Calculating the mole fraction in the liquid phase is a fundamental concept in chemistry, essential for understanding the composition and behavior of liquid mixtures and solutions. It represents the proportion of a specific component's moles relative to the total number of moles of all components present in the liquid mixture.
The mole fraction of a component is always a dimensionless quantity between 0 and 1, inclusive. It is calculated by dividing the number of moles of a particular component by the total number of moles of all substances in the mixture.
Understanding Mole Fraction (χ or x)
Mole fraction, commonly symbolized by the Greek letter chi (χ) or 'x' (especially for liquid phase compositions), offers a concentration measure that remains unaffected by temperature or pressure changes, unlike other measures like molarity. For a liquid mixture containing components A, B, C, and so on, the mole fraction of component A ($x_A$) is defined by the following formula:
$$
x_A = \frac{\text{Moles of Component A}}{\text{Total Moles of all Components in the Mixture}} = \frac{nA}{n{\text{total}}} = \frac{n_A}{n_A + n_B + n_C + \dots}
$$
Where:
- $n_A$ = number of moles of component A
- $n_B$, $n_C$, etc. = number of moles of other components (B, C, etc.)
- $n_{\text{total}}$ = total number of moles of all components in the mixture ($n_A + n_B + n_C + \dots$)
An important property of mole fractions is that the sum of the mole fractions for all components in a mixture must always equal 1:
$$
x_A + x_B + x_C + \dots = 1
$$
For a more general understanding of mole fraction, you can refer to Mole Fraction on Wikipedia.
Step-by-Step Calculation Guide
To determine the mole fraction of each component in a liquid mixture, follow these steps:
- Determine the Mass of Each Component: Begin by knowing the mass of each individual component present in your liquid mixture. If given volumes and densities, convert them to mass.
- Convert Mass to Moles: For every component, use its molar mass ($M$) to convert the mass ($m$) into moles ($n$) using the relationship: $n = m / M$.
- You will need the chemical formula for each component to calculate its molar mass from the atomic masses listed on the periodic table.
- Calculate Total Moles: Sum up the moles of all the individual components you calculated in the previous step to find the total moles in the mixture ($n_{\text{total}}$).
- Calculate Mole Fraction for Each Component: Divide the moles of each individual component by the total moles calculated in the previous step to get its respective mole fraction.
Practical Example: Ethanol-Water Mixture
Let's illustrate the calculation of mole fractions using a common liquid mixture: ethanol and water. Suppose you have a mixture containing 46.0 grams of ethanol ($C_2H_5OH$) and 54.0 grams of water ($H_2O$).
Given Data:
- Mass of ethanol ($m_{ethanol}$) = 46.0 g
- Mass of water ($m_{water}$) = 54.0 g
Step 1: Determine Molar Masses
- Molar mass of ethanol ($C_2H_5OH$):
- (2 × 12.01 g/mol C) + (6 × 1.008 g/mol H) + (1 × 16.00 g/mol O) = 46.068 g/mol ≈ 46.07 g/mol
- Molar mass of water ($H_2O$):
- (2 × 1.008 g/mol H) + (1 × 16.00 g/mol O) = 18.016 g/mol ≈ 18.02 g/mol
Step 2: Convert Mass to Moles
- Moles of ethanol ($n_{ethanol}$):
- $n_{ethanol} = \frac{46.0 \text{ g}}{46.07 \text{ g/mol}} \approx 0.9985 \text{ mol}$
- Moles of water ($n_{water}$):
- $n_{water} = \frac{54.0 \text{ g}}{18.02 \text{ g/mol}} \approx 2.9967 \text{ mol}$
Step 3: Calculate Total Moles
- $n{\text{total}} = n{ethanol} + n_{water} = 0.9985 \text{ mol} + 2.9967 \text{ mol} = 3.9952 \text{ mol}$
Step 4: Calculate Mole Fraction for Each Component
- Mole fraction of ethanol ($x_{ethanol}$):
- $x{ethanol} = \frac{n{ethanol}}{n_{\text{total}}} = \frac{0.9985 \text{ mol}}{3.9952 \text{ mol}} \approx 0.250$
- Mole fraction of water ($x_{water}$):
- $x{water} = \frac{n{water}}{n_{\text{total}}} = \frac{2.9967 \text{ mol}}{3.9952 \text{ mol}} \approx 0.750$
Verification:
- $x{ethanol} + x{water} = 0.250 + 0.750 = 1.000$ (The sum equals 1, confirming the accuracy of the calculation).
Importance in Vapor-Liquid Equilibrium
The mole fraction in the liquid phase (often denoted as 'x' or $x_i$ for component 'i') is profoundly important in understanding the behavior of liquid mixtures, particularly in the context of vapor-liquid equilibrium (VLE). For ideal solutions, Raoult's Law provides a direct relationship between the partial vapor pressure of a component above a liquid mixture and its mole fraction in that liquid phase.
According to Raoult's Law:
$$
P_i = x_i \cdot P_i^0
$$
Where:
- $P_i$ = partial vapor pressure of component 'i' above the solution
- $x_i$ = mole fraction of component 'i' in the liquid phase
- $P_i^0$ = vapor pressure of pure component 'i' at the same temperature
This relationship highlights that the higher the mole fraction of a volatile component in the liquid phase, the greater its partial pressure above the liquid. This partial pressure contributes to the overall vapor pressure of the mixture and, through Dalton's Law of Partial Pressures, helps determine the mole fraction of that component in the vapor phase (often denoted as 'y' or $y_i$). The vapor phase mole fraction ($y_i$) can be calculated as:
$$
y_i = \frac{Pi}{P{\text{total}}}
$$
Where $P_{\text{total}}$ is the total pressure of the vapor mixture. Therefore, comprehending the mole fraction in the liquid phase is essential for predicting the composition of the vapor phase in equilibrium with a liquid mixture, which is a critical aspect in various chemical engineering applications like distillation and evaporation. For more on this relationship, you can explore Raoult's Law on LibreTexts.
Summary of Key Terms and Notations
| Term | Symbol | Description | Units |
|---|---|---|---|
| Mole Fraction (Liquid) | $x_i$ (or χ_i) | Moles of component 'i' divided by total moles in liquid | Dimensionless |
| Moles of Component | $n_i$ | Amount of substance (quantity) of component 'i' | mol |
| Total Moles | $n_{\text{total}}$ | Sum of moles of all components in the mixture | mol |
| Partial Vapor Pressure | $P_i$ | Pressure exerted by component 'i' in the vapor phase | e.g., atm, Pa |
| Pure Vapor Pressure | $P_i^0$ | Vapor pressure of pure component 'i' at a given temperature | e.g., atm, Pa |
| Mole Fraction (Vapor) | $y_i$ | Moles of component 'i' divided by total moles in the vapor phase | Dimensionless |
