Raoult’s Law: Derivation and Deviations from Ideal Behavior

Raoult’s law was proposed by French chemist Francois Marie Raoult in 1886. It states that the vapor pressure of any volatile substance in a solution is equal to the vapor pressure of a pure substance multiplied by the mole fraction of that substance at a given temperature.

It also states that the relative lowering of the vapor pressure of a dilute solution is equal to the mole fraction of solute present in the dilute solution. This is an empirical relationship between the relative lowering of vapor pressure and the concentration of solute in a solution.

For example, when a non-volatile solute is added to a pure solvent, molecules of solute block the surface of the solvent. This affects the evaporation of the solvent which results in decreased vapor pressure. Let’s say P is the vapor pressure of the solvent and P_s is the vapor pressure of the solution. The lowering of vapor pressure will be (P – P_s).

Relative lowering of vapor pressure = P – P_s / P

Mathematically, Raoult’s law can be expressed as

P – P_s / P = n / n + N

where,

n = molecules of solute

N = molecules of solvent

Derivation

Pure solvent converts into vapor when its molecules evaporate from the surface. The greater the number of evaporating molecules, the greater the vapor pressure. However, when a nonvolatile solute is added to it, the vapor pressure goes down.

The vapor pressure of the solution (p_s) can be determined by the molecules of solvent (N) at the surface which is directly proportional to the mole fraction.

P_s ∝ N / n + N

P_s = k N / n + N

k is the proportionality constant.

Let, in the case of the pure solvent, n = 0

Mole fraction of solvent = N / n + N = N / 0 + N = 1

Now the proportionality constant k = P, so

P_s = P N / n + N

P_s / P = N / n + N

1 – P_s / P = 1 – N / n + N

P – P_s / P = n / n + N

Ideal solutions and Raoult’s law

The ideal solutions are those solutions that have uniform cohesive forces. In other words, molecules of an ideal solution experience the same force from each side.

Let A and B be the two components of an ideal solution. The intermolecular forces between the components A and A, B and B, and A and B, will be the same.

There are the following characteristics of an ideal solution:

The ideal solution must obey Rault’s law.
When two pure components are mixed together to form an ideal solution, there is no heat absorbed or released.
The overall volume of an ideal solution is equal to the sum of its component’s volume.
The molecules of an ideal solution exert the same forces from each side.

There are some examples of ideal liquids pairs such as,

Benzene and toluene
n-hexane and n-heptane
Ethylene bromide and propylene bromide
Benzene and ether

In order to explain Raoult’s law, suppose an ideal solution of two liquids A and B. These liquids are completely miscible with each other. X_A and X_B are the mole fraction of liquids A and B respectively in that solution. And P^o_A and P^o_B are the vapor pressure of the pure liquids A and B respectively.

The partial vapor pressure will be,

P_A = P^o_A X_A

P_B = P^o_B X_B

The total vapor pressure of the solution is,

P = P_A + P_B

P = P^o_A X_A + P^o_B X_B (i)

Since

X_A + X_B = 1

X_A = 1 – X_B

So equation (i) can be written as,

P = P^o_A(1 – X_B) – P^o_B X_B

P = P^o_A – P^o_AX_B + P^o_B X_B

P = P^o_A + (P^o_B – P^o_A) X_B

The above equation explains the relationship between the solution composition and its vapor pressure. The P^o_A and P^o_B are the constants at a specific temperature. It shows that total pressure is a linear function of the mole fraction X_B or X_A.

When there is a graph plotted between vapor pressure P against mole fraction X_B or X_A, it results in a straight line. The bold lines explain the total pressure and mole fraction in the diagram while the dotted lines show the partial pressure vs composition. When dotted lines pass near the origin that indicates both components are behaving ideally.

raoult's law - vapor pressure compositon diagram of ideal solution

Case 1:

When X_A = 0, X_B = 1. The result is a solution that contains only B.

P = P^o_AX_A + P^o_BX_B

P = P^o_A x 0 + P^o_BX_B = P^o_B

Case 2:

When X_A = 1, X_B = 0. The solutions contains only A.

P = P^o_A

Therefore, at any concentration, the total pressure is the sum of the partial pressure of the two components of the solution.

Non-ideal solutions and Raoult’s law

Non-ideal solutions are the real solutions. They do not obey Raoult’s law. These solutions contain molecules that interact with each other.

There are the following properties of real or nonideal solutions:

They show deviation from Raoult’s law.
The solution’s component activity is not equal to its mole fraction.
The volume changes upon mixing the solute and solvent. Also, they show compression or expansion when dissolving.
Heat effects were also observed during the mixing of substances. Heat can either be absorbed or evolved. In other words, unlike ideal solutions, the enthalpy change is not zero.

Deviations from ideal behavior

There are many cases when two miscible liquids mix with each other and form a non-ideal solution. Such solutions show two types of deviation from Raoult’s law (ideal behavior): Positive and negative deviations. These deviations are due to the reason that both liquids have different molecular structures and so different intermolecular forces.

Positive deviation from Raoult’s law

When two liquids A and B are mixed together, a solution is obtained. Cohesive forces between the molecules of the liquid (A and B) in a solution are weaker as compared to molecules of pure liquids (A-A or B-B). Consequently, liquids A and B have a high tendency to escape from the solution. Whereas, pure liquids (A-A or B-B) have a low tendency to escape. So, they have higher partial pressures than expected by Raoult’s law. As a result, there is a positive deviation from ideal behavior.

Here are examples of liquids that show positive deviation:

Acetone and carbon disulfide
Acetone and ethyl alcohol
Acetone and ether

raoult's law positive deviation

In the above graph, dotted lines show the ideal behavior. Whereas, bold lines are the actual vapor pressure of the component and solution.

Negative Deviation from Raoult’s law

When the molecules of A and B have a greater attraction for each other than pure liquids, then they have a lesser tendency to escape from the solution. They have less partial pressure as compared to partial pressure predicted by Raoult’s law. This results in a negative deviation from Raoult’s law.

The following are the examples of negative deviation of Raoult’s law:

Acetone and chloroform
Acetone and methanol
Acetic acid and pyridine etc.

raoult's law negative deviation

The above graph shows bold and dotted lines. The dotted lines are indications of ideal behavior while bold lines are the actual vapor pressure of components.

Limitations of Raoult’s law

That solution which ideally behaves obeys Raoult’s law
They are applicable only for very dilute solutions
That solution which contains nonvolatile solute obeys Raoult’s law
Those solutes that dissociate or associate with a solution do not obey Raoult’s law

Concepts Berg

What is the equation of Raoult’s law?

The equation for Raoult’s law is

P – P_s / P = n / n + N

where

n = molecules of solute
N = molecules of solvent
P = vapor pressure of the solvent
P_s = vapor pressure of the solution

What is Henry’s law?

It states that the mass of gas dissolved in a liquid is directly proportional to the pressure of the gas. This means that the greater the mass of gas, the greater the pressure of that gas.

What is the significance of Raoult’s law?

By applying Raoult’s law we can study the behavior of the solution. Those solutions that obey Raoult’s law are the ideal solutions. While the solutions that do not obey Raoult’s law are nonideal or real solutions.

Why is Raoult’s law independent of temperature?

Because Raoult’s law is used to calculate the vapor pressure of the ideal solution at a constant temperature.

Why is Raoult’s law not applicable for immiscible liquid?

The mixture of immiscible liquids has increased vapor pressure and is equal to the vapor pressure of two solvents.

Which solution shows positive and negative deviation?

Non-ideal or real solutions show deviation from Raoult’s law. There are two types of deviations: positive and negative deviation.

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