Question

The concentrated sulfuric acid we use in the laboratory is 98.0 percent \(\mathrm{H}_{2} \mathrm{SO}_{4}\) by mass. Calculate the molality and molarity of the acid solution. The density of the solution is \(1.83 \mathrm{~g} / \mathrm{mL}\)

Short answer

Molality is 500 mol/kg, and molarity is 18.30 M.

Step-by-step solution

Define the Given Information

The given concentration is 98% \(\mathrm{H}_2\mathrm{SO}_4\) by mass. This means there are 98 grams of sulfuric acid in 100 grams of the solution. The density of the solution is \(1.83\, \mathrm{g/mL}\).

Calculate Moles of Sulfuric Acid

The molar mass of \(\mathrm{H}_2\mathrm{SO}_4\) is approximately 98.09 g/mol. Calculate the moles of sulfuric acid:\[\text{Moles of } \mathrm{H}_2\mathrm{SO}_4 = \frac{98\, \text{g}}{98.09\, \text{g/mol}} \approx 1\, \text{mol}\]

Calculate Mass of Solvent (Water)

Since the total mass of the solution is 100 g, the mass of the solvent (water) is:\[100\, \text{g} - 98\, \text{g} = 2\, \text{g}\]Convert this mass to kilograms:\[2\, \text{g} = 0.002\, \text{kg}\]

Calculate Molality

Molality (\(m\)) is calculated using the formula:\[m = \frac{\text{moles of solute}}{\text{kg of solvent}} = \frac{1\, \text{mol}}{0.002\, \text{kg}} = 500\, \text{mol/kg}\]Thus, the molality is 500 mol/kg.

Calculate Volume of Solution

The total mass of the solution is 100 g. Using the density, calculate the volume:\[\text{Volume} = \frac{100\, \text{g}}{1.83\, \text{g/mL}} \approx 54.64\, \text{mL}\]Convert to liters:\[54.64\, \text{mL} = 0.05464\, \text{L}\]

Calculate Molarity

Molarity (\(M\)) is calculated using the formula:\[M = \frac{\text{moles of solute}}{\text{liters of solution}} = \frac{1\, \text{mol}}{0.05464\, \text{L}} \approx 18.30\, \text{mol/L}\]Thus, the molarity is approximately 18.30 M.

Key concepts

Molality

Molality is a way of expressing the concentration of a solution, focusing on the ratio of solute to solvent by mass. Unlike molarity, which uses volume, molality is expressed in terms of the solvent's mass, so it does not change with temperature. This makes it a useful measurement in situations involving temperature fluctuations.

To calculate molality, use the formula:

• \( m = \frac{\text{moles of solute}}{\text{kg of solvent}} \)

In our exercise, we calculated the molality of sulfuric acid to be 500 mol/kg.
This involved determining that we have 1 mole of \( \mathrm{H}_2\mathrm{SO}_4 \) dissolved in 0.002 kg of water.

The molality gives us a clear idea of how concentrated the acid is in relation to the mass of the solvent, and it helps when we need precise calculations that are independent of temperature changes.

Molarity

Molarity is one of the most common ways to express chemical concentration.
It represents the number of moles of a solute per liter of solution. Molarity changes with temperature because it depends on volume, which can expand or contract with temperature changes.
To find molarity, follow the formula:

• \( M = \frac{\text{moles of solute}}{\text{liters of solution}} \)

In this case, the molarity of sulfuric acid was calculated to be approximately 18.30 M.
Such a high molarity indicates a concentrated solution, which is quite common in laboratory settings. This measure helps chemists know how many moles are dissolved in the solution, allowing them to scale reactions accordingly.

Density of Solutions

The density of a solution is an important property that links mass and volume. It's measured in units like grams per milliliter (g/mL) and is crucial for converting between mass-based and volume-based concentrations, like molality and molarity.

Density tells us how much mass is contained in a given volume. For concentrated sulfuric acid, the density is 1.83 g/mL.
Using this density, we calculated the volume of our 100 g solution, which was essential in determining the molarity.
Density helps bridge the gap between different concentration calculations and is especially useful when working with concentrated solutions, as it enables precise volume determinations from a known mass. Understanding density is key when conversions between mass and volume are required in chemical solutions.