Question

You are given a solid that is a mixture of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) and \(\mathrm{K}_{2} \overline{\mathrm{SO}}_{4}\). A \(0.205-\mathrm{g}\) sample of the mixture is dissolved in water. An excess of an aqueous solution of \(\mathrm{BaCl}_{2}\) is added. The \(\mathrm{BaSO}_{4}\) that is formed is filtered, dried, and weighed. Its mass is \(0.298 \mathrm{~g}\). What mass of \(\mathrm{SO}_{4}{ }^{2-}\) ion is in the sample? What is the mass percent of \(\mathrm{SO}_{4}{ }^{2-}\) ion in the sample? What are the percent compositions by mass of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) and \(\mathrm{K}_{2} \mathrm{SO}_{4}\) in the sample?

Short answer

The mass of \(\mathrm{SO}_{4}{ }^{2-}\) ions in the sample is approximately \(0.1227\,g\), and the mass percent of \(\mathrm{SO}_{4}{ }^{2-}\) ions in the sample is approximately 59.85%. The percent compositions by mass of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) and \(\mathrm{K}_{2} \mathrm{SO}_{4}\) in the sample are also both approximately 59.85%.

Step-by-step solution

Determine mole ratio of \(\mathrm{BaSO}_{4}\) to \(\mathrm{SO}_{4}{ }^{2-}\)-ions

The balanced equation for the reaction of \(\mathrm{SO}_{4}{ }^{2-}\) ions with \(\mathrm{BaCl}_{2}\) to form \(\mathrm{BaSO}_{4}\) is: \[ \mathrm{SO}_{4}^{2-} + \mathrm{Ba}^{2+} (from\:\mathrm{BaCl}_2) ⟶ \mathrm{BaSO}_4 \] From this equation, we can see that 1 mole of \(\mathrm{BaSO}_{4}\) is produced from 1 mole of \(\mathrm{SO}_{4}{ }^{2-}\) ions.

Determine moles of \(\mathrm{BaSO}_{4}\) produced

First, we find moles of \(\mathrm{BaSO}_{4}\) in the precipitate: \[\mathrm{moles\ of\ BaSO}_{4} = \frac{mass}{molar\ mass}\] Given the mass of \(\mathrm{BaSO}_{4}\) as \(0.298\,g\), and the molar mass of \(\mathrm{BaSO}_{4}\) as \(137.327+32.07+4(16)=233.43\,g/mol\), \[\mathrm{moles\ of\ BaSO}_{4} = \frac{0.298\,g}{233.43\, g/mol}≈1.277×10^{-3}\: moles\]

Determine moles of \(\mathrm{SO}_{4}{ }^{2-}\) ions in the sample

Since the mole ratio of \(\mathrm{BaSO}_{4}\) to \(\mathrm{SO}_{4}{ }^{2-}\) is 1:1, we have the same number of moles of \(\mathrm{SO}_{4}{ }^{2-}\) ions in the sample as moles of \(\mathrm{BaSO}_{4}\) produced: \[ \mathrm{moles\ of\ SO_{4}^{2-}}= 1.277×10^{-3}\:moles \]

Calculate mass of \(\mathrm{SO}_{4}{ }^{2-}\) ions in the sample

Using the moles of \(\mathrm{SO}_{4}{ }^{2-}\) ions and the molar mass of \(\mathrm{SO}_{4}{ }^{2-}\), we can calculate its mass: \[ mass = \mathrm{moles} × \mathrm{molar\ mass} \] With the molar mass of \(\mathrm{SO}_{4}{ }^{2-}\) as \(32.07+4(16)=96.07\,g/mol\), \[ mass = (1.277×10^{-3}\:moles) × (96.07\,g/mol) \approx 0.1227\,g \] So, the mass of \(\mathrm{SO}_{4}{ }^{2-}\) ions in the sample is approximately \(0.1227\,g\).

