Solutions

Solutions chemistry deals with the properties of mixtures and how they differ from pure substances.

Overview

graph TD
    A[Solutions] --> B[Concentration Units]
    A --> C[Raoult's Law]
    A --> D[Colligative Properties]
    D --> D1[Vapor Pressure Lowering]
    D --> D2[Boiling Point Elevation]
    D --> D3[Freezing Point Depression]
    D --> D4[Osmotic Pressure]

Concentration Units

Unit	Formula
Molarity (M)	$\frac{\text{mol solute}}{\text{L solution}}$
Molality (m)	$\frac{\text{mol solute}}{\text{kg solvent}}$
Mole Fraction (χ)	$\frac{n_i}{\sum n}$
Mass % (w/w)	$\frac{\text{mass solute}}{\text{mass solution}} \times 100$
ppm	$\frac{\text{mass solute}}{\text{mass solution}} \times 10^6$

Relation Between M and m

$$M = \frac{m \times d}{1 + \frac{m \times M_{solute}}{1000}}$$

where d = density of solution

Raoult’s Law

For ideal solutions:

$$\boxed{P_A = \chi_A \times P_A^0}$$

Total Vapor Pressure

$$P_{total} = P_A + P_B = \chi_A P_A^0 + \chi_B P_B^0$$

Composition of Vapor

$$y_A = \frac{P_A}{P_{total}}$$

Ideal vs Non-Ideal Solutions

Property	Ideal	Positive Deviation	Negative Deviation
ΔH_mix	0	> 0	< 0
ΔV_mix	0	> 0	< 0
P_total	Raoult’s law	> Raoult’s	< Raoult’s
A-B forces	= A-A, B-B	< A-A, B-B	> A-A, B-B
Example	Benzene-Toluene	Ethanol-Water	Chloroform-Acetone

Azeotropes

• Minimum boiling: Positive deviation (Ethanol-Water, 95.4%)
• Maximum boiling: Negative deviation (HCl-Water, 20.2%)

Cannot be separated by fractional distillation.

Colligative Properties

Properties depending only on the number of solute particles, not their nature.

1. Relative Lowering of Vapor Pressure

$$\boxed{\frac{P^0 - P}{P^0} = \chi_{solute} = \frac{n_2}{n_1 + n_2}}$$

For dilute solutions: $\frac{\Delta P}{P^0} \approx \frac{n_2}{n_1}$

2. Elevation of Boiling Point

$$\boxed{\Delta T_b = K_b \times m}$$ $$\boxed{\Delta T_b = \frac{K_b \times w_2 \times 1000}{M_2 \times w_1}}$$

where:

• $K_b$ = molal elevation constant
• $m$ = molality
• $w_2$, $M_2$ = mass and molar mass of solute
• $w_1$ = mass of solvent

3. Depression of Freezing Point

$$\boxed{\Delta T_f = K_f \times m}$$ $$\boxed{\Delta T_f = \frac{K_f \times w_2 \times 1000}{M_2 \times w_1}}$$

4. Osmotic Pressure

$$\boxed{\pi = CRT = \frac{n}{V}RT}$$ $$\pi = \frac{w_2 RT}{M_2 V}$$

Van’t Hoff Factor (i)

Accounts for dissociation or association:

$$\boxed{i = \frac{\text{observed colligative property}}{\text{calculated colligative property}}}$$

For Electrolytes

$$i = 1 + (n-1)\alpha$$

where:

• n = number of ions produced
• α = degree of dissociation

Examples:

• NaCl (n=2): $i = 1 + \alpha$
• BaCl₂ (n=3): $i = 1 + 2\alpha$
• Al₂(SO₄)₃ (n=5): $i = 1 + 4\alpha$

For Association

$$i = 1 - \alpha + \frac{\alpha}{n}$$

where n = number of molecules associating

Modified Colligative Property Equations

$$\Delta T_b = i \times K_b \times m$$ $$\Delta T_f = i \times K_f \times m$$ $$\pi = i \times CRT$$

Determination of Molar Mass

From any colligative property:

$$M_2 = \frac{K_b \times w_2 \times 1000}{\Delta T_b \times w_1}$$ $$M_2 = \frac{w_2 RT}{\pi V}$$

Abnormal Molar Mass

Observation	Cause	i value
M_observed < M_actual	Dissociation	> 1
M_observed > M_actual	Association	< 1

Henry’s Law

For gas dissolved in liquid:

$$\boxed{P = K_H \times \chi_{gas}}$$

where $K_H$ = Henry’s law constant

Higher $K_H$ → Lower solubility

Applications:

• Carbonated beverages
• Deep sea diving (bends)
• Respiration at high altitude

Practice Problems

1. Calculate the molality of a 20% glucose solution.

2. Calculate the boiling point of a solution containing 1 mol glucose in 1 kg water. ($K_b$ = 0.52 K⋅kg/mol)

3. Calculate the osmotic pressure of 0.1 M NaCl at 27°C. (Assume complete dissociation)

4. The freezing point depression of 0.01 m aqueous solution of urea is 0.0186°C. What is $K_f$ of water?

Further Reading

• Equilibrium
• Thermodynamics