Define molarity, molality, mole fraction and mass percentage. Derive the relationship between molarity and molality.

📘 Complete Explanation & Derivation

🔵 1. Molarity (M)

Molarity is defined as the number of moles of solute present in one litre (1000 mL) of solution.

M = Moles of solute ÷ Volume of solution in litre

M = n / V

Molarity depends on the volume of solution, so it can change with temperature.

Unit: mol L⁻¹ or M

🟢 2. Molality (m)

Molality is defined as the number of moles of solute present in 1 kg of solvent.

m = Moles of solute ÷ Mass of solvent in kg

m = n / W

Unit: mol kg⁻¹ or m

Molality does not depend on volume and is therefore essentially independent of temperature.

🟣 3. Mole Fraction (χ)

Mole fraction is the ratio of the number of moles of one component to the total number of moles of all components in the solution.

χ_A = n_A ÷ (n_A + n_B + ...)

For a binary solution:

χ_A + χ_B = 1

Mole fraction has no unit.

🟠 4. Mass Percentage

Mass percentage represents the mass of solute present in 100 g of solution.

Mass % = (Mass of solute ÷ Mass of solution) × 100

Mass % is expressed as a percentage (%).

📊 Quick Comparison

Molarity

moles / L solution

Unit: M

Molality

moles / kg solvent

Unit: m

Mole Fraction

moles of component / total moles

No unit

Mass %

mass solute / mass solution × 100

Unit: %

🔴 Derivation: Relationship Between Molarity and Molality

Let:

Molarity = M
Molality = m
Molar mass of solute = M_B g mol⁻¹
Mass percentage of solution = w%
Density of solution = d g mL⁻¹

Step 1: Consider 1 L of solution

For 1 litre (1000 mL) of solution:

Mass of solution = Volume × Density

= 1000 × d

= 1000d g

Step 2: Calculate mass of solute

If the solution contains w% solute:

Mass of solute = (w/100) × 1000d

= 10dw g

Step 3: Calculate mass of solvent

Mass of solvent = Mass of solution − Mass of solute

= 1000d − 10dw

= 10d(100 − w) g

Step 4: Calculate moles of solute

Moles of solute = Mass of solute ÷ Molar mass

= 10dw ÷ M_B

But for 1 L of solution, the number of moles of solute is equal to molarity M.

M = 10dw / M_B

Step 5: Calculate molality

m = Moles of solute ÷ Mass of solvent in kg

= (10dw / M_B) ÷ [10d(100 − w) / 1000]

Therefore:

m = 1000M ÷ [1000d − M M_B]

⭐ Final Relationship

m = 1000M / [1000d − M M_B]

📌 Common Form Using Mass Percentage

Since:

M = 10dw / M_B

The relationship can also be written as:

M = 1000d m ÷ [1000 + mM_B]

where d is density in g mL⁻¹ and M_B is molar mass of solute.

🎬 3D-Style Concentration Animation

Solute particles are distributed throughout the solvent.

Solute Solvent Solute Solvent Solvent Solute

Molarity → Volume of Solution

Molality → Mass of Solvent

Mole Fraction → Number of Moles

Mass % → Mass of Solution

✅ Final Summary

M = Moles of solute / Volume of solution (L)

m = Moles of solute / Mass of solvent (kg)

χ = Moles of component / Total moles

Mass % = (Mass of solute / Mass of solution) × 100

m = 1000M / [1000d − MM_B]

🎯 Exam Point

Molarity depends on volume and can change with temperature.
Molality depends on mass of solvent and is independent of temperature.
Mole fraction has no unit.
Mass percentage is expressed as %.