Solutions can be ideal or non-ideal, affecting their behavior. Ideal solutions follow Raoult's and Henry's laws, assuming no interactions between components. Non-ideal solutions deviate from these laws due to molecular interactions, requiring activity coefficients and excess properties to describe their behavior.

Models like regular solutions, Van Laar, and Margules equations help predict non-ideal solution behavior. These models use parameters to account for deviations from ideality, allowing for more accurate predictions of thermodynamic properties in real-world applications.

Ideal Solutions

Raoult's Law

• Describes the vapor pressure of an ideal solution
• States that the partial vapor pressure of each component in an ideal solution is equal to the vapor pressure of the pure component multiplied by its mole fraction in the solution
• Mathematically expressed as $p_i = x_i p_i^$, where $p_i$ is the partial vapor pressure of component $i$, $x_i$ is the mole fraction of component $i$, and $p_i^$ is the vapor pressure of pure component $i$
• Assumes no intermolecular interactions between different components in the solution
• Applicable to solutions where the components have similar molecular sizes and intermolecular forces (ethanol and water)

Henry's Law

• Describes the solubility of a gas in a liquid at a given temperature
• States that the amount of dissolved gas is directly proportional to the partial pressure of the gas above the liquid
• Mathematically expressed as $c = kP$, where $c$ is the concentration of the dissolved gas, $P$ is the partial pressure of the gas, and $k$ is Henry's law constant, which depends on the solute, solvent, and temperature
• Assumes that the gas molecules do not interact with each other in the solution
• Applicable to dilute solutions where the solute is a gas and the solvent is a liquid (carbon dioxide in water)

Non-Ideal Solutions

Activity Coefficient

• A factor that accounts for the deviation of a solution from ideal behavior
• Defined as the ratio of the actual fugacity (or activity) of a component to its ideal fugacity (or activity)
• Mathematically expressed as $\gamma_i = \frac{f_i}{x_i f_i^}$, where $\gamma_i$ is the activity coefficient of component $i$, $f_i$ is the fugacity of component $i$ in the solution, $x_i$ is the mole fraction of component $i$, and $f_i^$ is the fugacity of pure component $i$
• A value of 1 indicates ideal behavior, while values greater than 1 indicate positive deviations and values less than 1 indicate negative deviations from ideality
• Depends on the composition of the solution and the intermolecular interactions between components (ethanol and water at high concentrations)

Excess Properties

• Thermodynamic properties that describe the deviation of a solution from ideal behavior
• Defined as the difference between the actual value of a property and the value it would have in an ideal solution
• Examples include excess Gibbs free energy ($G^E$), excess enthalpy ($H^E$), and excess entropy ($S^E$)
• Mathematically expressed as $M^E = M - M^{id}$, where $M^E$ is the excess property, $M$ is the actual value of the property, and $M^{id}$ is the value of the property in an ideal solution
• Provide insights into the nature and strength of intermolecular interactions in the solution (mixing of ethanol and water results in a negative excess enthalpy)

Deviation from Ideality

• Occurs when the properties of a solution deviate from those predicted by ideal solution laws (Raoult's law and Henry's law)
• Can be caused by differences in molecular size, shape, or intermolecular forces between components
• Positive deviations occur when the intermolecular attractions between unlike molecules are weaker than those between like molecules, leading to higher vapor pressures and lower boiling points than predicted (acetone and chloroform)
• Negative deviations occur when the intermolecular attractions between unlike molecules are stronger than those between like molecules, leading to lower vapor pressures and higher boiling points than predicted (chloroform and ethanol)
• The extent of deviation depends on the nature and concentration of the components in the solution

Models for Non-Ideal Solutions

Regular Solutions

• A model that accounts for the non-ideality of solutions caused by differences in intermolecular forces between components
• Assumes that the excess entropy of mixing is zero ($S^E = 0$) and that the excess enthalpy of mixing is proportional to the product of the mole fractions of the components
• Mathematically expressed as $G^E = x_1 x_2 \Lambda$, where $G^E$ is the excess Gibbs free energy, $x_1$ and $x_2$ are the mole fractions of components 1 and 2, and $\Lambda$ is a parameter that depends on the intermolecular interactions between the components
• Provides a simple way to estimate the activity coefficients and vapor-liquid equilibrium of non-ideal solutions (hexane and benzene)
• Limited to solutions where the components have similar molecular sizes and shapes

Van Laar Equation

• An empirical model that describes the excess Gibbs free energy of non-ideal solutions
• Assumes that the excess Gibbs free energy is a function of the mole fractions and two adjustable parameters, $A$ and $B$
• Mathematically expressed as $\frac{G^E}{RT} = \frac{A x_1 x_2}{x_1 + B x_2} + \frac{B x_1 x_2}{B x_1 + x_2}$, where $G^E$ is the excess Gibbs free energy, $R$ is the gas constant, $T$ is the temperature, $x_1$ and $x_2$ are the mole fractions of components 1 and 2, and $A$ and $B$ are adjustable parameters that depend on the components and temperature
• Provides a better fit to experimental data than the regular solution model for solutions with moderate deviations from ideality (ethanol and water)
• Requires the determination of the adjustable parameters from experimental data

Margules Equation

• An empirical model that describes the excess Gibbs free energy of non-ideal solutions
• Assumes that the excess Gibbs free energy is a polynomial function of the mole fractions and adjustable parameters
• The two-suffix Margules equation is mathematically expressed as $\frac{G^E}{RT} = x_1 x_2 (A_{12} + B_{12} (x_1 - x_2))$, where $G^E$ is the excess Gibbs free energy, $R$ is the gas constant, $T$ is the temperature, $x_1$ and $x_2$ are the mole fractions of components 1 and 2, and $A_{12}$ and $B_{12}$ are adjustable parameters that depend on the components and temperature
• The three-suffix Margules equation includes an additional term with a third adjustable parameter to better describe solutions with large deviations from ideality (acetone and chloroform)
• Provides a flexible and accurate way to model the thermodynamic properties of non-ideal solutions
• Requires the determination of the adjustable parameters from experimental data