It took \(150 . \mathrm{s}\) for a current of \(1.25 \mathrm{~A}\) to plate out \(0.109 \mathrm{~g}\) of \(\mathrm{a}\) metal from a solution containing its cations. Show that it is not possible for the cations to have a charge of \(1+\).

Electroplating involves depositing a layer of metal onto a conductive surface by passing an electric current through an electrolytic solution containing the metal's cations. This process is widely used for various applications, such as improving corrosion resistance, enhancing appearance, and increasing thickness of parts.

To understand electroplating, consider a piece of jewelry being coated with silver. The jewelry serves as the cathode (negative electrode) and a piece of silver as the anode (positive electrode). When the circuit is completed and the current flows through the solution, silver ions in the solution gain electrons (are reduced) and deposit onto the jewelry in a thin, uniform layer.

The thickness and quality of the plating depend on several factors, including the duration of electrolysis, the concentration of metal ions, the current's magnitude, and the solution's temperature and pH. By manipulating these conditions, craftsmen can ensure high-quality, durable coatings that meet specific aesthetic or functional requirements.

During electrolysis, electric current serves as the driving force that instigates chemical reactions at the electrodes. To better understand this, we can equate electric current to water flowing through a pipe; it is a stream of charged particles. In electrolysis, this stream consists of electrons that move from the anode to the cathode through an external circuit.

Faraday's laws of electrolysis govern the relationship between the current and the amount of substance that undergoes the electrochemical change. One key relationship defined by these laws is the amount of substance produced or consumed at each electrode in direct proportion to the quantity of electricity that passes through the electrolyte.

For example, the exercise demonstrates how we use the formula Q = I * t to calculate the charge in coulombs (Q), where I is the current in amperes, and t is time in seconds. This relationship enables us to estimate how much of a substance will be deposited (in the case of electroplating) or dissolved (in the case of metal purification), giving us a comprehensive insight into the process efficiency and the required operational conditions.

Molar mass calculation is a fundamental aspect of electrolysis since it relates the amount of substance deposited at an electrode to the quantity of electric charge passed through the electrolyte. According to Faraday's first law of electrolysis, the mass of the substance altered at an electrode is directly proportional to the quantity of electricity used. This is demonstrated by the equation m = (M * Q) / (n * F), where M denotes molar mass, and F is the Faraday constant, approximately 96485 C/mol.

In the given exercise, when attempting to determine the metal's molar mass based on the mass plated out, we see that Faraday's law helps to conclude that the cation's charge cannot be 1+ if the calculated molar mass exceeds the known molar masses of metals with 1+ charge.

Moreover, understanding how to reorganize this equation to isolate and solve for the molar mass (M) aids in analyzing electrochemical processes. This skill is crucial in various scientific fields, including metallurgy, electrochemistry, and materials science, where accurate measurements and conversions enable professionals to predict and control electrolytic reactions effectively.