Solutions B.Sc. 2nd Year Physical Chemistry Notes

Solubility of Gases in Liquids

Gases dissolve in liquids to form homogeneous solutions. The solubility of a gas in a liquid depends upon the following factors:

1. The nature of the substance (solute)
2. The nature of the solvent
3. Temperature of the solution
4. Pressure

Some gases like N₂, H₂, and O₂ dissolve in water to a very small extent, whereas gases like NH₃ and HCl are highly soluble in water. The most soluble gases are those which chemically react with the liquid solvent.

The solubility of gases in liquids is greatly influenced by pressure and temperature:

1. Effect of temperature: The solubility of gases decreases with an increase in temperature because gases dissolve in a liquid with the evolution of heat (i.e., an exothermic process). Therefore, in accordance with Le Chatelier's principle, an increase in temperature will result in a decrease in the solubility of the gas.

2. Effect of pressure: The solubility of gases increases with an increase in pressure. This is also in accordance with Le Chatelier's principle.

Henry's Law

The solubility of a gas in a liquid is quantitatively explained by Henry's law. Henry's law states that the solubility of a gas increases with increasing pressure at a constant temperature.

Henry's law also states that the partial pressure of a gas in the vapour phase (p) is proportional to the mole fraction of the gas (X) in the solution:

p ∝ X
p = K_H · X

Where K_H is the Henry's law constant, which depends on the nature of the gas and the temperature. K_H has a constant value for a specific gas at a given temperature. The value of K_H increases with increasing temperature. Thus, the value of K_H is inversely proportional to the solubility of the gas in a liquid.

Raoult’s Law

Raoult’s law states that the relative lowering of vapour pressure of a solution containing a non-volatile solute is equal to the mole fraction of the solute.

If the vapour pressure of the pure solvent is P and that of the solution is P_o, then:

Relative lowering of vapour pressure = (P - Po) / P

If the mole fraction of the solute is X_solute, then according to Raoult's law:

(P - P_o) / P = X_solute

If w grams of solute (with molecular mass m) is dissolved in W grams of solvent (with molecular mass M), the mole fraction of the solute is given by:

X_solute = (w / m) / [ (w / m) + (W / M) ]

For highly dilute solutions, the number of moles of solute (w / m) is much smaller than the number of moles of solvent (W / M). Therefore, the term w / m in the denominator can be neglected, simplifying the expression to:

X_solute ≈ (w / m) / (W / M) = (w · M) / (W · m)

Substituting this back into the Raoult's law equation:

(P - P_o) / P = (w · M) / (W · m)

Rearranging the equation to solve for the molecular mass of the unknown solute (m):

m = (w · M · P) / [ W · (P - P_o) ]

Using this final equation, we can easily calculate the molecular mass of a non-volatile solute experimentally.

Duhem–Margules Equation

The Duhem–Margules equation is a thermodynamic statement showing the relationship between the partial vapour pressures of the two components in a binary liquid mixture, where the vapour mixture behaves as an ideal gas.

Let us consider a binary liquid mixture of two components, A and B, in equilibrium with their vapour at a constant temperature and pressure. According to the Gibbs–Duhem equation:

n_Adμ_A + n_Bdμ_B = 0 -----(Equation 1)

Where nA and nB are the number of moles of components A and B, while μA and μB represent their respective chemical potentials.

Dividing Equation 1 by the total number of moles (nA + nB), we get:

[ n_A / (n_A + n_B) ] dμ_A + [ n_B / (n_A + n_B) ] dμ_B = 0

X_Adμ_A + X_Bdμ_B = 0 -----(Equation 2)

Where XA and XB are the mole fractions of components A and B.

The chemical potential of any component in a mixture depends upon temperature, pressure, and the composition of the mixture. If temperature and pressure are kept constant, the differential chemical potentials can be written as:

dμ_A = (∂μ_A / ∂X_A)_T,P · dX_A -----(Equation 3)
dμ_B = (∂μ_B / ∂X_B)_T,P · dX_B -----(Equation 4)

Substituting Equations 3 and 4 into Equation 2 gives:

X_A(∂μ_A / ∂X_A)_T,P · dX_A + X_B(∂μ_B / ∂X_B)_T,P · dX_B = 0 -----(Equation 5)

We know that the sum of the mole fractions of all components in a system is equal to unity:

X_A + X_B = 1 ⇒ dX_A + dX_B = 0 ⇒ dX_A = -dX_B

Therefore, Equation 5 simplifies to:

