Problem 146 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

A quantity of \(10 \mathrm{~g}\) of acetic acid is dissolved in \(100 \mathrm{~g}\) of each of the following solvents. In which solvent, the mole fraction of solute is maximum? Assume no any dissociation or association of acetic acid in the solvent. (a) Water (b) Ethanol (c) Benzene (d) Same in all solvents

Step-by-Step Solution

Answer

The mole fraction of acetic acid will be maximum in the solvent with the greatest molar mass because this lowers the total number of moles of solvent adding 100 g. Since the molar mass of benzene is highest among the given solvents, the mole fraction of acetic acid will be maximum in benzene.

Step 1: Calculate the molar mass of acetic acid

To find the mole fraction, first calculate the molar mass of acetic acid (CH3COOH). Molar mass is the sum of the atomic masses of all atoms present in the molecule. Carbon (C) has an atomic mass of approximately 12.01 g/mol, hydrogen (H) is about 1.01 g/mol, and oxygen (O) is about 16.00 g/mol. Thus, the molar mass of acetic acid is: 2*(12.01 g/mol) + 4*(1.01 g/mol) + 2*(16.00 g/mol) = 60.05 g/mol.

Step 2: Calculate moles of acetic acid

Using the molar mass calculated in Step 1, find the number of moles of acetic acid by dividing the mass of acetic acid by its molar mass: Number of moles = Mass of acetic acid / Molar mass of acetic acid = 10 g / 60.05 g/mol.

Step 3: Analyze mole fraction in each solvent

The mole fraction of acetic acid in a solution is given by the formula: mole fraction = moles of solute / (moles of solute + moles of solvent). Since we have 100 g of each solvent, we need to calculate the moles of each solvent using their respective molar masses.

Step 4: Calculate mole fraction for solution (a) - Water

The molar mass of water (H2O) is approximately 18.02 g/mol. Find the moles of water: 100 g / 18.02 g/mol. Then, calculate the mole fraction of acetic acid: mole fraction (a) = moles of acetic acid / (moles of acetic acid + moles of water).

Step 5: Calculate mole fraction for solution (b) - Ethanol

The molar mass of ethanol (C2H5OH) is about 46.07 g/mol. Find the moles of ethanol: 100 g / 46.07 g/mol. Calculate the mole fraction of acetic acid: mole fraction (b) = moles of acetic acid / (moles of acetic acid + moles of ethanol).

Step 6: Calculate mole fraction for solution (c) - Benzene

The molar mass of benzene (C6H6) is approximately 78.11 g/mol. Find the moles of benzene: 100 g / 78.11 g/mol. Calculate the mole fraction of acetic acid: mole fraction (c) = moles of acetic acid / (moles of acetic acid + moles of benzene).

Step 7: Conclusion - Compare mole fractions

Compare the mole fractions calculated for the solvents in Step 4, Step 5, and Step 6. The solvent that gives the highest mole fraction will be the one where acetic acid has the maximum mole fraction.

Key Concepts

Molar Mass and its Role in Determining Mole Fraction

The concept of molar mass is fundamental when dealing with mole fraction calculations. Molar mass, essentially, is the weight of one mole of a substance, typically measured in grams per mole (\text{g/mol}). One mole corresponds to Avogadro's number (\(6.022 \times 10^{23}\) entities) of atoms, molecules, or formula units of the substance.

Understanding the molar mass of acetic acid (\text{CH}_3\text{COOH}) is crucial because it allows us to convert the mass of acetic acid into moles, which is a required step in calculating its mole fraction in different solutions. The calculated molar mass is based on the atomic masses of carbon, hydrogen, and oxygen and their respective quantity in a molecule of acetic acid.

Solution Concentration and Mole Fraction

When discussing solution concentration, mole fraction is a comparative measure of the concentration of a particular component within a mixture. The mole fraction is the ratio of the number of moles of that component to the total number of moles of all components in the mixture.

To determine the mole fraction of acetic acid when it is dissolved in various solvents, it's important to also know the molar masses of the solvents. By calculating the moles of solvent, then adding it to the moles of acetic acid, we find the total number of moles in the solution. The mole fraction is thus calculated and assessed. It's this ratio that dictates the 'concentration' of the solute (acetic acid) in comparison to the solvent (such as water, ethanol, or benzene), indicating which solution has a stronger or weaker presence of acetic acid.

Chemical Solubility and Its Effect on Mole Fraction

Chemical solubility is an expression of the ability of a substance to dissolve in a solvent. It can impact mole fraction significantly, even though the exercise assumes no change in the state of acetic acid. In real-world scenarios, different solvents exhibit varying capabilities to dissolve the same solute due to molecular interactions.

However, since the solubility factor is not considered in the provided exercise, the mole fraction solely depends on the quantity and molar mass of the solvent. In cases where solubility does play a role, the amount of a substance that can dissolve in a given amount of a solvent would directly influence the mole fraction — the higher the solubility, the greater the potential mole fraction of the solute in that solvent, assuming equal amounts of solutes are being compared.