Vapour Pressure of Solutions : Vapour Pressure of Liquid-Liquid Solutions

 

Liquid Solutions

Introduction

• A solution is a homogeneous mixture of two or more substances.
• When the solvent is a liquid, the solute can be a gas, liquid, or solid.
• This section focuses on solutions where the solute is a liquid or a solid.

Types of Liquid Solutions

Binary solutions (solutions containing only two components) are classified as:

1. Liquid-Liquid Solutions

• Both solute and solvent are liquids.
• Example: Ethanol in water, benzene in toluene.

2. Solid-Liquid Solutions

• The solute is a solid, and the solvent is a liquid.
• Example: Salt in water, sugar in water.

Nature of Components

• Volatility of Components:
  • The solvent is generally volatile (evaporates easily).
  • The solute may be volatile or non-volatile.
• These solutions may contain one or more volatile components.

Equilibrium in Binary Solutions

• When a solution is kept in a closed vessel, the solvent and solute (if volatile) evaporate.
• Eventually, an equilibrium is established between the liquid phase and the vapour phase.
• The total vapour pressure (p$_{total}$) of the solution depends on the partial pressures of its components.

Raoult’s Law

• Given by Francois Marte Raoult (1886).
• Statement: The partial vapour pressure of each component in a solution of volatile liquids is directly proportional to its mole fraction.

Mathematical Expression:

$$p_1 = x_1 p_1^0$$ $$p_2 = x_2 p_2^0$$

where,

• $p_1, p_2$ = partial vapour pressures of components 1 and 2, respectively.
• $x_1, x_2$ = mole fractions of components 1 and 2, respectively.
• $p_1^0, p_2^0$ = vapour pressures of pure components 1 and 2, respectively.

Total Vapour Pressure of the Solution:

$$p_{\text{total}} = p_1 + p_2 = x_1 p_1^0 + x_2 p_2^0$$

• This equation is valid for ideal solutions where intermolecular forces between solute and solvent are similar to those in the pure components

DETAILS :-👇

Raoult’s Law and Vapour Pressure of Solutions

1. Raoult’s Law

• Definition: The partial vapour pressure of each component in a solution is directly proportional to its mole fraction in the solution.
• Mathematical Representation:

For Component 1:

$$p_1 \propto x_1$$ $$p_1 = p_1^0 x_1$$

where:

• $p_1$ = partial vapour pressure of component 1
• $p_1^0$ = vapour pressure of pure component 1 at the same temperature
• $x_1$ = mole fraction of component 1

For Component 2:

$$p_2 = p_2^0 x_2$$

where:

• $p_2$ = partial vapour pressure of component 2
• $p_2^0$ = vapour pressure of pure component 2
• $x_2$ = mole fraction of component 2

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2. Total Vapour Pressure of the Solution

• According to Dalton’s Law of Partial Pressures, the total vapour pressure over a solution is the sum of the partial vapour pressures of all components.

$$p_{\text{total}} = p_1 + p_2$$

Substituting Raoult’s Law Equations:

$$p_{\text{total}} = x_1 p_1^0 + x_2 p_2^0$$

Since $x_1+x_2=\mathrm{1<span\; class="katex"><br></span>,\; we\; can\; express\; the\; equation\; as:}$

$$p_{\text{total}} = (1 - x_2) p_1^0 + x_2 p_2^0$$ $$p_{\text{total}} = p_1^0 + (p_2^0 - p_1^0) x_2$$

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3. Key Conclusions from Raoult’s Law

1. Total vapour pressure depends on the mole fraction of any one component.
2. The total vapour pressure varies linearly with the mole fraction of component 2.
3. Effect of increasing mole fraction of a component:
• If $p_2^0 > p_1^0$ then increasing $x_2$ increases $p_{\text{total}}$.
• If $p_2^0 < p_1^0$ then increasing $x_2$ decreases $p_{\text{total}}$.

Graphical Representation of Raoult’s Law

1. Vapour Pressure vs. Mole Fraction

• A plot of partial vapour pressure ($p_1$ or $p_2$) against mole fractions ($x_1$ and $x_2$) results in a linear graph.
• Lines I and II represent the variation of partial vapour pressures ($p_1$ and $p_2$) with mole fractions.
• Each line passes through the point where the mole fraction of that component is unity           ($x_1=\mathrm{1<span\; class="katex"\; style="font-weight:\; bold;"> </span>}$$x_2 = 1$).
• This confirms the direct proportionality of vapour pressure to mole fraction, as stated in Raoult’s Law.

2. Total Vapour Pressure vs. Mole Fraction

• A plot of total vapour pressure ($p_{\text{total}}$) versus mole fraction $x_2$ also gives a linear graph (Line III).
• The minimum value of total vapour pressure is $p_1^0$ (vapour pressure of pure component 1).
• The maximum value of total vapour pressure is $p_2^0$ (vapour pressure of pure component 2).
• If component 1 is less volatile than component 2, then: $$p_1^0 < p_2^0$$

3. Key Observations

• For an ideal solution, the graphs are straight lines.
• The slope of the total vapour pressure graph depends on the relative volatilities of the two components.
• The higher the volatility of a component, the higher its contribution to total vapour pressure.

Composition of Vapour Phase in Equilibrium with Solution

1. Determination of Vapour Phase Composition

• The composition of the vapour phase in equilibrium with a solution is determined by the partial pressures of the components in the solution.
• Dalton’s Law of Partial Pressures states that the partial pressure of a component in a gas mixture is the product of its mole fraction in the vapour phase and the total vapour pressure of the solution.

2. Mathematical Representation

Let:

• $y_1$ and $y_2$ be the mole fractions of components 1 and 2 in the vapour phase.
• $p_1$ and $p_2$ be the partial vapour pressures of components 1 and 2.
• $p_{\text{total}}$ be the total vapour pressure of the solution.

According to Dalton’s Law:

$$p_1 = y_1 \cdot p_{\text{total}} \quad \text{(Equation 1.17)}$$
$$p_2 = y_2 \cdot p_{\text{total}} \quad \text{(Equation 1.18)}$$
$$p_i = y_i \cdot p_{\text{total}} \quad \text{(General Form, Equation 1.19)}$$

where:

• $p_i$ is the partial vapour pressure of component $i$.
• $y_i$ is the mole fraction of component $i$ in the vapour phase.

3. Key Points

• The vapour phase composition is different from the liquid phase composition because more volatile components contribute more to the vapour pressure.
• The component with higher volatility (higher vapour pressure) will have a higher mole fraction in the vapour phase compared to the liquid phase.
• This principle is important in distillation since the vapour phase is richer in the more volatile component.