Calculate the mole fractions of methanol, \(\mathrm{CH}_{3} \mathrm{OH} ;\) ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} ;\) and water in a solution that is \(40 \%\) methanol, 40\% ethanol, and 20\% water by mass. (Assume the data are good to two significant figures.)

Understanding the molar mass of a substance is crucial for converting between mass and moles. Molar mass is defined as the mass of one mole of a substance, typically measured in grams per mole (g/mol). This concept is essential for many calculations in chemistry, including the boiling point elevation, osmotic pressure, and mole fractions in a solution.

For example, in the provided exercise, molar mass allows us to convert the mass of methanol, ethanol, and water into moles. This is done by dividing the mass of each substance by its respective molar mass. It's the fundamental step that leads to the calculation of mole fractions in a solution, setting the stage for understanding solution composition. Each compound has a unique molar mass, reflecting the combined atomic masses of the elements in the molecular formula.

The mole concept is a bridge between the microscopic world of atoms and molecules and the macroscopic world we can measure. One mole is Avogadro's number (approximately 6.022 x 10^23) of particles, whether they are atoms, molecules, ions, or electrons. This number provides a way to count entities on the molecular scale using a quantity we can measure - mass.

In practice, if we take the molar mass of a substance and weigh out that amount in grams, we have one mole of that substance. The exercise at hand involves using the mole concept to express the composition of a methanol, ethanol, and water mixture, not in terms of mass, but in terms of the number of molecules present, represented as moles.

The conversion from mass percentage to moles is a conversion from a mass-based description to a number-based description of a substance's amount. To do this, we assume a convenient mass of the mixture which allows us to use the percentage directly as mass. In the exercise, the solution has been assumed to be 100 grams, so the mass percentages can be directly treated as mass in grams.

Once the masses are established, they are divided by the molar masses of the respective substances to convert those masses into moles. This method simplifies calculations significantly, and it provides a reliable way to compare the amounts of different components in a mixture, as required for finding mole fractions.

Understanding the composition of solutions is a key aspect in chemistry. The composition is often expressed in terms of mole fractions, which provide a way to describe the proportion of each component in a solution. The mole fraction is the number of moles of a particular component divided by the total number of moles of all components in the solution.

In the exercise provided, once the total moles are calculated, the mole fraction of each component can be determined. This fraction represents the contribution of each substance to the overall solution and is critical in determining physical properties of the solution such as vapor pressure and boiling point. Mole fractions are dimensionless numbers that sum up to one, reflecting the fact that they are fractions of a whole. They give us a clear and precise understanding of the solution composition without the influence of volume, which can be affected by temperature and pressure.