Problem 93 - General Chemistry: Principles and Modern Applications

Question

The density of a mixture of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) and water is 1.78 g/mL. The percent composition of the mixture is to be determined by converting \(\mathrm{H}_{2} \mathrm{SO}_{4}\) to \(\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4}\) If \(32.0 \mathrm{mL}\) of the mixture gives \(65.2 \mathrm{g}\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4}\) then what is the percent composition of the mixture?

Step-by-Step Solution

Answer

The percent composition of sulfuric acid in the mixture is calculated to be the ratio of the mass of the sulfuric acid to the total mass of the mixture, each calculated in their respective steps, and multiplied by 100.

Step 1: Calculate Mass of Sulfuric Acid from Mass of Ammonium Sulfate

Given that 65.2g of \((\mathrm{NH}_{4})_{2} \mathrm{SO}_{4}\) is obtained from the mixture, we can calculate the mass of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) from its molar mass and the molar mass of \((\mathrm{NH}_{4})_{2} \mathrm{SO}_{4}\). \(\mathrm{H}_{2} \mathrm{SO}_{4}\) and \((\mathrm{NH}_{4})_{2} \mathrm{SO}_{4}\) are equivalent in terms of moles of sulfur, as each molecule contains one atom of sulfur. Therefore, \(\mathrm{H}_{2} \mathrm{SO}_{4}\) (98.08 g/mol) : \((\mathrm{NH}_{4})_{2} \mathrm{SO}_{4}\) (132.14 g/mol) = 65.2g : x. Solve the proportion for \(x\), which is the mass of sulfuric acid.

Step 2: Calculate Total Mass of Mixture

Using the density of the mixture (1.78 g/mL) and the volume of the mixture (32.0 mL), we can calculate the total mass of the mixture. Multiplying the density by the volume will give the mass in grams.

Step 3: Calculate Percent Composition

Finally, the percent composition of sulfuric acid in the mixture can be calculated by dividing the mass of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) (calculated in step 1) by the total mass of the mixture (calculated in step 2) and then multiplying by 100 to convert it to percentage.

Key Concepts

Density Calculation

Density is a measure of how much mass is contained within a given volume. It allows us to determine the mass of a substance based on how much space it takes up. In this exercise, the density of the sulfuric acid and water mixture is given as 1.78 g/mL. This means that 1 milliliter of the mixture has a mass of 1.78 grams. To find the total mass of any volume of this mixture, you can multiply the volume by the density. For example, with a 32.0 mL volume, the mass calculation will be:

• Total Mass = Density x Volume = 1.78 g/mL x 32.0 mL
• Total Mass = 56.96 g

This calculation is crucial because it gives you the full mass of the mixture, which is needed to find the percent composition of the components within it.

Molar Mass

Molar mass is the mass of one mole of a specific substance and is expressed in grams per mole (g/mol). It is a crucial concept for converting between grams of a substance and moles of a substance. In this problem, we use the molar masses of sulfuric acid and ammonium sulfate to determine how much sulfuric acid is needed to produce the given amount of ammonium sulfate.

• The molar mass of \(\mathrm{H}_2 \mathrm{SO}_4\) is 98.08 g/mol.
• The molar mass of \(\left(\mathrm{NH}_4\right)_2 \mathrm{SO}_4\) is 132.14 g/mol.

Since each compound contains one sulfur atom, we can directly relate the mass of ammonium sulfate to the mass of sulfuric acid by using proportions. This proportional relationship allows us to calculate the mass of sulfuric acid from the mass of ammonium sulfate produced:

• Given: 65.2 g of \(\left(\mathrm{NH}_4\right)_2 \mathrm{SO}_4\)
• Mass of \(\mathrm{H}_2 \mathrm{SO}_4\) = \(\frac{65.2 \text{ g} \times 98.08 \text{ g/mol}}{132.14 \text{ g/mol}}\)
• This calculation gives the mass of sulfuric acid that was in the original mixture.

Conversion of Substances

Converting substances from one form to another is an essential skill in chemistry. It involves using given data to transition between different chemical compounds. In this exercise, we convert the mass of \(\left(\mathrm{NH}_4\right)_2 \mathrm{SO}_4\) back to \(\mathrm{H}_2 \mathrm{SO}_4\) by focusing on individual components like sulfur.
Initially, we start with the mass of ammonium sulfate and then use the proportional relationship with sulfuric acid to determine how much \(\mathrm{H}_2 \mathrm{SO}_4\) was present in the original mixture. This conversion is not about changing the substance physically but understanding how much of each component corresponds to equivalent quantities in chemical reactions.

• It involves calculating how much sulfuric acid would yield the equivalent moles of ammonium sulfate.
• Once the mass of \(\mathrm{H}_2 \mathrm{SO}_4\) is determined, it helps in finding the percent composition of the original mixture when related to the total mass.

This powerful tool helps chemists infer the original conditions and components in a sample just by understanding the final products and their relationships.