Question

Consider the reaction of \(19.0 \mathrm{~g}\) of zinc with excess silver nitrite to produce silver metal and zinc nitrite. The reaction is stopped before all the zinc metal has reacted and \(29.0 \mathrm{~g}\) of solid metal is present. Calculate the mass of each metal in the \(29.0-\mathrm{g}\) mixture.

Short answer

In the \(29.0-\mathrm{g}\) mixture, there are approximately \(9.387\, \mathrm{g}\) of zinc and \(19.613\, \mathrm{g}\) of silver.

Step-by-step solution

Write The Balanced Chemical Equation

The balanced chemical equation for the reaction between zinc (Zn) and silver nitrite (AgNO2) is: \[2 \mathrm{AgNO_2} + \mathrm{Zn} \rightarrow 2 \mathrm{Ag} + \mathrm{Zn(NO_2)_2}\]

Determine The Initial Number of Moles of Zinc

To determine the initial number of moles of zinc, we will first find the molar mass of Zn. The atomic mass of Zn is approximately 65.38 g/mol. Therefore, the initial number of moles of Zn is: \[\text{moles of Zn} = \frac{\text{mass of Zn}}{\text{molar mass of Zn}}\] \[\text{moles of Zn} = \frac{19.0 \mathrm{~g}}{65.38 \mathrm{~g/mol}}\] \[\text{moles of Zn} \approx 0.2906 \mathrm{~mol}\]

Determine The Final Mass of Zinc in the Mixture

If not all of the zinc has reacted, then the final mass of zinc in the \(29.0 \mathrm{~g}\) metal mixture will be the initial mass of zinc minus the mass of zinc that reacted. Let's denote the mass of reacted zinc as \(x\). Then the final mass of zinc will be: \[\text{final mass of Zn} = 19.0 \mathrm{~g} - x\] The moles of Zn that reacted can be found using the stoichiometry of the balanced equation: \[\text{moles of reacted Zn} = \frac{x}{65.38 \mathrm{~g/mol}}\] Since 2 moles of silver are produced for every mole of zinc that reacted: \[\text{moles of produced Ag} = 2 \times \frac{x}{65.38 \mathrm{~g/mol}}\] \[\text{mass of produced Ag} = \text{moles of produced Ag} \times \text{molar mass of Ag}\] \[\text{mass of produced Ag} = 2 \times \frac{x}{65.38 \mathrm{~g/mol}} \times 107.87 \mathrm{~g/mol}\] Using the conservation of mass: \[\text{mass of produced Ag} + \text{final mass of Zn} = 29.0 \mathrm{~g}\] \[2 \times \frac{x}{65.38 \mathrm{~g/mol}} \times 107.87 \mathrm{~g/mol} + 19.0 \mathrm{~g} - x = 29.0 \mathrm{~g}\] Now, we need to solve the equation for x value, the mass of the reacted Zn.

Solve the Equation and Determine the Mass of Silver in the Mixture

Solving for x, we get: \[x \approx 9.613\mathrm{~g}\] Now we can calculate the final mass of Zn: \[\text{final mass of Zn} = 19.0 \mathrm{~g} - 9.613 \mathrm{~g} \approx 9.387 \mathrm{~g}\] To find the mass of silver in the mixture, substitute the value of x back into the equation for mass of produced Ag: \[\text{mass of produced Ag} = 2 \times \frac{9.613 \mathrm{~g}}{65.38 \mathrm{~g/mol}} \times 107.87 \mathrm{~g/mol} \approx 19.613 \mathrm{~g}\] So, in the \(29.0-\mathrm{g}\) mixture, there are approximately \(9.387\, \mathrm{g}\) of zinc and \(19.613\, \mathrm{g}\) of silver.

Key concepts

Balanced Chemical Equation

A balanced chemical equation is fundamental to understanding chemical reactions. It represents the relationship between reactants and products in terms of their moles. In our exercise, the reaction involves zinc (Zn) and silver nitrite (AgNO₂). The balanced chemical equation is:
\[2 ext{AgNO}_2 + ext{Zn} ightarrow 2 ext{Ag} + ext{Zn(NO}_2 ext{)}_2\]
This equation shows how two molecules of silver nitrite react with one atom of zinc to produce two atoms of silver and one molecule of zinc nitrite. Balancing equations ensures that the number of atoms for each element is maintained on both the reactant and product sides, highlighting the conservation of mass in the reaction.

Molar Mass Calculation

Understanding molar mass is essential for converting between mass and moles of a substance. The molar mass is defined as the mass of one mole of a substance and is usually expressed in grams per mole (g/mol). For Zinc (Zn), the molar mass is approximately 65.38 g/mol.
To calculate the number of moles of zinc, use the formula: \[\text{Moles of Zn} = \frac{\text{Mass of Zn}}{\text{Molar Mass of Zn}} = \frac{19.0 \, \text{g}}{65.38 \, \text{g/mol}} \approx 0.2906 \, \text{mol}\]
This calculation converts the given mass into moles, to determine how much zinc is initially involved in the reaction.

Stoichiometric Coefficients

Stoichiometric coefficients are the numbers placed in front of compounds in a balanced chemical equation. They indicate the proportion of moles of each substance involved in the reaction. In our balanced equation:
\[2 ext{AgNO}_2 + ext{Zn} ightarrow 2 ext{Ag} + ext{Zn(NO}_2 ext{)}_2\]
The coefficient '2' in front of both AgNO₂ and Ag indicates that two moles of silver nitrite yield two moles of silver. Similarly, one mole of zinc and one mole of zinc nitrite each appear in the reaction.

• These coefficients are crucial for calculating the amounts of products formed from given reactants.
• They help in determining the theoretical yield of a reaction.

This relationship allows us to understand how reactants are converted into products scientifically.

Conservation of Mass

The principle of conservation of mass states that mass is neither created nor destroyed in a chemical reaction. This idea is fundamental when writing and balancing chemical equations, ensuring that the mass of reactants equals the mass of products.
In the exercise given: * Mass of reactants = Mass of products
This is illustrated as the initial and final masses of zinc and silver being calculated and verified. The balanced equation provided in the exercise serves as a tool to apply this principle effectively, making sure that the mass of zinc (19.0 g) and the mass formed of silver and residual zinc accounts for the total observed mass in the reaction process. Using the equation:\[2 \times \frac{x}{65.38 \, \text{g/mol}} \times 107.87 \, \text{g/mol} + 19.0 \, \text{g} - x = 29.0 \, \text{g}\]
By solving this, we adhere to the law of conservation, ensuring the calculation reflects the mass truthfully.