Problem 87 - Chemistry and Chemical Reactivity

Question

Saccharin, an artificial sweetener, has the formula \(\mathrm{C}_{7} \mathrm{H}_{5} \mathrm{NO}_{3} \mathrm{S}\). Suppose you have a sample of a saccharin-containing sweetener with a mass of \(0.2140 \mathrm{g} .\) After decomposition to free the sulfur and convert it to the \(\mathrm{SO}_{4}^{2-}\) ion, the sulfate ion is trapped as water-insoluble \(\mathrm{BaSO}_{4}\) (Figure 4.4). The quantity of \(\mathrm{BaSO}_{4}\) obtained is 0.2070 g. What is the mass percent of saccharin in the sample of sweetener?

Step-by-Step Solution

Answer

The mass percent of saccharin in the sweetener sample is approximately 76.0%.

Step 1: Determine the molar mass of BaSO4

To solve this problem, start by determining the molar mass of \( \mathrm{BaSO}_{4} \). The atomic masses for the elements are: Ba = 137.33 g/mol, S = 32.07 g/mol, and O = 16.00 g/mol. Thus, the molar mass of \( \mathrm{BaSO}_{4} \) is: \[ \text{BaSO}_{4} = 137.33 + 32.07 + 4(16.00) = 233.39 \text{ g/mol} \]

Step 2: Calculate moles of BaSO4 produced

Next, calculate the moles of \( \mathrm{BaSO}_{4} \) using its mass. The mass of \( \mathrm{BaSO}_{4} \) obtained is given as 0.2070 g. Use the formula: \[ \text{moles of BaSO}_{4} = \frac{0.2070}{233.39} \approx 0.000887 \text{ moles} \]

Step 3: Determine moles of sulfur in BaSO4

Since each mole of \( \mathrm{BaSO}_{4} \) contains one mole of sulfur, the moles of sulfur in \( \mathrm{BaSO}_{4} \) are also 0.000887 moles.

Step 4: Calculate mass of sulfur in saccharin

To find the mass of sulfur in the original sample, use the moles of sulfur and its atomic mass (32.07 g/mol): \[ \text{mass of sulfur} = 0.000887 \times 32.07 \approx 0.02845 \text{ g} \]

Step 5: Calculate molar mass of saccharin

The molar mass of saccharin (\( \mathrm{C}_{7}\mathrm{H}_{5}\mathrm{NO}_{3}\mathrm{S} \)) is: \[ 7(12.01) + 5(1.008) + 14.01 + 3(16.00) + 32.07 = 183.18 \text{ g/mol} \]

Step 6: Calculate mass of saccharin in the sample

Given that sulfur accounts for 32.07 g of the \( 183.18 \) g/mol in saccharin, use the proportion to find the mass of saccharin: \[ \text{mass saccharin} = \frac{183.18}{32.07} \times 0.02845 \approx 0.1626 \text{ g} \]

Step 7: Calculate mass percent of saccharin in sample

Finally, calculate the mass percent of saccharin in the sweetener sample: \[ \text{Mass percent} = \left(\frac{0.1626}{0.2140} \right) \times 100 \approx 76.0 \% \]

Key Concepts

Molar Mass Calculation

Molar mass calculation is a fundamental concept in chemistry that helps to determine the mass of one mole of a substance. To calculate the molar mass, you sum the atomic masses of all the atoms in the molecule. This is crucial when you need to convert between the mass of a compound and the number of moles, which is often required in stoichiometric calculations.
Let's take the example of barium sulfate, \(\mathrm{BaSO}_4\). To find its molar mass, you add the atomic masses: barium (\(\mathrm{Ba}\)) is 137.33 g/mol, sulfur (\(\mathrm{S}\)) is 32.07 g/mol, and oxygen (\(\mathrm{O}\)), which has four atoms in the molecule, is \(4 \times 16.00 = 64.00\) g/mol. Summing these gives a molar mass for \(\mathrm{BaSO}_4\) of 233.39 g/mol.

• Find atomic masses of each element.
• Multiply by the number of each type of atom in the chemical formula.
• Sum these amounts to find the total molar mass.

Understanding molar mass is critical for translating between the macroscopic world of grams and the molecular world of moles, enabling accurate stoichiometric analyses.

Stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. It uses the principles of conservation of mass and the concept of moles to quantify substances' proportions in chemical reactions. In this exercise, stoichiometry helps determine how much sulfur is present in the \(\mathrm{BaSO}_4\) you obtained from the sweetener.
First, calculate the moles of \(\mathrm{BaSO}_4\) using the mass and its molar mass calculated previously. For a sample weighing 0.2070 g, you'd find:\[\text{moles of } \mathrm{BaSO}_4 = \frac{0.2070}{233.39} \approx 0.000887 \text{ moles}\]Since the formula \(\mathrm{BaSO}_4\) implies one mole of \(\mathrm{S}\) for each mole of \(\mathrm{BaSO}_4\), you also have 0.000887 moles of sulfur.

• Relate moles of reactants to moles of products.
• Ensure mole ratios reflect the coefficients in the chemical equations.

This connection is vital for finding how much of one substance will react with a given quantity of another, or what amount of product will result from specific amounts of reactants.

Mass Percent Calculation

The mass percent calculation is a way to express the concentration of an element within a compound or the compound's concentration within a mixture. In this exercise, you're asked to determine the mass percent of saccharin within a sweetener sample.
After finding the moles of sulfur and calculating the equivalent mass of saccharin using its molar mass, you can find the mass percent of saccharin. You found that \(0.1626 \text{ g}\) of saccharin was calculated from the sulfur content. To find the mass percent, use:

• Formula: \[\text{Mass percent} = \left(\frac{\text{mass of component}}{\text{total mass of mixture}}\right) \times 100\]
• Apply the values: \[\frac{0.1626}{0.2140} \times 100 \approx 76.0\text{\%}\]

Mass percent is an important way to quantify the concentration of a component within a mixture. It's especially useful in industries that need precise compositions of mixtures, such as food, pharmaceuticals, and materials.