Question

Silver is electroplated at the cathode of an electrolysis cell by this half- reaction. $$ \mathrm{Ag}^{+}(a q)+\mathrm{e}^{-} \longrightarrow \mathrm{Ag}(\mathrm{s}) $$ How many moles of electrons are required to electroplate \(5.8 \mathrm{~g}\) of \(\mathrm{Ag}\) ?

Short answer

5.8 g of Ag requires approximately 0.0538 moles of electrons.

Step-by-step solution

Calculate moles of Ag

Use the molar mass of silver to convert grams of Ag to moles. The molar mass of Ag is approximately 107.9 g/mol. The formula to calculate moles is: \[ \text{moles} = \frac{\text{mass in grams}}{\text{molar mass}} \]

Analyze the electroplating reaction

For the given half-reaction: \[ \mathrm{Ag}^{+}(aq)+ \mathrm{e}^{-} \rightarrow \mathrm{Ag}(s) \] One mole of electrons (e-) reduces one mole of Ag+ ions to produce one mole of solid Ag.

Calculate moles of electrons needed

Since the reaction shows a 1:1 ratio between Ag+ ions and electrons, the moles of electrons will be equal to the moles of Ag calculated from the mass of silver electroplated.

Key concepts

Stoichiometry

Stoichiometry is a branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It helps us understand how much of each substance is needed and how much will be produced. In the case of electroplating silver, stoichiometry informs us of the simple 1:1 ratio between silver ions \(\mathrm{Ag^{+}}\) and electrons \(\mathrm{e^{-}}\) required for the deposition of solid silver \(\mathrm{Ag(s)}\). In practical terms, this means if we know the amount of silver we want to plate, we can directly calculate the necessary amount of electrons for the process.

For students tackling these problems, a useful tip is to always start with a balanced chemical equation and use it as a blueprint to figure out the relationships between different entities involved in the reaction.

Molar Mass

Molar mass is the mass of one mole of a given substance and is expressed in grams per mole \(\text{g/mol}\). It is a vital constant that connects the weight of a material to its stoichiometry in a reaction. For silver \(\text{Ag}\), the molar mass is approximately 107.9 \(\text{g/mol}\). In electroplating chemistry, understanding the molar mass allows us to convert the mass of silver (as given in the exercise) to moles, providing a link to the amount of electricity needed to carry out the electroplating process. Knowledge of the molar mass is a critical step in moles calculation and subsequently, in determining the stoichiometry of the electrochemical reaction.

Electrochemical Reactions

Electrochemical reactions involve the transfer of electrons from one substance to another and are driven by a device called an electrolytic cell. During electroplating, an electrochemical reaction occurs at the cathode where metal ions in solution gain electrons to form a metal coat on the cathode's surface. The equation \(\mathrm{Ag^{+}(aq) + e^{-} \rightarrow Ag(s)}\) represents a half-reaction that occurs at the cathode of this cell. The simplicity of this reaction indicates that each silver ion requires exactly one electron to be reduced to metallic silver. Understanding the nature of electrochemical reactions is key to grasping the relationship between the amount of electrons and the formation of the metal layer in the electroplating process.

Important for students to note is the conservation of charge in these reactions, which ensures that the number of electrons lost by one substance equals the number of electrons gained by another.

Moles Calculation

Moles calculation is essential in converting the mass of substances to a common unit that facilitates the application of stoichiometry in chemical equations. Through the formula \(\text{moles} = \frac{\text{mass in grams}}{\text{molar mass}}\), one can determine the number of moles from the mass of the substance and its molar mass. In the provided exercise, after finding the moles of silver, we use the stoichiometry of the reaction (1:1 ratio between \(\mathrm{Ag^{+}}\) and \(\mathrm{e^{-}}\)) to deduce that the moles of electrons needed for electroplating are equal to the moles of silver. Hence, in this context, moles calculation allows us to quantify the exact amount of charge in the form of electrons required to electroplate a specific mass of silver.

Students should remember to keep track of their units during calculations, ensuring that they cancel properly and leave the answer in moles, which is the desired unit for expressing the quantity in chemical reactions.