# Raoult’s Law

## Raoult’s Law Definition

The pressure at which vapor is formed above a solid or liquid at a particular temperature is called the vapor pressure. Vapor and solid or liquid are in dynamic equilibrium at this temperature. In 1980s Rault discovered when a non-volatile solute is dissolved in a solvent the vapor pressure decreases. The lowering of vapor pressure depends on the mole fraction of the solute dissolved and the vapor pressure of the pure solvent. Thus the Rault’s stated as the relative lowering of vapor pressure of a dilute solution containing non-volatile solute is equal to the mole fraction of that solute and the vapor pressure of the pure solvent.

## Explanation

If P is the vapor pressure of pure solvent and Ps is the vapor pressure of the solution. The lowering of vapor pressure is (P – Ps). So,

the relative lowering of vapor pressure = $$\frac { P-{ P }_{ s } }{ P }$$

The Raoult’s law can be expressed mathematically as,

$$\frac { P-{ P }_{ s } }{ P } =\quad \frac { n }{ n+N }$$

Here n = number of moles of solute
N = number of moles of solvent.

## Derivation of Raoult’s law

The vapor pressure depends on the number of molecules evaporate from the surface. If non volatile solute dissolved in a solvent the non volatile molecule blocks the fraction of solvent molecules to evaporate. Thus as the less number of molecules evaporate from the surface the vapor pressure also lesser than the pure solvent.

This vapor pressure of the solution can be determined by the number of solvent molecules present on the surface or the mole fraction of solvent. Thus
$${ P }_{ s }\quad \propto \quad \frac { N }{ n+N } \\ { P }_{ s }\quad =\quad k\quad \frac { N }{ n+N } \quad \\ [Here\quad K\quad is\quad a\quad proportionality\quad constant]$$ $$If\quad n\quad =\quad 0,\quad { P }_{ s }\quad =\quad P\quad \\ \frac { N }{ n+N } \quad =\quad \frac { N }{ 0+N } \quad =\quad 1,\quad \\ so\quad the\quad vapour\quad pressure\quad \\ P\quad =\quad k$$ $$Thus\\ { P }_{ s }\quad =\quad P\quad \frac { N }{ n+N } \\ \frac { { P }_{ s } }{ { P } } \quad =\quad \frac { N }{ n+N } \\ 1-\frac { { P }_{ s } }{ { P } } \quad =\quad 1-\frac { N }{ n+N } \\ \frac { { P-P }_{ s } }{ { P } } \quad =\quad \frac { n }{ n+N }$$

## Determination of molecular mass

If the weight of solute and solvent are known, the molecular weight of the non volatile solute can be determined from the equation of the Raoult’s law. Assume w grams of solute dissolved in W grams of solvent. M and m is the molecular weight of solvent and solute. Then,

$$No.\quad of\quad moles\quad of\quad solute\quad (n)\quad =\quad \frac { w }{ m } \\ No.\quad of\quad moles\quad of\quad solvent\quad (N)\quad =\quad \frac { W }{ M }$$ $$Substituting\quad these\quad value\quad in\quad \\ Raoult’s\quad law\quad equation\\ \frac { { P-P }_{ s } }{ { P } } \quad =\quad \frac { n }{ n+N } \\ \frac { { P-P }_{ s } }{ { P } } \quad =\quad \frac { w/m }{ w/m+W/M }$$

As the Raoult’s law is applied to a very dilute solution. w/m is very small and negligible as a denominator. Thus the equation can be rewritten as

$$\frac { { P-P }_{ s } }{ { P } } \quad =\quad \frac { w/m }{ W/M }$$

## Raoult’s law for ideal solution

A solution which strictly obeys Raoult’s law is known as ideal solution. A non ideal solution shows deviation from Raoult’s law. Suppose A and B are the molecule of solvent and solute. So,

ϒAB = the attractive force between the molecule A and B

ϒAA = the attractive force between the molecule A and A

If ϒAB = ϒAA, the solution will obey the Raoult’s law. Thus it is an idea solution.

But if ϒAB < ϒAA, then the molecule will escape more readily than the pure solvent. The vapour pressure of the solution will be higher and positive deviation from Raoult’s law will be observed.

Again if ϒAB > ϒAA, then the molecule will escape less readily than the pure solvent. The vapour pressure of the solution will be lower and negative deviation from Raoult’s law will be observed.

As in very dilute solution the molecular size and attraction of solute and solvent are similar. Thus these solutions obey Raoult’s law fairly accurately.