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Chapter 3: Problem 186

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

Short Answer

Expert verified
The mass percentage of potassium chloride (KCl) in the original mixture is \(32.80\%\).

Step by step solution

01

Determine the mass percentages of potassium in the compounds.

In order to solve this problem, we need to determine the mass percentages of potassium in both compounds. To do this, we can divide the molar mass of potassium (K) by the molar mass of each compound. Recall that the molar mass of potassium is \(39.10\ g/mol\). For KCl (Potassium chloride): \(KCl = 39.10\ g/mol + 35.45\ g/mol = 74.55\ g/mol \) Percentage of potassium in KCl: \(\dfrac{39.10}{74.55} × 100\% = 52.46\%\) For KNO3 (Potassium nitrate): \(KNO3 = 39.10\ g/mol + 14.01\ g/mol + 3(16.00\ g/mol) = 101.10\ g/mol \) Percentage of potassium in KNO3: \(\dfrac{39.10}{101.10} × 100\% = 38.68\%\)
02

Write a system of equations.

Now, we'll create a system of two equations using the given information. Let x represent the mass percentage of KCl, and let y represent the mass percentage of KNO3. The mass percentages will add up to 100% (100% of the mixture): Equation 1: \(x + y = 100\%\) We know that the mixture is 43.2% potassium. We can write this in terms of the potassium content of KCl and KNO3 as follows: Equation 2: \(0.5246x + 0.3868y = 43.20\)
03

Solve the system of equations.

In order to find the percentage of KCl in the mixture, we'll solve the system of equations. We can start by solving Equation 1 for x: \(x = 100 - y\) Now we can replace x in Equation 2 with the expression we found: \(0.5246(100 - y) + 0.3868y = 43.20\) This simplifies to: \(52.46 - 0.5246y + 0.3868y = 43.20\) Combining like terms: \(0.1378y = 9.26\) Finally, solve for y: \(y = \dfrac{9.26}{0.1378} = 67.20\%\) Now, substitute the value of y back into the equation for x: \(x = 100 - 67.20\% = 32.80\%\)
04

Interpret the result.

The mass percentage of KCl (potassium chloride) in the original mixture is 32.80%.

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