Problem 72 - Chemistry and Chemical Reactivity

Question

An old method of measuring the current flowing in a circuit was to use a "silver coulometer." The current passed first through a solution of \(\mathrm{Ag}^{+}(\mathrm{aq})\) and then into another solution containing an electroactive species. The amount of silver metal deposited at the cathode was weighed. From the mass of silver, the number of atoms of silver was calculated. since the reduction of a silver ion requires one electron, this value equalled the number of electrons passing through the circuit. If the time was noted, the average current could be calculated. If, in such an experiment, \(0.052 \mathrm{g}\) of \(\mathrm{Ag}\) is deposited during \(450 \mathrm{s}\), what was the current flowing in the circuit?

Step-by-Step Solution

Answer

The current flowing in the circuit was approximately 0.103 A.

Step 1: Determine Moles of Silver Deposited

First, calculate the number of moles of silver deposited. Use the relation \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \). The molar mass of silver (Ag) is approximately 107.87 g/mol. Therefore, the moles of Ag deposited are: \( \frac{0.052 \text{ g}}{107.87 \text{ g/mol}} = 4.82 \times 10^{-4} \text{ mol} \).

Step 2: Calculate Number of Electrons Passed Through the Circuit

Since the reduction of one silver ion \((\mathrm{Ag}^+)\) to silver metal \((\mathrm{Ag})\) requires one electron, the number of moles of electrons is equal to the moles of silver deposited. Thus, \( 4.82 \times 10^{-4} \text{ mol} \) of electrons have passed through the circuit.

Step 3: Convert Moles of Electrons to Charge (Coulombs)

Use Faraday's constant, which is approximately 96,485 C/mol, to convert the moles of electrons to total charge. The total charge is \( 4.82 \times 10^{-4} \text{ mol} \times 96485 \text{ C/mol} = 46.5 \text{ C} \).

Step 4: Calculate the Current

Current \( I \) is defined as the charge \( Q \) flowing through the circuit per unit time \( t \). Use the formula \( I = \frac{Q}{t} \), where \( Q = 46.5 \text{ C} \) and \( t = 450 \text{ s} \). Thus, \( I = \frac{46.5 \text{ C}}{450 \text{ s}} = 0.103 \text{ A} \).

Key Concepts

Silver Coulometer

A silver coulometer is a classic device used in electrochemistry to measure the current flowing through an electrical circuit. It operates by depositing a specific metal, in this case, silver, onto an electrode, which is usually the cathode. As electrons flow through the circuit, silver ions in solution gain these electrons and form metallic silver. The amount of silver deposited directly relates to the number of electrons that have passed through the circuit. This number of electrons can be quantified by weighing the deposited silver. This concept utilizes the principle that a precise number of electrons corresponds to a specific amount of a substance, as dictated by Faraday's laws of electrolysis. This method is particularly beneficial for understanding the underlying mechanics of electrochemical reactions, which are crucial in battery technology and other fields of electrochemistry.

Current Calculation

Calculating the current in a circuit involves understanding how charge and time relate. Current, often denoted by the symbol 'I', represents the flow of electric charge over a given period of time, and it is measured in amperes (A). The foundational formula for current is \( I = \frac{Q}{t} \), where \(Q\) represents the total charge in coulombs and \(t\) is time in seconds.
In the context of a silver coulometer, the charge is determined by the amount of silver deposited. First, you calculate the moles of silver, then use Faraday's constant to convert these moles into coulombs of charge. After the total charge is found, it is divided by the total time during which the deposition occurred to find the average current. This method not only gives insight into the electrical properties of the circuit but also enhances our understanding of electrochemical processes.

Electrochemistry

Electrochemistry is a branch of chemistry that focuses on the relationship between electrical energy and chemical changes. It involves the study of reactions where electrons are transferred between molecules, often with the use of an external circuit. Key topics in electrochemistry include:

• Electrode reactions - where oxidation and reduction occur.
• Electrolytes - substances that conduct electricity when dissolved in water.
• Electroplating - the process of depositing a metal onto a surface using an electric current.
• Batteries - devices that convert stored chemical energy into electrical energy.

Overall, understanding these concepts is essential not only for academics studying chemical processes but also for industries working with batteries, corrosion prevention, and electrolysis.

Faraday's Constant

Faraday's constant is a vital number in electrochemistry that relates the amount of charge carried by one mole of electrons. Its approximate value is 96,485 coulombs per mole (C/mol). This constant allows chemists to convert between moles of electrons and the charge in coulombs. For instance, in a silver coulometer experiment, to find out how many coulombs correspond to a certain mass of deposited silver, Faraday's constant is used. You multiply the number of moles of electrons by 96,485 C/mol to obtain the charge in coulombs.
Faraday's constant is named after Michael Faraday, a pioneering scientist in the field of electromagnetism and electrochemistry, and it remains a cornerstone for calculating and understanding the quantitative aspects of electrochemical reactions.