How does \(\Delta H_{\text {soln }}\) relate to deviations from Raoult's law? Explain.

Raoult's law states that the partial vapor pressure of a component in a solution is proportional to its mole fraction and its pure component vapor pressure at a given temperature. Mathematically, it can be represented as: \(P_i = x_iP^*_i\) 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 pure component vapor pressure of component i. Raoult's law holds for ideal solutions.

Deviations from Raoult's law occur when the solution shows non-ideal behavior, meaning the intermolecular interactions between the solvent and solute molecules are different from those in the pure components. In such cases, the measured vapor pressures of the components in the solution do not follow the linear relationship predicted by Raoult's law, leading to either positive or negative deviations. Positive deviations occur when the intermolecular interactions between the solute and solvent are weaker than the interactions within the pure components. This results in a higher-than-expected vapor pressure of the solution and hence an increased vapor pressure of each component relative to the value predicted by Raoult's law. Negative deviations occur when the intermolecular interactions between the solute and solvent are stronger than the interactions within the pure components. This results in a lower-than-expected vapor pressure of the solution and hence a decreased vapor pressure of each component relative to the value predicted by Raoult's law.

The enthalpy of solution (\(\Delta H_{\text {soln}}\)) is a measure of the energy change associated with the formation of a solution, which includes breaking the intermolecular forces within the solute and solvent and forming new intermolecular interactions between them. Deviations from Raoult's law depend on the overall strength of these newly-formed intermolecular attractions compared to those in the pure components. If the enthalpy of solution is endothermic (\(\Delta H_{\text {soln}} > 0\)), this means that the overall energy cost of breaking the intermolecular forces within the pure components is greater than the energy gained by forming new interactions between the solute and solvent. This results in positive deviations from Raoult's law, as the weaker solute-solvent interactions contribute to a higher vapor pressure in the solution. If the enthalpy of solution is exothermic (\(\Delta H_{\text {soln}} < 0\)), this means that the overall energy gained by forming new solute-solvent interactions is greater than the energy cost of breaking the intermolecular forces within the pure components. This results in negative deviations from Raoult's law, as the stronger solute-solvent interactions contribute to a lower vapor pressure in the solution. In summary, a positive enthalpy of solution correlates with positive deviations from Raoult's law, while a negative enthalpy of solution correlates with negative deviations. This connection helps identify non-ideal behavior in solutions based on the enthalpy change that occurs upon formation of the solution.