:

\[ \Delta_{rxn} G^{o} = -RT \sum_i \ln (a_i^{eq})^{\nu_i} = -RT \ln K\]

Since the activity, $a_i$, for each species can be written as the product of the activity coefficient, $\gamma_i$, and the concentration ratio relative to a standard concentration ($\frac{c_i}{c^{\circ}}$), we expand this expression:

\[ K = \prod_i (a_i^{eq})^{\nu_i} = \prod_i (\gamma_i^{eq})^{\nu_i} \cdot \left( \frac{c_i}{c^{\circ}} \right)^{\nu_i} \]

This expression allows the equilibrium constant K to account for non-ideal behavior via the activity coefficients. In ideal solutions, where $\gamma_i = 1 $, the expression for $K$ simplifies to depend solely on the concentrations of the species involved.

This generalized form of the equilibrium constant can be applied to gases, liquids, solids, and dissolved species. For gases, we use fugacity, which is the effective pressure accounting for non-ideality. The equilibrium constant’s value depends on the chosen standard state.