Question

When elemental copper is placed in a solution of silver nitrate, the following oxidation-reduction reaction takes place, forming elemental silver: $$\mathrm{Cu}(s)+2 \mathrm{AgNO}_{3}(a q) \rightarrow \mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2}(a q)+2 \mathrm{Ag}(s)$$ What mass of copper is required to remove all the silver from a silver nitrate solution containing 1.95 mg of silver nitrate?

Short answer

The mass of copper required to remove all the silver from a silver nitrate solution containing 1.95 mg of silver nitrate is approximately \(3.65 \times 10^{-4} \: g\).

Step-by-step solution

Write the balanced equation for the reaction

The balanced equation for the reaction is given as: \[\mathrm{Cu}(s)+2 \mathrm{AgNO}_{3}(a \: q) \rightarrow \mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2}(a \: q)+2 \mathrm{Ag}(s)\]

Calculate the moles of silver nitrate

To determine the mass of copper required, we first need to know the moles of silver nitrate in the solution. We are given the mass of silver nitrate as 1.95 mg, and we know the molar mass of silver nitrate is \(169.87 \: g/mol\). First, we will convert the mass from milligrams to grams and then find the moles of silver nitrate: \[\text{mass of silver nitrate} = 1.95 \: mg = 0.00195 \: g\] \[\text{moles of silver nitrate} = \frac{\text{mass of silver nitrate}}{\text{molar mass of silver nitrate}}= \frac{0.00195 \: g}{169.87 \: \frac{g}{mol}} = 1.148 \times 10^{-5} \: mol\]

Use stoichiometry to find the moles of copper

Next, we will use the stoichiometry of the balanced equation to find the moles of copper needed. From the balanced equation, we can see that 1 mole of copper reacts with 2 moles of silver nitrate. So, we can derive the ratio of moles of copper to moles of silver nitrate as follows: \[\frac{\text{moles of Cu}}{\text{moles of AgNO}_{3}} = \frac{1}{2}\] Now, we can determine the moles of copper needed by multiplying the moles of silver nitrate by the ratio: \[\text{moles of Cu} = \text{moles of AgNO}_{3} \times \frac{1}{2} = 1.148 \times 10^{-5} \: mol \times \frac{1}{2} = 5.74 \times 10^{-6} \: mol\]

Calculate the mass of copper required

Lastly, we will find the mass of copper needed using the moles of copper and its molar mass. The molar mass of copper is \(63.55 \: g/mol\). To find the mass of copper required, multiply the moles of copper by its molar mass: \[\text{mass of Cu} = \text{moles of Cu} \times \text{molar mass of Cu} = 5.74 \times 10^{-6} \: mol \times 63.55 \: \frac{g}{mol} = 3.65 \times 10^{-4} \: g\] The mass of copper required to remove all the silver from the silver nitrate solution containing 1.95 mg of silver nitrate is approximately \(3.65 \times 10^{-4} \: g\).

Key concepts

Stoichiometry

Understanding stoichiometry is essential for solving problems involving chemical reactions. In the context of the copper and silver nitrate reaction, stoichiometry is the quantitative relationship between reactants and products in a chemical equation. It can also be seen as the 'recipe' for a chemical reaction, telling us how much of each substance is involved.

Stoichiometry relies on the law of conservation of mass, which states that matter is neither created nor destroyed in a chemical reaction. Thus, when we balance the chemical equation, we ensure that the number of atoms for each element is the same on both sides of the equation. This balancing act is critical because it establishes the mole ratios between reactants and products, which are used to convert between masses of different substances, in our case, between copper and silver nitrate.

Stoichiometric Coefficients and Mole Ratios

In our exercise, stoichiometry comes into play after balancing the chemical equation. For instance, we deduced that one mole of copper reacts with two moles of silver nitrate, and this fixed ratio allows us to calculate the exact amount of copper needed to react with a given amount of silver nitrate. Without this understanding, it would be impossible to accurately determine the mass of copper required. Always remember that stoichiometry is the foundation for practically all quantitative aspects of chemistry.

Chemical Reaction Balancing

Balancing chemical reactions is critical for any stoichiometry-related problem. A balanced chemical equation follows the law of conservation of mass, ensuring that the same number of atoms of each element are present on both sides of the equation.

Let's break down the process, as seen in the given exercise. We started with the unbalanced formula for the reaction between copper and silver nitrate. By adjusting the coefficients—the numbers placed in front of compounds—in the reaction, we achieved a balanced equation that reflects the stoichiometry of the reaction. In this case, the chemical equation shows that one copper atom reacts with two molecules of silver nitrate to form one molecule of copper(II) nitrate and two atoms of silver.

Importance of a Balanced Equation

The importance of balancing lies in its utility for predicting amounts of reactants needed or products formed. When the equation isn't balanced, the stoichiometry won't yield correct results, leading to errors in empirical experimentation or theoretical calculations, like figuring out how much copper is needed to react with a certain mass of silver nitrate.

Molar Mass

Molar mass links the microscopic world of atoms and molecules to the macroscopic world we can measure. In chemical reactions, knowing the molar mass of each substance lets you convert between grams and moles, an essential step in stoichiometry.

The molar mass is the weight of one mole of a substance, usually expressed in grams per mole (g/mol). One mole contains Avogadro's number of particles (atoms, molecules, or ions), which is approximately 6.022 x 10^23. In the exercise regarding copper and silver nitrate, we've utilized the molar mass of silver nitrate (169.87 g/mol) to convert the given mass of silver nitrate in milligrams to moles. This step is fundamental because it is the moles that participate in the reaction, not the mass. Similarly, by knowing the molar mass of copper, we were able to find out the mass of copper required from the moles of copper calculated.

Interconversion between Mass and Moles

Understanding molar mass allows students to switch between the mass of a sample and the number of moles it represents. This interconversion is pivotal in calculating the mass of substances involved in chemical reactions, particularly when dealing with laboratory quantities and theoretical exercises.