You are given a solid that is a mixture of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) and \(\mathrm{K}_{2} \mathrm{SO}_{4}\) 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 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?

First, we have to find the molar mass of BaSO4. Using the periodic table, we find that the molar mass of BaSO4 is \(137.33 + (32.07 + 4(16.00)) = 233.43\) g/mol. Next, we have to find the number of moles of BaSO4 formed, which is given by \(\frac{\text{mass of BaSO4}}{\text{molar mass of BaSO4}}\). So, we have: Number of moles of BaSO4 = \(\frac{0.298 \,g}{233.43 \,g/mol} = 1.28 \times 10^{-3}\) mol. Since the stoichiometric ratio of BaSO4 to SO4^2- is 1:1, this means that the same number of moles of SO4^2- are present in the sample. Now, we find the mass of SO4^2- by using the molar mass of SO4^2-, which is given by \(32.07 + 4(16.00) = 96.07\) g/mol: Mass of SO4^2- = (number of moles of SO4^2-) × (molar mass of SO4^2-) = \(1.28 \times 10^{-3} mol \times 96.07 g/mol = 0.123\) g.

To find the mass percent of SO4^2- ion in the sample, we can use the formula: Mass percent of SO4^2- = \(\frac{\text{mass of SO4}^{2-}}{\text{total mass of sample}} \times 100\%\). Mass percent of SO4^2- = \(\frac{0.123\,g}{0.205\,g} \times 100\% = 60.0\%\)

We'll start by determining the mass of the Na2SO4 and K2SO4 part in the sample. Since the sample is comprised of only Na2SO4 and K2SO4, the mass of these parts is the difference between the total mass of the sample and the mass of SO4^2- ion: Mass of Na2SO4+K2SO4 = Total mass - Mass of SO4^2- = \(0.205\,g - 0.123\,g = 0.082\,g\) Now, we know that the mass percent of SO4^2- ion in the sample is 60.0%. Therefore, the mass percent of Na2SO4+K2SO4 is the remaining percentage: Mass percent of Na2SO4+K2SO4 = \(100\% - 60.0\% = 40.0\%\) Finally, we don't have enough information to further break down the percentage composition for Na2SO4 or K2SO4 individually. So, we can only report that the percent composition by mass of the mixture of Na2SO4 and K2SO4 is 40.0%. In summary: - The mass of SO4^2- ion in the sample is 0.123g - The mass percent of SO4^2- ion in the sample is 60.0% - The percent composition by mass of the mixture of Na2SO4 and K2SO4 in the sample is 40.0%