Chemistry · Ch 1 — Solutions

Vapour Pressure of Liquid-Liquid Solutions

1.4.1

Vapour Pressure of Liquid-Liquid Solutions

Consider a binary solution of two volatile liquids, labelled component 1 and component 2. When this solution is kept in a closed vessel, both liquids evaporate. In time a dynamic equilibrium is set up between the liquid phase and the vapour phase above it.

At this equilibrium let:

$p_{total}$ = total vapour pressure over the solution,
$p_1, p_2$ = partial vapour pressures of components 1 and 2,
$x_1, x_2$ = mole fractions of components 1 and 2 in the liquid.

The question is: how does each partial pressure depend on the composition of the liquid?

Raoult's law for volatile liquids

The French chemist François Marie Raoult (1886) answered this. Raoult's law states that for a solution of volatile liquids, the partial vapour pressure of each component is directly proportional to its mole fraction in the solution.

For component 1:

p_1 \propto x_1

and

p_1 = p_1^{0}\, x_1

For component 2:

p_2 = p_2^{0}\, x_2

where

$p_1^{0}$ = vapour pressure of pure component 1 (at that temperature),
$p_2^{0}$ = vapour pressure of pure component 2 (at that temperature).

Each partial pressure grows linearly from zero (when that component is absent) up to the pure-liquid value (when it is the only component present).

Total vapour pressure

By Dalton's law of partial pressures, the total pressure over the solution is the sum of the partial pressures:

p_{total} = p_1 + p_2

Substituting Raoult's law for each component:

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

Since the mole fractions add to one, $x_1 = 1 - x_2$ . Replacing $x_1$ :

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

Expanding and grouping the $x_2$ terms gives the linear form:

p_{total} = p_1^{0} + \left(p_2^{0} - p_1^{0}\right) x_2

What the linear form tells us

Three conclusions follow directly:

The total vapour pressure can be expressed through the mole fraction of any one component.
$p_{total}$ varies linearly with the mole fraction $x_2$ of component 2.
Depending on the values of $p_1^{0}$ and $p_2^{0}$ , the total pressure rises or falls as $x_2$ increases — it falls with increasing $x_1$ when component 1 is the less volatile one.

A plot of $p_1$ or $p_2$ against mole fraction is a straight line, and the plot of $p_{total}$ against $x_2$ is also a straight line. If component 1 is less volatile than component 2 (that is, $p_1^{0} < p_2^{0}$ ), then $p_{total}$ ranges between a minimum of $p_1^{0}$ (pure 1) and a maximum of $p_2^{0}$ (pure 2).

Figure 1.3The plot of vapour pressure and mole fraction of an ideal solution at constant temperature. The dashed lines I and II represent the partial pressure of the components; the total vapour pressure is line III.

$Fig. 1.3 — The plot of vapour pressure and mole fraction of an ideal solution at constant temperature. The dashed lines I and II represent the partial pressure of the components; the total vapour pressure is line III.$

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What the Figure Shows

The plot has vapour pressure on the vertical axis and mole fraction on the horizontal axis. The horizontal axis runs from pure component 1 (mole fraction $x_1 = 1$ , $x_2 = 0$ ) on the left to pure component 2 ( $x_1 = 0$ , $x_2 = 1$ ) on the right. At the left edge, the vapour pressure of pure component 1, $p_1^0$ , is marked; at the right edge, the vapour pressure of pure component 2, $p_2^0$ , is marked.

Three lines appear:

Dashed line I (labelled $p_1$ ): starts at $p_1^0$ on the left and falls linearly to zero at the right. This shows that the partial pressure of component 1 is directly proportional to its mole fraction $x_1$ .
Dashed line II (labelled $p_2$ ): starts at zero on the left and rises linearly to $p_2^0$ on the right. This shows that the partial pressure of component 2 is directly proportional to its mole fraction $x_2$ .
Solid line III (labelled $p_{\text{total}}$ ): is a straight line joining $p_1^0$ on the left to $p_2^0$ on the right. It represents the sum of the two dashed lines at every composition.

The Physical Idea

The figure teaches Raoult's law for an ideal binary liquid solution: the partial vapour pressure of each volatile component in the solution is proportional to its mole fraction in the liquid phase. Because the total vapour pressure is the sum of the partial pressures, it also varies linearly with composition. The dashed lines show the individual contributions; the solid line shows the overall vapour pressure of the solution.

Key Formulas Developed from the Figure

For component 1:

p_1 = p_1^0 \, x_1

where $p_1$ is the partial vapour pressure of component 1 above the solution, $p_1^0$ is the vapour pressure of pure component 1 at the same temperature, and $x_1$ is its mole fraction in the liquid.

For component 2:

p_2 = p_2^0 \, x_2

with analogous meanings.

The total vapour pressure is:

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

Composition of the vapour phase

The vapour in equilibrium with the solution has its own composition, fixed by the partial pressures. If $y_1$ and $y_2$ are the mole fractions of components 1 and 2 in the vapour, then again by Dalton's law: …

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