Mole Fractions

The mole fraction expresses the concentration of a component in a mixture or solution. It is the ratio of the moles of one component to the total moles of all components in the mixture. The mole fraction is especially useful in fields like chemistry, physics, and engineering, where precise concentrations are important.

1.0Definition and Formula

The mole fraction (often represented by X) of component A in a solution or mixture is calculated as:

Mole fraction of A, 

$X_A=\frac{\text{moles of A}}{\text{total moles of all components}}$

In a mixture of two components, A and B:

$X_A=\frac{n_A}{n_A+n_B}$

$X_B=\frac{n_B}{n_A+n_B}$

Where:

• n_A​ is the number of moles of component A,
• n_B​ is the number of moles of component B.

For a mixture of multiple components, the general formula is:

$X_i=\frac{n_i}{\sum n_j}$

Where:

• X_i is the mole fraction of component iii,
• n_i is the number of moles of component iii,
• ∑n_j is the sum of moles of all components.

2.0Properties of Mole Fractions

1. Dimensionless Quantity: Mole fraction has no units, as it is a ratio of moles to moles.
2. Range: The mole fraction of each component ranges from 0 to 1.
3. The sum of Mole Fractions: In any mixture or solution, the sum of mole fractions of all components is always equal to 1.
   X_A + X_B + X_C +.... =1

Example 1: In a solution with 2 moles of ethanol (C₂H₅OH) and 3 moles of water (H₂O):

• Total moles = 2 + 3 = 5
• Mole fraction of ethanol, X_C2H5OH =2/5=0.4
• Mole fraction of water, X_H2O =3/5=0.6

Example 2: For a mixture of 1 mole of nitrogen (N₂), 2 moles of oxygen (O₂), and 4 moles of argon (Ar):

• Total moles = 1 + 2 + 4 = 7
• Mole fraction of N₂, X_N2 = 1/7≈0.143
• Mole fraction of O₂, X_O2 = 2/7≈0.286
• Mole fraction of Ar, X_Ar = 4/7≈0.571

3.0Applications of Mole Fractions

$Pi=\chi i\cdot P_i={\chi }_i\cdot P_{\text{total}}$

1. Ideal Gases: In gases, mole fraction directly relates to partial pressure, as the partial pressure of a gas is its mole fraction multiplied by the total pressure.

2. Raoult’s Law in Solutions: Mole fraction is used in Raoult’s law to calculate the vapour pressure of components in ideal solutions.

$P_A={\chi }_A\cdot P_A^0$

3. Where P_A is the partial vapour pressure of component A, and P^0_A​ is the vapour pressure of pure A.

4. Colligative Properties: Mole fraction is important in calculating colligative properties like boiling point elevation and freezing point depression.

5. Mole Fraction in Chemistry and Engineering: Used in reaction stoichiometry, calculating mole balances, and for solutions in laboratory settings.

Advantages of Using Mole Fractions

• Temperature Independence: Unlike molarity or molality, mole fraction is independent of temperature, as it doesn’t rely on volume.
• Simplicity in Calculations: Mole fractions simplify calculations, especially in gas mixtures where they relate directly to partial pressures.

4.0Solved Example

1. Calculate the mole fraction of sodium chloride (NaCl) if the solution contains 0.3 moles of NaCl in 180 mL of water.

Solution:

Given data:

• Moles of NaCl = 0.3 moles
• Volume of water = 180 mL
• Density of water = 1 g/mL
• Molar mass of water (H₂O) = 18 g/mol

2. If 4 moles of sulfur dioxide (SO₂) and 6 moles of oxygen (O₂) are placed in a closed container, what will be the mole fraction of SO₂ after the reaction?

A) 0
B) 0.4
C) 0.6
D) 1

Solution:

Since all 4 moles of SO₂ react completely, none remains in the gas phase. The only gases left are 4 moles of O₂, so the mole fraction of SO₂ is 0.