Problem 87 - General Chemistry: Principles and Modern Applications

Question

To deposit exactly one mole of \(\mathrm{Ag}\) from an aqueous solution containing \(\mathrm{Ag}^{+}\) requires a quantity of electricity known as one faraday (F). The electrodeposition requires that each \(\mathrm{Ag}^{+}\) ion gain one electron to become an Ag atom. Use appropriate physical constants listed on the inside back cover to obtain a precise value of the Avogadro constant, \(N_{A}\).

Step-by-Step Solution

Answer

The Avogadro constant \(N_{A}\), the number of entities per mole, calculated from the Faraday's laws of electrolysis, is approximately \(6.022 × 10^{23} mol^{-1}\).

Step 1: Understand Faraday's laws of electrolysis

Faraday's first law of electrolysis states that the amount of material transformed at an electrode during electrolysis is directly proportional to the quantity of electricity that passes through the solution. One mole of electrons, known as one faraday (F), is equivalent to \(96485.3 C/mol\). This is the Faraday constant.

Step 2: Write the electrodeposition reaction for Ag

Silver is deposited from its aqueous solution according to the following reaction: \(Ag^{+} + e^{-} -> Ag\). In this reaction, each silver ion \(\mathrm{Ag}^{+}\) gains one electron to become a silver atom, Ag. Therefore, if we have 1 mol of silver ions, they will require 1 faraday of charge to be fully reduced to silver atoms.

Step 3: Calculate the Avogadro constant

We know that 1 faraday corresponds to 1 mol of electrons, and the charge of a faraday is given by the Faraday constant \(96485.3 C/mol\). Therefore, the number of electrons in one mole (also the Avogadro's number, \(N_{A}\)) would be given by \(N_{A} = F/e\), where e is the elementary charge, \(1.602176634 × 10^{-19} C\). Using these values, we find that \(N_{A} = 96485.3 C/mol / 1.602176634 × 10^{-19} C = 6.02214076 × 10^{23} mol^{-1}\).

Step 4: Summarize the results

So, the Avogadro constant, the number of entities per mole, is approximately \(6.022 × 10^{23} mol^{-1}\). One mole of silver ions requires one faraday (1 mole of electrons) to be deposited as silver atoms.

Key Concepts

Faraday's Laws of Electrolysis

Faraday's laws of electrolysis are fundamental principles that help us understand how materials are transformed during electrolysis. There are two main laws.

The **first law** states that the amount of a substance that is deposited or dissolved at an electrode is directly proportional to the amount of electricity that passes through the circuit. This means that the more electricity we use, the more material we will see change at the electrode.

- **Electricity** refers to the flow of electrons. - **Electrodes** are the sites where electrons enter or leave in the process.

• The second law is limited to elaborated explanations of proportionality with chemical equivalents per mole of electrons.

This law tells us that one mole of electrons corresponds to one 'faraday' of charge, equivalent to approximately 96485.3 coulombs.
These are vital for understanding reactions like that for silver deposition.

Electrodeposition of Silver

Electrodeposition is a fascinating process where a metal ion in a solution is reduced and deposited as a solid metal on an electrode. For silver, this process involves turning silver ions into silver atoms.

The chemical reaction involved in silver electrodeposition can be described as:
\[ \text{Ag}^{+} + \text{e}^{-} \rightarrow \text{Ag} \]

Here, a silver ion (\( \text{Ag}^+ \)) in the solution gains an electron (\( \text{e}^- \)) to become a silver atom (\( \text{Ag} \)). This reaction occurs on the electrode surface.

• The electrode acts as a platform where electrons facilitate the transformation of ions to solid metals.
• Only one electron is needed per silver ion to achieve this transformation.

This means that one mole of silver ions will use one mole of electrons, or one faraday of electric charge, to be fully converted into solid silver.

Faraday Constant

The Faraday constant is a crucial concept in electrochemistry and is essential for calculating the amount of substance being transformed in electrolysis. It is defined as the charge of one mole of electrons.

The value for the Faraday constant is approximately 96485.3 coulombs per mole (\( C/mol \)). This means:

• To deposit one mole of electrons on an electrode, 96485.3 coulombs of electric charge are needed.
• This one mole of electrons directly corresponds to the number of atoms transformed or deposited.

Using the Faraday constant allows precise calculations of other related constants, such as Avogadro's number, by connecting moles of substances with electric charge needed.

Elementary Charge

The elementary charge is a fundamental physical constant that signifies the electric charge of a single electron.

Given as \( 1.602176634 \times 10^{-19} \ C \), it bridges the gap between the macroscopic world of electrochemistry and the microscopic realm of individual particles.

• Each electron carries this minimal unit of charge.
• It is the building block measurement for understanding electricity on a molecular level.

When calculating the Avogadro constant (\( N_A \)), we use the elementary charge along with the Faraday constant.

For example, dividing the Faraday constant by the elementary charge results in Avogadro’s number (\( N_A = \frac{96485.3 \ C/mol}{1.602176634 \times 10^{-19} \ C} \)), leading us to approximately \( 6.022 \times 10^{23} \) entities per mole.