Question

A \(0.8870-\mathrm{g}\) sample of a mixture of \(\mathrm{NaCl}\) and \(\mathrm{KCl}\) is dissolved in water, and the solution is then treated with an excess of \(\mathrm{AgNO}_{3}\) to yield \(1.913 \mathrm{~g}\) of \(\mathrm{AgCl}\). Calculate the percent by mass of each compound in the mixture.

Short answer

\(\mathrm{NaCl}\) is 44.32% and \(\mathrm{KCl}\) is 55.59% of the mixture by mass.

Step-by-step solution

Write the Chemical Reactions

The reactions occurring are:1. \( \mathrm{NaCl} + \mathrm{AgNO}_3 \rightarrow \mathrm{AgCl} + \mathrm{NaNO}_3 \)2. \( \mathrm{KCl} + \mathrm{AgNO}_3 \rightarrow \mathrm{AgCl} + \mathrm{KNO}_3 \) These reactions indicate that both \( \mathrm{NaCl} \) and \( \mathrm{KCl} \) react with \( \mathrm{AgNO}_3 \) to produce \( \mathrm{AgCl} \).

Calculate Moles of AgCl Formed

Given that the mass of \( \mathrm{AgCl} \) is \(1.913 \mathrm{~g}\), we first find moles of \( \mathrm{AgCl} \).The molar mass of \( \mathrm{AgCl} \) is approximately \( 143.32 \mathrm{~g/mol} \).\[\text{Moles of } \mathrm{AgCl} = \frac{1.913 \mathrm{~g}}{143.32 \mathrm{~g/mol}} \approx 0.013347 \text{ mol}\]

Set Up Equations for Moles of NaCl and KCl

Let \(x\) be the moles of \(\mathrm{NaCl}\) and \(y\) be the moles of \(\mathrm{KCl}\).From the stoichiometry, we have:\[x + y = 0.013347\] (because each mole of NaCl or KCl gives one mole of AgCl)Also considering the mass of the mixture:\[ \text{For NaCl: } 58.44 \mathrm{~g/mol} \times x + \text{For KCl: } 74.55 \mathrm{~g/mol} \times y = 0.8870 \mathrm{~g} \]

Solve the System of Equations

We have the system of equations:\[ \begin{align*} x + y &= 0.013347 \58.44x + 74.55y &= 0.8870 \end{align*}\]Substitute \(y = 0.013347 - x\) into the second equation:\[58.44x + 74.55(0.013347 - x) = 0.8870\]Solve for \(x\):\[58.44x + 0.99546085 - 74.55x = 0.8870\]Rearrange:\[-16.11x = -0.10846085\]\[x \approx 0.00673 \]

Calculate Moles and Mass of KCl

Substitute \(x\) back into the equation for \(y\):\[y = 0.013347 - 0.00673 = 0.006617 \]Calculate the mass of \( \mathrm{NaCl} \) and \( \mathrm{KCl} \):\[\text{Mass of } \mathrm{NaCl} = 58.44 \times 0.00673 \approx 0.393\,\mathrm{g}\]\[\text{Mass of } \mathrm{KCl} = 74.55 \times 0.006617 \approx 0.493\,\mathrm{g} \]

Calculate Percent by Mass for Each Compound

Finally, calculate the percent by mass:\[\text{Percent of } \mathrm{NaCl} = \left( \frac{0.393}{0.8870} \right) \times 100 \approx 44.32\%\]\[\text{Percent of } \mathrm{KCl} = \left( \frac{0.493}{0.8870} \right) \times 100 \approx 55.59\% \]

Key concepts

Chemical Reactions

Chemical reactions are processes where substances known as reactants transform into new substances called products. In this exercise, the reactions involve common salts, sodium chloride (NaCl) and potassium chloride (KCl), reacting with silver nitrate (AgNO₃). Here are the equations for the reactions: - NaCl + AgNO₃ → AgCl + NaNO₃ - KCl + AgNO₃ → AgCl + KNO₃ These are examples of precipitation reactions where solid silver chloride (AgCl) is formed from the solution. Such reactions are crucial in chemistry for separating and analyzing compounds.

Moles Calculation

Calculating moles allows us to determine how much of a substance is involved in a reaction, based on its mass and molar mass. The mole is a basic unit in chemistry, used to express amounts of a chemical substance. In our problem, we calculate the moles of silver chloride (AgCl) produced. Given its mass is 1.913 grams and the molar mass is 143.32 g/mol: \[\text{Moles of AgCl} = \frac{1.913}{143.32} \approx 0.013347\text{ moles}\] This calculation is vital to understand how much reactant has been used or product formed, which is necessary for determining the composition of mixtures.

Percent Composition

Percent composition is used to express how much of each component is present in a compound or mixture. It helps us see the ratio of different substances after a chemical reaction. In this exercise, once we know the mass of NaCl and KCl in the mixture, we can find their percent composition by using the formula: \[\text{Percent by mass} = \left( \frac{\text{mass of component}}{\text{total mass of mixture}} \right) \times 100 \] - Percent of NaCl = \(\frac{0.393}{0.8870} \times 100 \approx 44.32\%\) - Percent of KCl = \(\frac{0.493}{0.8870} \times 100 \approx 55.59\%\) These percentages indicate the proportion of each salt in the original mixture.

Mass Calculation

In stoichiometry, mass calculation entails determining the mass of substances involved in or produced by chemical reactions. This provides insight into the proportions necessary for reactions to occur as described. From the previous steps, we calculated the mass of NaCl and KCl after determining the moles: - Mass of NaCl = 58.44 g/mol * 0.00673 mol \(\approx 0.393\text{ g}\) - Mass of KCl = 74.55 g/mol * 0.006617 mol \(\approx 0.493\text{ g}\) These masses summed up should equal the initial mass of the mixture, verifying our calculations and assisting in solving for the composition using known molar masses.