Problem 176 - Chemistry An Atoms First Approach

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?

Step-by-Step Solution

Answer

The original mixture is 100% potassium chloride (KCl) by mass. This is determined by setting up an equation using the given mass composition of potassium in the mixture (43.2%) and the mass composition of potassium in both KCl and KNO3. Solving for the mass fraction of KCl in the mixture gives a value of 1, indicating that the entire mixture is composed of KCl.

Step 1: Determine mass composition of potassium in KCl and KNO3

First, calculate the mass composition of potassium in each compound: For potassium chloride (KCl): - Potassium (K) has a molar mass of 39.10 g/mol - Chlorine (Cl) has a molar mass of 35.45 g/mol - The molar mass of KCl = 39.10 g/mol (K) + 35.45 g/mol (Cl) = 74.55 g/mol The mass composition of potassium in KCl is \(\frac{39.10}{74.55}\) or 0.5244 (52.44%) For potassium nitrate (KNO3): - Potassium (K) has a molar mass of 39.10 g/mol - Nitrogen (N) has a molar mass of 14.01 g/mol - Oxygen (O) has a molar mass of 16.00 g/mol (and there are 3 O atoms in KNO3) - The molar mass of KNO3 = 39.10 g/mol (K) + 14.01 g/mol (N) + 3(16.00 g/mol) (O) = 101.11 g/mol The mass composition of potassium in KNO3 is \(\frac{39.10}{101.11}\) or 0.3866 (38.66%)

Step 2: Set up and solve the equation to find mass fraction of KCl

Let x be the mass fraction of KCl in the mixture. Then the mass fraction of KNO3 in the mixture is (1-x). We can set up the following equation using the given mass composition of potassium in the mixture (43.2%) and the mass composition of potassium in both compounds: \(0.432 = (0.5244)(x) + (0.3866)(1-x)\) Now we can solve for x: \(0.432 = 0.5244x + 0.3866 - 0.3866x\) \(0.0454x = 0.0454\) \(x = 1\) So, the mass fraction of KCl in the mixture is 1 (100%).

Step 3: Calculate the percent of KCl by mass in the original mixture

The mass fraction of KCl in the mixture is 1, which means the entire mixture is composed of KCl. So, the percent of KCl by mass in the original mixture is: Percent KCl by mass = (mass fraction of KCl) × 100 = 1 × 100 = 100% Therefore, the original mixture is 100% potassium chloride (KCl) by mass.