Question

A solution of \(49.0 \% \mathrm{H}_{2} \mathrm{SO}_{4}\) by mass has a density of \(1.39 \mathrm{~g} / \mathrm{cm}^{3}\) at 293 K. \(A 25.0-\mathrm{cm}^{3}\) sample of this solution is mixed with enough water to increase the volume of the solution to \(99.8 \mathrm{~cm}^{3}\). Find the molarity of sulfuric acid in this solution.

Short answer

The molarity of sulfuric acid in the 99.8 cm³ solution is approximately 6.13 M.

Step-by-step solution

Calculate Mass of Sulfuric Acid in Sample

Determine the mass of the sulfuric acid in the original sample by using the percentage by mass and the sample's volume. The mass of H2SO4 (m_H2SO4) can be calculated using the density of the solution: mass of solution = density × volume. Then, find the mass of H2SO4 using the percentage by mass: m_H2SO4 = mass of solution × (49.0/100).

Calculate Total Mass of the Solution

The total mass of the diluted solution is equal to the mass of the original solution plus the mass of the added water. Since the added water will only affect the volume, not the mass, the total mass of the solution after dilution remains the same.

Find Moles of Sulfuric Acid

To determine the moles of sulfuric acid, use the molar mass of H2SO4 (98.079 g/mol). Moles of H2SO4 (n_H2SO4) can be calculated by dividing the mass of H2SO4 by its molar mass: n_H2SO4 = m_H2SO4 / Molar mass of H2SO4.

Calculate Molarity of the Solution

Molarity (M) is defined as the number of moles of solute per liter of solution. To calculate the molarity of H2SO4 in the diluted solution, divide the moles of H2SO4 by the final volume of the solution in liters: M = n_H2SO4 / final volume in liters.

Key concepts

Understanding Mass Percent Concentration

When dealing with solutions, the mass percent concentration is a commonly used way of expressing the concentration of a solute in a solution. It is defined as the mass of the solute divided by the total mass of the solution, multiplied by 100%. Simplified, it represents how many grams of the solute are present in 100 grams of the solution.

For instance, in the given exercise, a solution with a 49.0% mass percent concentration of sulfuric acid means that in every 100 grams of the solution, there are 49 grams of sulfuric acid. In practical terms, to calculate the mass of sulfuric acid in a specific sample, you simply apply this percentage to the mass of the sample which is obtained by multiplying its volume by its density.

Solution Density and Its Importance

Solution density is crucial when you need to convert between volume and mass, especially in chemistry problems like the one provided. Density is defined as the mass of substance per unit volume. In our exercise, the density of the sulfuric acid solution is given as 1.39 g/cm³. This information is key for converting the 25.0 cm³ of the solution to its mass, which is later used to determine the mass of the solute, sulfuric acid.

Knowledge of the solution's density also affects the calculation of the final concentration after dilution, as the mass of the solution doesn't change during the dilution process.

The Role of Molar Mass of Solute in Calculations

The molar mass of a solute is the weight of one mole of that substance, expressed in grams per mole (g/mol). In the case of our exercise, the molar mass of sulfuric acid (H2SO4) is 98.079 g/mol. Understanding the molar mass is fundamental because it allows the conversion of mass into moles, which is a step in calculating the molarity of a solution.

When given the mass of a solute, you find the number of moles by dividing its mass by the molar mass. This is an integral step in determining the molarity of the solution. The precision of molar mass values is essential in ensuring the accuracy of these types of calculations.

Dilution of Solutions and Concentration Changes

Dilution is the process of reducing the concentration of a solute in a solution, typically by adding more solvent. In the context of the given problem, we are adding water to the sulfuric acid solution. The essential point to understand is that the moles of solute remain constant during dilution; only the volume changes.

This is important when calculating the new molarity after dilution. The molarity, a measure of the concentration of the solution, is calculated by dividing the moles of solute by the final volume of the solution in liters. When you add water to the original solution, you increase its total volume and, by extension, decrease its concentration. Accurately calculating the final molarity requires an understanding of this inverse relationship between dilution and concentration.