1.2.5. Mole Fraction

Commonly used symbol for mole fraction is x and subscript used on the right hand side of x denotes the component. It is defined as:

Mole fraction of a component = $\frac{\text{Number of moles of the component}}{\text{Total number of moles of all the components}}$

For example, in a binary mixture, if the number of moles of A and B are n_A and n_B respectively, the mole fraction of A will be:

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

For a solution containing i number of components, we have:

$x_i = \frac{n_i}{\sum_{j=1}^{i}n_j}$

It can be shown that in a given solution the sum of all the mole fractions is unity, i.e.

$x_1 + x_2 + \ldots + x_i = 1$

Mole fraction is a way to express the concentration of a component in a solution. It represents the ratio of the number of moles of a particular substance to the total number of moles of all substances in the mixture. For instance, if we have 2 moles of component A and 3 moles of component B, the mole fraction of A is 2/(2+3)=0.4 and for B it is 3/(2+3)=0.6. It's important to note that the sum of the mole fractions of all components in a mixture always equals 1.

Think of mole fractions like slices of a pizza. If you have a pizza divided into 10 slices, and you take 4 slices of cheese and 6 slices of pepperoni, the fraction of cheese slices to the total pizza is 4/10 or 0.4, while for pepperoni, it is 6/10 or 0.6. This reflects the same concept as mole fractions in a solution.

Mole fraction unit is very useful in relating some physical properties of solutions, say vapour pressure with the concentration of the solution and quite useful in describing the calculations involving gas mixtures.

Mole fractions are particularly valuable in calculations that involve physical properties like vapour pressure, where the mole fraction can help predict how a solution behaves. For example, the vapour pressure of a solvent in a solution can be determined using the mole fraction of the solute. In gas mixtures, mole fractions allow chemists to understand how different gases contribute to the overall properties of the mixture, such as total pressure.

Imagine you have a tank filled with oxygen and nitrogen. If you measure the mole fraction of oxygen, you can understand how much oxygen is present in relation to the total amount of gas, which can help in understanding how the mixture behaves under certain pressure conditions, like in a scuba tank for divers.

Example 1: Calculate the mole fraction of ethylene glycol (C₂H₆O₂) in a solution containing 20% of C₂H₆O₂ by mass. Assume that we have 100 g of solution (one can start with any amount of solution because the results obtained will be the same). Solution will contain 20 g of ethylene glycol and 80 g of water. Molar mass of C₂H₆O₂ = 12 × 2 + 1 × 6 + 16 × 2 = 62 g mol^–1. 20 g\nMoles of C₂H₆O₂ = \frac{20 g}{62 g mol^–1} = 0.322 mol\n80 g\nMoles of water = \frac{80 g}{18 g mol^–1} = 4.444 mol\nmoles of C₂H₆O₂\nx_C2H6O2 = \frac{0.322 mol}{0.322 mol + 4.444 mol} = 0.068\nmoles of water\nx_water = \frac{0.322 mol}{0.322 mol + 4.444 mol} = 0.932.

In this example, we start with a solution, which contains 20 g of ethylene glycol and 80 g of water, making it a total of 100 g. We calculate the number of moles of each component by dividing the mass by their respective molar masses. After determining the moles, we find the mole fractions by taking the moles of each component and dividing by the total moles. This gives us the mole fraction of ethylene glycol as 0.068 and of water as 0.932.

Consider a recipe where you mix 20g of sugar (like ethylene glycol) in 80g of water. If you wanted to know how "sweet" your water is relative to the total amount in the mixture, you'd use mole fractions, showing that even a small amount of sugar has a significant role in determining the overall taste (properties) of the mixture.

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