Concentration Terms: Mole Fraction

When chemists study mixtures — whether gases, liquids, or solutions — they need precise ways to describe how much of each component is present. One of the most fundamental and versatile concentration expressions is the mole fraction, symbolized by the Greek letter χ (chi). The mole fraction of a component is defined as the ratio of the number of moles of that component to the total number of moles of all components in the mixture. Because it is a ratio of moles to moles, the mole fraction is dimensionless — it carries no units — and all mole fractions in a mixture must sum to exactly 1. Mole fraction is particularly powerful because it connects directly to physical laws governing gas mixtures and vapor pressures of solutions. Dalton's Law of Partial Pressures states that the partial pressure of any gas in a mixture equals its mole fraction multiplied by the total pressure. Raoult's Law, which describes ideal solutions, states that the vapor pressure of a solvent above a solution is equal to the mole fraction of the solvent multiplied by the vapor pressure of the pure solvent. These relationships make mole fraction indispensable in industrial chemistry, atmospheric science, and the study of colligative properties. Unlike molarity (moles per liter) or mass percent (grams per 100 grams), mole fraction does not change with temperature, because neither the number of moles nor their ratio changes when a system is heated or cooled (assuming no reaction or evaporation). This temperature independence makes mole fraction the preferred concentration unit in thermodynamic calculations, including those involving entropy of mixing, activity coefficients, and phase diagrams. Mastering mole fraction calculations is an essential step toward understanding physical chemistry at a deeper level.

Key takeaways
  • Define the System and List All Components
  • Calculate Total Moles
  • Compute Each Mole Fraction
  • Apply Dalton's Law to Gas Mixtures
  • Apply Raoult's Law to Liquid Solutions

Define the System and List All Components

Identify every component in the mixture. For a binary mixture (two components A and B), you need n_A (moles of A) and n_B (moles of B). For a multicomponent mixture, list n_1, n_2, ..., n_k for each component. If masses are given, convert each mass to moles using the molar mass: n = m / M.

Calculate Total Moles

Sum all individual mole quantities: n_total = n_A + n_B + n_C + ... This is the denominator in every mole fraction calculation for that mixture. Always double-check that you have included every component, including the solvent in a solution.

Compute Each Mole Fraction

For each component i, the mole fraction is χ_i = n_i / n_total. Calculate one fraction for each component. As a required check: the sum of all mole fractions must equal exactly 1.00 (χ_A + χ_B + χ_C + ... = 1). Any deviation beyond rounding error indicates a calculation mistake.

Apply Dalton's Law to Gas Mixtures

For a gas mixture at total pressure P_total, the partial pressure of component i is: P_i = χ_i × P_total. For example, dry air is approximately 78% N₂ and 21% O₂ by mole. At 1 atm total pressure, P(N₂) ≈ 0.78 × 1 atm = 0.78 atm, and P(O₂) ≈ 0.21 × 1 atm = 0.21 atm. The remaining ~1% is mostly Ar and CO₂.

Apply Raoult's Law to Liquid Solutions

For an ideal solution, the vapor pressure of solvent A above the solution is: P_A = χ_A × P°_A, where P°_A is the vapor pressure of pure solvent A at the same temperature. Adding a non-volatile solute (such as glucose in water) lowers χ_solvent, which lowers the vapor pressure. This vapor pressure lowering is one of the four colligative properties and drives important phenomena like boiling point elevation and freezing point depression.

The Key Equation

χi=nin1+n2+n3+⋯+nk=ni∑j=1knj\chi_i = \frac{n_i}{n_1 + n_2 + n_3 + \cdots + n_k} = \frac{n_i}{\sum_{j=1}^{k} n_j}
  • \chi_i — Mole fraction of component i (dimensionless): A number between 0 and 1 expressing the proportion of component i in the mixture by amount of substance.
  • n_i — Moles of component i (mol): The amount of substance of the specific component whose mole fraction is being calculated.
  • \sum_{j=1}^{k} n_j — Total moles in mixture (mol): The sum of moles of all k components present in the mixture; serves as the denominator.

Worked Example

Problem. A solution is prepared by dissolving 18.0 g of glucose (C₆H₁₂O₆, molar mass = 180.0 g/mol) in 90.0 g of water (H₂O, molar mass = 18.0 g/mol). (a) Calculate the mole fraction of glucose (χ_glucose) and of water (χ_water). (b) Using Raoult's Law, calculate the vapor pressure of water above this solution at 25°C, given that the vapor pressure of pure water at 25°C is 23.8 mmHg.

χglucose=0.0196,χwater=0.980,Pwater=0.9804×23.8 mmHg=23.3 mmHg\chi_{\text{glucose}} = 0.0196,\quad \chi_{\text{water}} = 0.980,\quad P_{\text{water}} = 0.9804 \times 23.8\ \text{mmHg} = 23.3\ \text{mmHg}

Answer. χ_glucose = 0.0196, χ_water = 0.980; vapor pressure of water above solution = 23.3 mmHg.

Figure
Diagram
Raoult's Law: Vapor Pressure vs. Mole Fraction of SolventMole Fraction of Solvent, χ_solventVapor Pressure (mmHg)00.250.500.751.00015304560P° = 23.8P°_water = 23.8 mmHg (pure solvent)Solution point(χ=0.98, P=23.3)Ideal (Raoult's Law) lineP_A = χ_A × P°_AGlucose-Waterχ_water = 0.980P_water = 23.3 mmHg
Origins · 1 minFrançois-Marie Raoult and how this was discovered
FR
François-Marie Raoult
1830–1901 · French

François-Marie Raoult was a French chemist who, between 1878 and 1882, systematically investigated the vapor pressures of solutions. He discovered that the vapor pressure lowering of a solvent is proportional to the mole fraction of the solute — the relationship now known as Raoult's Law. He also independently discovered the relationships for freezing point depression and boiling point elevation, laying the experimental foundation for the study of colligative properties and for the concept of mole fraction as a concentration measure in physical chemistry. Raoult worked without a university laboratory for most of his early career, conducting painstaking experiments in modest conditions at the University of Grenoble. He measured the freezing points of hundreds of solutions by hand to confirm his law's generality.

  • Dalton's Law of Partial Pressures — Dalton's Law directly uses mole fractions: the partial pressure of each gas in a mixture equals its mole fraction times the total pressure.
  • Molarity and Solution Concentration — Molarity is another key concentration term; understanding how it differs from mole fraction (temperature dependence, volume vs. moles) is essential for choosing the right expression in calculations.
  • Colligative Properties — Vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure all depend on the mole fraction (or closely related molality) of solute particles, not their chemical identity.
  • Ideal Gas Law — The ideal gas law underlies Dalton's Law. For ideal gases, each component behaves independently, and mole fraction directly gives the fraction of total pressure contributed by that gas.