X_A(∂μ_A / ∂X_A)_T,P = X_B(∂μ_B / ∂X_B)_T,P -----(Equation 6)

The chemical potential of a volatile component in an ideal gas vapour mixture is given by:

μ = μ_o + RT ln(P)

Differentiating this expression with respect to its mole fraction (X), we obtain:

dμ / dX = RT · (d ln(P) / dX)

Applying this specifically to our two components, A and B:

∂μ_A / ∂X_A = RT · (d ln(P_A) / dX_A)
∂μ_B / ∂X_B = RT · (d ln(P_B) / dX_B)

Substituting these dynamic values back into Equation 6 allows us to cancel out the constant RT term:

X_A · (d ln(P_A) / dX_A) = X_B · (d ln(P_B) / dX_B)

Using the mathematical identity dX / X = d ln(X), the expression yields:

(d ln(P_A) / d ln(X_A))_T,P = (d ln(P_B) / d ln(X_B))_T,P -----(Equation 7)

Equation 7 is the final, generalized form of the Duhem–Margules equation.

Azeotrope Mixtures

An azeotrope (or constant boiling point mixture) is a mixture of two or more liquids whose proportions cannot be altered or separated by simple fractional distillation. When an azeotrope is boiled, the generated vapour has the exact same proportions of constituents as the unboiled liquid phase.

Classification of Azeotropes

• Homogeneous Azeotropes: Formed when the liquid constituents are completely miscible with each other in all proportions (e.g., Ethanol and Water).
• Heterogeneous Azeotropes (Heteroazeotropes): Formed when the liquid constituents are not completely miscible, leading to a phase separation inside a miscibility gap. A heteroazeotropic distillation system contains two coexisting liquid phases.
• Binary & Ternary Azeotropes: Azeotropes consisting of two components are called binary (e.g., Benzene and Hexafluorobenzene), whereas mixtures consisting of three distinct elements are termed ternary azeotropes (e.g., Acetone / Methanol / Chloroform).

Types of Boiling Behaviors

Based on their deviations from Raoult's Law, azeotropes are split into two performance classes:

1. Minimum Boiling Azeotropes: Solutions that exhibit a large positive deviation from Raoult's law form minimum boiling azeotropes at a specific composition. For example, an ethanol-water mixture on fractional distillation yields a solution containing a maximum of 97.2% ethanol by volume. Once reached, liquid and vapor compositions lock, causing them to boil at a temperature lower than either pure ethanol or pure water.

2. Maximum Boiling Azeotropes: Solutions that exhibit a large negative deviation from Raoult's law form maximum boiling azeotropes. A classic example is a Nitric Acid and Water mixture. At a mass composition of roughly 68% nitric acid and 32% water, the system forms an azeotrope that boils at a high constant temperature of 393.5 K.

Steam Distillation

Steam distillation is a physical separation technique applied to isolate substances that are steam-volatile and completely immiscible with water. During this operation, steam generated from an external boiler is passed continuously through a heated flask containing the compound mixture.

The vaporized mix of steam and the volatile organic compound condenses and collects in a receiver flask, where they can be easily separated cleanly into individual layers via a separating funnel.

A liquid mixture boils when the combined total vapour pressure of the organic compound (P₁) and water (P₂) reaches equilibrium with the atmospheric system pressure (P):

P = P₁ + P₂

Because the required partial pressure P₁ is lower than the total atmospheric value, the organic compound vaporizes at a temperature well below its true standalone boiling point. For instance, an insoluble system will boil close to, but safely below, 373 K (100°C), protecting temperature-sensitive elements like aniline from thermal degradation.

Industrial Applications

• Essential Oils & Perfumery: Extensively utilized to extract volatile botanical oils (e.g., large-scale industrial extraction of pure orange oil).
• Petroleum Refining: Applied to separate distinct fatty acid fractions within processing units.

Fractional Distillation

Fractional distillation is applied when the boiling point difference between two miscible liquids in a mixture is narrow (typically less than 25°C). In these instances, a simple distillation setup cannot successfully isolate them because their vapors transition and condense concurrently.

To remedy this, a fractionating column is fitted over the boiling flask. As hot vapors ascend, components with higher boiling points condense early on the column's surfaces and drip downward, while the lower-boiling, more volatile vapors continue rising toward the top, gaining high purity along the way.

Each successive cycle of condensation and vaporization occurring within the column replicates an individual distillation step and is referred to as a theoretical plate. Modern industrial columns use hundreds of these plates to refine complex materials, most notably separating crude oil fractions into gasoline, kerosene, and diesel within petroleum industries.