Question

A mixture contains only sodium chloride and potassium chloride. A \(0.1586-\mathrm{g}\) sample of the mixture was dissolved in water. It took \(22.90 \mathrm{~mL}\) of \(0.1000 \mathrm{M} \mathrm{AgNO}_{3}\) to completely precipitate all the chloride present. What is the composition (by mass percent) of the mixture?

Short answer

The composition of the mixture by mass percent is approximately 67.16% NaCl and 32.84% KCl.

Step-by-step solution

Calculate the moles of AgNO3 used

We are given the volume (22.90 mL) and the concentration (0.1000 M) of AgNO3. We can use these values to find the moles of AgNO3 used in the reaction. Moles of AgNO3 = Volume × Concentration Moles of AgNO3 = \(22.90 \times 10^{-3} L \times 0.1000 \frac{moles}{L}\) Moles of AgNO3 = \(2.29 \times 10^{-3} moles\)

Calculate the moles of Cl- ions and the total mass of Cl- ions

Since the AgNO3 reacts with chloride ions (Cl-) in a 1:1 ratio, the moles of Cl- ions in the sample are equal to the moles of AgNO3 used. Moles of Cl- ions = Moles of AgNO3 = \(2.29 \times 10^{-3} moles\) Now, we can calculate the total mass of Cl- ions in the sample using the molar mass of Cl (35.45 g/mol): Total mass of Cl- ions = Moles of Cl- ions × Molar mass of Cl = \(2.29 \times 10^{-3} moles \times 35.45 \frac{g}{mole}\) Total mass of Cl- ions = 0.0811 g

Calculate the mass of NaCl and KCl in the sample

Let x be the mass of sodium chloride (NaCl) in the sample. Then, the mass of potassium chloride (KCl) in the sample is (0.1586 - x). The moles of Cl- ions in NaCl = \(\frac{x}{58.44}\) The moles of Cl- ions in KCl = \(\frac{0.1586 - x}{74.55}\) Since the total moles of Cl- ions in the sample are equal to 2.29 x 10^{-3}, we can write the equation: \(\frac{x}{58.44} + \frac{0.1586 - x}{74.55} = 2.29 \times 10^{-3}\)

Solve for x and calculate the mass percent composition

Now, we will solve the equation for x: \(x (1/58.44 - 1/74.55) = 2.29 \times 10^{-3} - 0.1586/74.55\) \(x \approx 0.1065 g\) The mass of NaCl = 0.1065 g, and the mass of KCl = (0.1586 - 0.1065) = 0.0521 g. Now, we can calculate the mass percent composition of the mixture: Mass percent of NaCl = \(\frac{0.1065}{0.1586} \times 100\% \approx 67.16\%\) Mass percent of KCl = \(\frac{0.0521}{0.1586} \times 100\% \approx 32.84\%\) Thus, the composition of the mixture by mass percent is approximately 67.16% NaCl and 32.84% KCl.

Key concepts

Mole Concept

Understanding the mole concept is essential when dealing with chemical equations and reactions. A mole is an SI unit for measuring the amount of a substance. Analogous to a dozen representing 12 items, a mole represents approximately \(6.022 \times 10^{23}\) entities, whether atoms, molecules, ions, or electrons. This number is known as Avogadro's number.

In stoichiometry problems such as the one given, the mole concept allows us to convert between the mass of a substance and the amount of entities (atoms, ions, or molecules) it contains. The molar mass, which is the mass of one mole of that substance, is expressed in grams per mole (g/mol). For instance, when we calculated the moles of \(AgNO_3\) used, we used the volume and molarity to determine that there were \(2.29 \times 10^{-3}\) moles of \(AgNO_3\), subsequently allowing us to find the equivalent moles of chloride ions.

Chemical Titration

Chemical titration is a technique used to determine the concentration of a specific substance by reacting it with a solution of known concentration, commonly referred to as a titrant. In the example problem, a titration was performed where a solution of \(AgNO_3\) with a known molarity of 0.1000 M was added to the mixture of sodium chloride and potassium chloride until all chloride ions reacted, forming an insoluble precipitate.

By measuring the volume of \(AgNO_3\) used in the titration, we can deduce the amount of chloride ions present in the original sample. This process uses the stoichiometry of the reaction, which in this case is a 1:1 ratio of \(AgNO_3\) to \(Cl^-\) ions.

Mass Percent Composition

Mass percent composition is a way of expressing the concentration of an element in a compound or a component in a mixture. It is calculated by taking the mass of the specific component divided by the total mass of the mixture and then multiplying by 100 to get a percentage.

In the context of our exercise, we determined the mass of individual components, NaCl and KCl, using stoichiometry and then divided these masses by the total mass of the mixture to calculate their mass percent compositions. The calculations showed that the mixture was composed of approximately 67.16% NaCl and 32.84% KCl.

This concept is useful not only in lab settings but also in understanding the formulation of mixtures and compounds used in various industries, ranging from food production to pharmaceuticals.