Problem 42 - Chemistry: The Central Science

Question

A solution is made containing \(20.8 \mathrm{~g}\) of phenol \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}\right)\) in \(425 \mathrm{~g}\) of ethanol \(\left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\right)\). Calculate (a) the mole fraction of phenol, (b) the mass percent of phenol, (c) the molality of phenol.

Step-by-Step Solution

Answer

a) Mole fraction of phenol: \(0.0234\) b) Mass percent of phenol: \(4.67\%\) c) Molality of phenol: \(0.52 \frac{mol}{kg}\)

Step 1: Calculate moles of phenol and ethanol

First, let's calculate the moles of phenol and ethanol. We would require their molar masses, which can be calculated as follows: Phenol (C6H5OH) - Molar mass = (6 * 12.01) + (5 * 1.01) + 16.00 + 1.01 = 94.11 g/mol Ethanol (CH3CH2OH) - Molar mass = (2 * 12.01) + (6 * 1.01) + 16.00 + 1.01 = 46.07 g/mol Now, we can calculate the moles: moles of phenol = mass / molar mass = 20.8 g / 94.11 g/mol = 0.221 mol moles of ethanol = mass / molar mass = 425 g / 46.07 g/mol = 9.23 mol

Step 2: Calculate the mole fraction of phenol

The mole fraction of phenol (X_phenol) can be calculated using the formula: X_phenol = moles of phenol / (moles of phenol + moles of ethanol) = 0.221 / (0.221 + 9.23) = \(0.0234\)

Step 3: Calculate the mass percent of phenol

The mass percent of phenol can be calculated using the formula: Mass percent of phenol = (mass of phenol / total mass of solution) * 100 = (20.8 g / (20.8 g + 425 g)) * 100 = \(4.67\%\)

Step 4: Calculate the molality of phenol

The molality of phenol can be calculated using the formula: Molality = moles of solute / kilograms of solvent = 0.221 mol / (425 g / 1000 g/kg) = \(0.52 \frac{mol}{kg}\) Summarizing the results: a) Mole fraction of phenol: \(0.0234\) b) Mass percent of phenol: \(4.67\%\) c) Molality of phenol: \(0.52 \frac{mol}{kg}\)

Key Concepts

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It is the ratio of the number of moles of a substance to the total number of moles of all substances present. For a two-component system like phenol in ethanol, the mole fraction of phenol is calculated by dividing the moles of phenol by the sum of the moles of phenol and ethanol.
In our example, the calculation yielded a mole fraction of 0.0234 for phenol. This number, while it seems small, gives a clear and temperature-independent measure of concentration. Understanding mole fraction is crucial because it is used in various applications like calculating vapor pressure and partial pressures in gases.

Mass Percent

Mass percent, also known as weight percent, is another common way to represent solution concentration. It is the mass of the solute divided by the total mass of the solution, multiplied by 100 to convert it into a percentage. In this example, the mass percent of phenol is found to be 4.67%, indicating that for every 100 grams of solution, there are 4.67 grams of phenol.
This concentration measure is practical in everyday situations, such as in cooking, where ingredients are often measured by weight. When dealing with solutions, mass percent provides a quick and easy way to understand how much solute is present relative to the amount of the overall solution.

Molality

Molality is the measure of the concentration of a solute in a solution defined as the number of moles of solute per kilogram of solvent. It differs from molarity, which is moles per liter of solution, in that temperature changes do not affect it because mass does not change with temperature (unlike volume).
For the given solution of phenol in ethanol, the molality is calculated to be 0.52 mol/kg. This means there are 0.52 moles of phenol for every kilogram of ethanol used. Molality is particularly useful in colligative properties calculations such as boiling point elevation and freezing point depression, where the effect of solute concentration on solvent properties is analyzed.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is a fundamental property that relates the mass of matter to the number of particles or moles. For instance, calculating the molar mass of phenol involves summing the atomic masses of each element in the compound (C6H5OH), resulting in 94.11 g/mol, and it is a critical step in all stoichiometric calculations.
Understanding molar mass is essential for converting between moles and grams, as seen in the solution we explored, which required knowing the molar masses of phenol and ethanol to determine the moles and subsequently, the mole fraction, mass percent, and molality.