Question

Consider a mixture of potassium chloride and potassium nitrate that is \(43.2 \%\) potassium by mass. What is the percent KCl by mass of the original mixture?

Short answer

The mass percent of KCl in the original mixture is found to be 98.5% by setting up an equation based on the given mass percent of potassium and considering the molar masses of KCl and KNO3. The equation is solved for the mass percent of KCl (x) and then converted into a percentage.

Step-by-step solution

1. Express the mass of potassium in terms of mass of KCl and KNO3.

Begin by writing an equation for the mass of potassium (K) in the original mixture. This can be written in terms of the mass percent of KCl and KNO3. Let x represent the mass percent of KCl in the mixture, which means that (1-x) represents the mass percent of KNO3. The mass percent of potassium in KCl is given by the formula: \(\frac{(molar \; mass \; of \; K)}{(molar \; mass \; of \; KCl)} \times 100\). Similarly, the mass percent of potassium in KNO3 is given by the formula: \(\frac{(molar \; mass \; of \; K)}{(molar \; mass \; of \; KNO3)} \times 100\). The given mass percent of potassium in the mixture is 43.2%. We can now create an equation for the mass percent of potassium in the mixture using the mass percents of KCl and KNO3: \(43.2 = x \cdot \frac{(molar\; mass \; of\; K)}{(molar \; mass \; of\; K Cl)} +(1-x) \cdot \frac{(molar\; mass \; of\; K)}{(molar \; mass \; of\; KNO3)}\)

2. Calculate the molar masses of KCl and KNO3.

To solve the equation, we need to determine the molar masses of KCl and KNO3. The molar masses can be calculated by adding the molar masses of the individual elements. Here are the molar masses of each element: Potassium (K): 39.10 g/mol Chlorine (Cl): 35.45 g/mol Nitrogen (N): 14.01 g/mol Oxygen (O): 16.00 g/mol Now, we can calculate the molar mass of KCl and KNO3: Molar Mass of KCl: Potassium (K) + Chlorine (Cl) = 39.10 g/mol + 35.45 g/mol = 74.55 g/mol Molar Mass of KNO3: Potassium (K) + Nitrogen (N) + 3 * Oxygen (O) = 39.10 g/mol + 14.01 g/mol + 3 * 16.00 g/mol = 101.11 g/mol

3. Substitute the molar masses in the equation and solve for x.

Now that we have calculated the needed molar masses, we can substitute these values into the equation: 43.2 = x * \(\frac{39.10}{74.55}\) + (1-x) * \(\frac{39.10}{101.11}\) Next, solve for x by following these steps: 1. Simplify the equation by dividing the molar masses: 43.2 = x * 0.5244 + (1 - x) * 0.3868 2. Distribute the coefficients: 43.2 = 0.5244x + 0.3868 - 0.3868x 3. Combine the x terms 43.2 = 0.1376x + 0.3868 4. Subtract 0.3868 from both sides of the equation: 42.8132 = 0.1376x 5. Divide by 0.1376 to solve for x: x = 0.985

4. Convert the decimal value of x to a percentage and check the result.

Now we simply need to convert the decimal value of x (0.985) into a percentage to obtain the mass percent of KCl in the original mixture: Mass percent of KCl = 0.985 * 100 = 98.5 % Thus, the mass percent of KCl in the original mixture is 98.5%. Since this result is close to 100%, it makes sense considering the given mass percent of potassium in the mixture is 43.2%, which is close to the mass percent of potassium in KCl.

Key concepts

Understanding Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It allows us to predict the amounts of substances consumed and produced in a reaction based on the coefficients of a balanced chemical equation. In the context of calculating percent composition by mass, stoichiometry is vital as it uses the molar ratios of compounds to express how much of each element contributes to the total mass.

For example, when determining the percent composition of a mixture like potassium chloride (KCl) and potassium nitrate (KNO₃), we rely on stoichiometry to relate the mass of potassium to the total mass of each compound. Knowing the molar mass of each substance, which is the sum of the atomic masses of its constituent atoms, we can calculate the percentage of each element. The challenge in this particular exercise lies in dissecting the mixture into its components, something stoichiometry equips us to do accurately and efficiently.

Molecular Mass and Its Role in Quantitative Analysis

Molecular mass, sometimes referred to as molecular weight, is the sum of the atomic masses of the atoms in a molecule and is expressed in atomic mass units (u) or grams per mole (g/mol). It is a fundamental concept in chemistry that provides a bridge between the microscopic world of atoms and molecules and the macroscopic world that we can measure.

To calculate the molecular mass, add the atomic masses of each atom in the molecule. This step is essential when exploring the composition of chemical compounds. For instance, to find the percent composition of potassium in a KCl and KNO₃ mixture, we first need to know the molecular mass of each compound, as demonstrated in the exercise's solution. The molecular mass enables us to express the mass fraction of each element in a compound and leads to understanding the mass percentage of the compound in a mixture.

Chemical Mixture Analysis: Percent Composition

Chemical mixture analysis involves determining the composition and quantities of substances within a mixture. Percent composition by mass is one of the key measurements in this analysis. It can provide insights into the purity of a substance or the ratio of components in a mixture, such as the problem presented, where we determine the percentage of KCl in a mixture with KNO₃ based on the percentage of potassium.

Calculating percent composition requires an understanding of the chemical's formula and its molar mass. By dividing the mass of the specific component (in this case, potassium) by the total molar mass of the compound and multiplying by 100, we calculate the percent by mass of an element within a compound. In mixtures, we extend this concept to consider the contributions of each compound in the overall mass percentage, as shown in the provided exercise. This technique is commonly used in laboratories for formulation and quality control, ensuring that products meet certain specifications and standards.