Raoult's Law and Vapour Pressure of Liquid-Liquid Solutions

Vapour Pressure of Solutions of Liquids in Liquids

• The study of binary solutions composed of two volatile liquids, denoted as $A_1$ and $A_2$, considers their behavior when placed in a closed container.
• Process of Vaporization: When such a solution is introduced into a closed environment, both liquid components undergo vaporization.
• Establishment of Equilibrium: Over time, a state of equilibrium is established between the vapour phase and the liquid phase of the system.
• Composition of the Vapour Phase: At equilibrium, both components ($A_1$ and $A_2$) are present in the vapour phase.
• Relationship of Pressures: The partial pressure of each component in the vapour phase is directly related to its respective mole fraction within the liquid solution.
• Raoult's Law: The specific mathematical relationship governing these conditions is defined by Raoult's law.

Raoult's Law

• Verbatim Definition: Raoult's Law states that "the partial vapour pressure of any volatile component of a solution is equal to the vapour pressure of the pure component multiplied by its mole fraction in the solution."
• Application to a Binary Solution: Consider a binary solution consisting of two volatile liquids, $A_1$ and $A_2$.
  • Let $P_1$ and $P_2$ represent the partial vapour pressures of the components.
  • Let $x_1$ and $x_2$ represent their respective mole fractions in the solution.
  • Let $P_1^0$ and $P_2^0$ represent the vapour pressures of the pure liquids $A_1$ and $A_2$, respectively.
• Mathematical Expressions (Equation 2.2):
  • For component 1: $P_1 = x_1 P_1^0$
  • For component 2: $P_2 = x_2 P_2^0$

Total Vapour Pressure and Dalton's Law

• Dalton's Law of Partial Pressures: According to this law, the total pressure ($P$) exerted by the vapour above the solution is the sum of the partial pressures of the individual volatile components.
• Calculation of Total Pressure (Equation 2.3):
  • $P = P_1 + P_2$
  • Substituting Raoult's Law expressions: $P = P_1^0 x_1 + P_2^0 x_2$
• Derivation in Terms of a Single Mole Fraction:
  • Given the relationship between mole fractions in a binary mixture: $x_1 + x_2 = 1$, it follows that $x_1 = 1 - x_2$.
  • Substituting this into the total pressure equation: $P = P_1^0 (1 - x_2) + P_2^0 x_2$
  • Expanding the terms: $P = P_1^0 - P_1^0 x_2 + P_2^0 x_2$
  • Factoring out $x_2$ (Equation 2.4): $P = (P_2^0 - P_1^0) x_2 + P_1^0$

Graphical Representation of Vapour Pressure

• Total Pressure Plot: Given that the pure component vapour pressures $P_1^0$ and $P_2^0$ are constants at a specific temperature, the plot of the total pressure ($P$) versus the mole fraction of the second component ($x_2$) results in a straight line.
• Component Pressure Plots: According to Equation 2.2, the plots of individual partial pressures against their respective mole fractions are also linear:
  • The plot of $P_1$ versus $x_1$ is a straight line.
  • The plot of $P_2$ versus $x_2$ is a straight line.
• Origin Correspondence: The individual plots for $P_1$ and $P_2$ are straight lines that pass through the origin (where the mole fraction of the component is zero).
• Reference to Figure 2.2: These relationship descriptions and the linearity of the partial and total pressures are illustrated in Figure 2.2 of the source material.