Question

A sulfuric acid solution containing \(571.6 \mathrm{~g}\) of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) per liter of solution has a density of \(1.329 \mathrm{~g} / \mathrm{cm}^{3} .\) Calculate (a) the mass percentage, (b) the mole fraction, (c) the molality, (d) the molarity of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) in this solution.

Short answer

The mass percentage of H2SO4 in the solution is \(43.00\% \), the mole fraction is \(0.121\), the molality is \(7.68 \: \text{mol/kg}\), and the molarity is \(5.82 \: \text{mol/L}\).

Step-by-step solution

Calculate the mass of the solution

We are given the density of the solution (1.329 g/cm³) and the mass of H2SO4 per liter (571.6 g). We can use this information to find the mass of the solution. Density (ρ) is defined as mass (m) divided by volume (V), which can be written as: ρ = m/V The volume of the solution is 1 L, which is equal to 1000 cm³. Using the density, we can calculate the mass of the solution: m = ρ * V = 1.329 g/cm³ * 1000 cm³ = 1329 g

Calculate the mass of the solvent

Now we need to find the mass of the solvent (water) in the solution. The mass of the solution is the sum of the mass of the solute (H2SO4) and the mass of the solvent (water). Using the given mass of H2SO4 (571.6 g), we can calculate the mass of water: mass of water = mass of solution - mass of solute = 1329 g - 571.6 g = 757.4 g

Calculate the number of moles of H2SO4

In order to calculate the mole fraction, molality, and molarity, we first need to establish the number of moles of H2SO4 in the solution. The molecular weight of H2SO4 can be found by adding the atomic masses of its constituent elements: H: 1 g/mol (2 atoms) => 2 g/mol S: 32.1 g/mol (1 atom) => 32.1 g/mol O: 16 g/mol (4 atoms) => 64 g/mol Molecular weight of H2SO4: 2 + 32.1 + 64 = 98.1 g/mol Now we can calculate the number of moles (n) of H2SO4: n = mass of solute (H2SO4) / molecular weight (H2SO4) = 571.6 g / 98.1 g/mol ≈ 5.82 mol

Calculate the mass percentage, mole fraction, molality, and molarity

(a) Mass percentage: mass percentage = (mass of solute / mass of solution) * 100 = (571.6 g / 1329 g) * 100 ≈ 43.00% (b) Mole fraction: First, we calculate the number of moles of water: Molecular weight of water (H2O) = 18.015 g/mol number of moles of water = mass of water / molecular weight of water = 757.4 g / 18.015 g/mol ≈ 42.07 mol Now calculate the mole fraction of H2SO4 (x_H2SO4): x_H2SO4 = moles of H2SO4 / (moles of H2SO4 + moles of water) = 5.82 mol / (5.82 mol + 42.07 mol) ≈ 0.121 (c) Molality (mol/kg): molality = moles of solute / mass of solvent (in kg) molality = 5.82 mol / (757.4 g * 1 kg / 1000 g) ≈ 7.68 mol/kg (d) Molarity (mol/L): molarity = moles of solute / volume of solution (in L) molarity = 5.82 mol / 1 L = 5.82 mol/L The final results are: (a) Mass percentage: 43.00% (b) Mole fraction: 0.121 (c) Molality: 7.68 mol/kg (d) Molarity: 5.82 mol/L

Key concepts

Mass Percentage

The mass percentage, often referred to as weight percent (w/w%), is a measure of concentration that indicates how much solute is contained in a solution relative to the total mass of the solution. It is calculated by taking the mass of the solute and dividing it by the total mass of the solution, then multiplying the result by 100 to get a percentage. This concept is critical for understanding the composition of solutions in chemistry. For example, if a solution has an overall mass of 100 grams and contains 5 grams of solute, its mass percentage would be \( 5/100 \times 100\text{%} = 5\text{%} \). Moreover, to improve student understanding, visual tools such as pie charts representing the mass distribution can be useful.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture or solution. It is defined as the number of moles of that component divided by the total number of moles of all substances present. Unlike mass percentage, mole fraction is a ratio without units and gives us a sense of proportionality between the components.For a better grasp of this concept, imagine you have a bag containing different colored balls. If 2 out of 10 balls are red, the mole fraction of red balls would be \( \frac{2}{10} = 0.2 \). This concept is widely used when dealing with mixtures in gas and solution phases. Making comparisons to everyday examples can often clarify the concept for learners who struggle with abstract ideas.

Molality

Molality, often symbolized as 'm', is a measure of the concentration of a solute in a solution, specifically in terms of the amount of substance in a given mass of solvent. It's defined as the number of moles of solute per kilogram of solvent. The formula is written as\( molality = \frac{n_{solute}}{mass_{solvent\text{(kg)}}} \).Molality is particularly useful in calculating changes in boiling points and freezing points of solutions (colligative properties) because it is not affected by temperature changes, unlike molarity. For illustration, imagine mixing exactly 1 mole of sugar (assuming it’s about 342 grams for sucrose) into 1 kilogram of water; this is a 1 molal sugar solution. Introducing analogies with cooking recipes, where specific amounts of ingredients (like a cup of sugar) are mixed into a certain mass of another ingredient (like dough), can sometimes make molality easier to understand.

Molarity

Molarity, represented by the symbol 'M', is one of the most common units of concentration in chemistry. It measures how many moles of a solute are present in a liter of solution. The formula for calculating molarity is\( molarity = \frac{n_{solute}}{V_{solution\text{(L)}}} \).When we talk about a 1 M solution, it means there's 1 mole of the solute dissolved per liter of the entire solution, which includes both the solute and the solvent it's dissolved in. To contextualize molarity, consider a pitcher of lemonade: if you dissolve the powder to make 2 moles of lemonade in 2 liters of water, you've created a 1 M lemonade solution. For students, understanding that molarity is akin to the concentration or 'strength' of the solution helps them relate to real-life scenarios, such as determining the potency of a medicinal drug in a solution.