Definition
The mole fraction (denoted by the Greek lowercase letter chi, \(\chi\), or capital \(X\)) is a rigorous thermodynamic expression of concentration. It is defined as the dimensionless ratio of the molar amount of a specific constituent to the total molar amount of all molecular species present within the homogeneous mixture.
Mathematical Expression
For a multicomponent system containing \(i\) distinct species, the mole fraction of component \(A\) (\(\chi_A\)) is mathematically defined as:
$$\chi_A = \frac{n_A}{n_{\text{total}}} = \frac{n_A}{n_A + n_B + n_C + \dots}$$Where:
- \(n_A\) represents the number of moles of component \(A\).
- \(n_{\text{total}}\) represents the sum of the molar quantities of all components.
A fundamental constraint of this expression is that the sum of the mole fractions for all components in a system must identically equal unity:
$$\sum_{i=1}^{m} \chi_i = 1$$Empirical Application
Consider a binary solution prepared by mixing \(2.00\text{ moles}\) of pure ethanol (\(\text{C}_2\text{H}_5\text{OH}\)) with \(8.00\text{ moles}\) of distilled water (\(\text{H}_2\text{O}\)).
The total molar quantity of the system (\(n_{\text{total}}\)) is calculated as:
$$n_{\text{total}} = n_{\text{ethanol}} + n_{\text{water}} = 2.00\text{ mol} + 8.00\text{ mol} = 10.00\text{ mol}$$Evaluating the independent mole fractions yields:
- Mole fraction of ethanol: $$\chi_{\text{ethanol}} = \frac{2.00\text{ mol}}{10.00\text{ mol}} = 0.200$$
- Mole fraction of water: $$\chi_{\text{water}} = \frac{8.00\text{ mol}}{10.00\text{ mol}} = 0.800$$
Verification of the boundary constraint confirms: \(\chi_{\text{ethanol}} + \chi_{\text{water}} = 0.200 + 0.800 = 1.000\).