Problem 44 - Chemistry and Chemical Reactivity

Question

In the electrolysis of a solution containing \(\mathrm{Ag}^{+}(\mathrm{aq})\) metallic Ag(s) deposits on the cathode. Using a current of 1.12 A for 2.40 hours, what mass of silver forms?

Step-by-Step Solution

Answer

10.81 g of silver forms.

Step 1: Understand the Electrolysis Equation

In electrolysis, silver ions \(\mathrm{Ag}^{+}\) gain electrons to form solid silver \(\mathrm{Ag(s)}\). The half-reaction is \(\mathrm{Ag}^{+} + e^{-} \rightarrow \mathrm{Ag(s)}\).

Step 2: Calculate Total Charge

Current (I) is related to charge (Q) and time (t) by \(Q = I \cdot t\). Here, \(I = 1.12\,\mathrm{A}\) and \(t = 2.40\,\text{hours} = 2.40 \times 3600\,\text{seconds}\). Calculate \(Q = 1.12 \times 2.40 \times 3600 = 9676.8\,\text{Coulombs}\).

Step 3: Relate Charge to Moles of Electrons

From Faraday's constant, 1 mole of electrons \((e^{-})\) corresponds to 96485 Coulombs. Calculate moles of electrons: \(\frac{9676.8\,\text{C}}{96485\,\text{C/mol}} = 0.1002\,\text{mol of } e^{-}\).

Step 4: Relate Moles of Electrons to Moles of Silver

From the half-reaction, 1 mole of electrons produces 1 mole of \(\mathrm{Ag}\). Therefore, 0.1002 moles of electrons produce 0.1002 moles of \(\mathrm{Ag}\).

Step 5: Calculate Mass of Silver

The molar mass of silver \(\mathrm{Ag}\) is 107.87 g/mol. Calculate the mass: \(0.1002\,\text{mol} \times 107.87 \text{g/mol} = 10.81\,\text{g}.\)

Key Concepts

Faraday's Constant

Faraday's constant is an important value in the field of electrochemistry. It represents the amount of electric charge carried by one mole of electrons. Specifically, the constant is valued at 96,485 Coulombs per mole (C/mol). This constant helps us bridge the gap between macroscopic measurements like current and time, with the microscopic world of moles of electrons.

Understanding Faraday's constant is essential for calculating the charge needed to complete electrochemical reactions. In layman's terms, it is like knowing how many electrons you need to "cash in" to cause a certain chemical change.

By using Faraday's constant, chemists can precisely calculate how much substance will be deposited, reacted, or evolved at an electrode during electrolysis. This is achieved by dividing the total charge passed through the solution by Faraday's constant to find the moles of electrons involved.

Cathode Deposition

Cathode deposition is a significant process in electrolysis where metal cations in a solution gain electrons and form solid metal on the electrode, known as the cathode. During the process, positively charged ions are attracted to the negatively charged cathode.

The equation \(\mathrm{Ag}^{+} + e^{-} \rightarrow \mathrm{Ag(s)}\) describes the deposition of silver ions onto the cathode. As the silver ions gain electrons, they convert from a dissolved state into solid metallic silver.

This reaction is specifically important in various industrial and research applications where pure metals need to be extracted or deposited. In our example, cathode deposition allows the transformation of silver ions into solid silver, resulting in a measurable mass of the metal, once the process is completed.

Electrochemistry

Electrochemistry is a branch of chemistry that studies the relationship between electricity and chemical changes. It plays a crucial role in everyday applications, such as batteries, corrosion, and electroplating.

Fundamental to electrochemistry is the system of oxidation-reduction (redox) reactions, where electrons are transferred between species. Electrolysis is a specific type of electrochemical process where an external electrical current causes a chemical reaction.

In the context of our exercise, during electrolysis, a current was applied to produce metallic silver from silver ions. This demonstrates how electrochemistry can be harnessed to effect a chemical change that would not occur without the input of electrical energy. Understanding electrochemistry allows scientists and engineers to innovate and improve the efficiency and effectiveness of such processes in technology and industry.

Moles of Electrons

Moles of electrons are a key concept when connecting electrical charge to chemical reactions. This concept leverages Avogadro's number to equate an electric charge with an amount of substance on the atomic scale.

To calculate the moles of electrons, the total charge is divided by Faraday's constant. This gives the quantity of electrons involved in the electrochemical process. For example, in our problem, we calculated that 9,676.8 Coulombs of charge corresponds to 0.1002 moles of electrons using Faraday's constant.

Understanding moles of electrons is vital for determining the amount of product formed during electrochemical reactions. It provides the link between electric measurements and the stoichiometry of chemical equations, allowing us to predict accurately how much of a substance will be produced or consumed.