Calculate mass percent of \(\mathrm{SO}_{4}{ }^{2-}\) ions in the sample

To find the mass percent of \(\mathrm{SO}_{4}{ }^{2-}\) ions in the sample, we use the following formula: \[ \mathrm{mass\ percent} = \frac{\mathrm{part}}{\mathrm{total}} × 100\% \] \[ \mathrm{mass\ percent\ of\ SO_{4}^{2-}\ ions\ in\ the\ sample} = \frac{0.1227\,g}{0.205\,g} × 100\% \approx 59.85\% \] Hence, the mass percent of \(\mathrm{SO}_{4}{ }^{2-}\) ions in the sample is approximately 59.85%.

Calculate percent compositions by mass of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) and \(\mathrm{K}_{2} \mathrm{SO}_{4}\) in the sample

Since we are given a mixture of \(\mathrm{Na}_{2}\mathrm{SO}_{4}\) and \(\mathrm{K}_{2}\overline{\mathrm{SO}}_{4}\) and the mass percent of \(\mathrm{SO}_{4}{ }^{2-}\) ions in the sample is approximately 59.85%, we can calculate the percent composition of the two salts by assuming that all the \(\mathrm{SO}_{4}{ }^{2-}\) ions come either from \(\mathrm{Na}_{2}\mathrm{SO}_{4}\) or from \(\mathrm{K}_{2}\overline{\mathrm{SO}}_{4}\). Since \(\mathrm{SO}_{4}{ }^{2-}\) ions are the same in both salts, the percent compositions by mass of \(\mathrm{Na}_{2}\mathrm{SO}_{4}\) and \(\mathrm{K}_{2}\overline{\mathrm{SO}}_{4}\) in the sample are also 59.85%. Hence, we have the following percent compositions by mass: - \(\mathrm{Na}_{2} \mathrm{SO}_{4} \approx 59.85 \%\) - \(\mathrm{K}_{2} \mathrm{SO}_{4} \approx 59.85 \%\)

Key concepts

Understanding Mole Ratio in Stoichiometry

The concept of mole ratio is crucial when delving into stoichiometry problems. It refers to the relationship between the amounts in moles of any two compounds involved in a chemical reaction. For instance, in the exercise where the reaction of \(\mathrm{SO}_{4}^{2-}\) ions with \(\mathrm{BaCl}_{2}\) forming \(\mathrm{BaSO}_{4}\) is given, the balanced chemical equation demonstrates a 1:1 mole ratio between the reactants and products.

This implies that for every mole of \(\mathrm{SO}_{4}^{2-}\) ions there is one mole of \(\mathrm{BaSO}_{4}\) produced. Grasping this concept is essential because it allows us to convert between moles of different substances in a chemical reaction, which is the foundational step in many stoichiometry calculations.

For students aiming to improve in solving these types of problems, it's important to remember that correctly identifying and using mole ratios is key to calculating the quantities of reactants or products in a chemical equation.

Gravimetric Analysis: A Quantitative Approach

Gravimetric analysis is a measurement method used in chemistry to determine the amount of an analyte based on mass. This technique was exemplified in our exercise where the solid sample is dissolved, reacted to form a precipitate, and the precipitate is then weighed. The weight of \(\mathrm{BaSO}_{4}\) precipitate allows us to calculate the moles of \(\mathrm{SO}_{4}^{2-}\) ions originally present in the sample.

By using a known molar mass of the precipitated compound, gravimetric analysis provides a direct measurement that can be very accurate. It's a powerful tool for students to understand because it doesn't rely on volume or concentration but on pure mass, making it less susceptible to error if carried out correctly. Practicing gravimetric analysis helps reinforce the relationship between mass, molar mass, and the number of moles in a substance.

Determining Percent Composition

Percent composition is a key concept when it comes to understanding the makeup of chemical compounds. It denotes the percent by mass of each element in a compound or a mix. In the provided exercise, calculating the percent composition by mass of \(\mathrm{SO}_{4}^{2-}\) ions in the sample involved dividing the mass of \(\mathrm{SO}_{4}^{2-}\) ions by the total mass of the sample and then multiplying by 100 to get a percentage.

In practical terms, knowledge of percent composition is invaluable for chemists when preparing solutions and for industries that need to create products with specific ratios of ingredients. For conceptual clarity and better problem-solving skills, students should practice determining percent composition since it not only applies to pure substances but also to solutions and mixtures, as illustrated in our exercise with a mixture of two different sulfates.