Problem 84 - Chemistry Principles and Practice

Question

What mass of cadmium is deposited from a \(\mathrm{CdCl}_{2}\) solution by passing a current of \(1.50 \mathrm{~A}\) for 38.0 minutes?

Step-by-Step Solution

Answer

Approximately 1.99 grams of cadmium is deposited.

Step 1: Calculate total charge

The total charge (Q) that passes through the circuit can be calculated using the formula \( Q = I \times t \). First, convert time into seconds: \( 38.0 \times 60 = 2280 \) seconds. Then, use the current \( I = 1.50 \) A to find \( Q = 1.50 \times 2280 = 3420 \) coulombs.

Step 2: Determine the moles of electrons

Using Faraday's constant (approximately \( 96485 \) C/mol of electrons), calculate the moles of electrons: \[ \text{moles of electrons} = \frac{3420}{96485} \approx 0.0354 \text{ mol} \].

Step 3: Identify the electrochemical reaction

In the electrolysis of \( \mathrm{CdCl}_{2} \), each cadmium ion charge \( \mathrm{Cd}^{2+} \) is reduced to \( \mathrm{Cd} \) solid by gaining 2 electrons: \( \mathrm{Cd}^{2+} + 2 \mathrm{e}^{-} \rightarrow \mathrm{Cd} \).

Step 4: Calculate moles of cadmium

From the balanced reaction, it takes 2 moles of electrons to deposit 1 mole of \( \mathrm{Cd} \). Thus, \( \text{moles of Cd} = \frac{0.0354}{2} = 0.0177 \text{ mol} \).

Step 5: Convert moles of cadmium to mass

The molar mass of cadmium (Cd) is approximately \( 112.41 \text{ g/mol} \). Therefore, the mass of deposited cadmium is: \( 0.0177 \times 112.41 \approx 1.99 \text{ g} \).

Key Concepts

Faraday's Law of Electrolysis

Faraday's Law of Electrolysis helps us understand how much substance is deposited in an electrochemical cell. It states that the amount of substance that is deposited or dissolved at an electrode is directly proportional to the quantity of electric charge passed through the electrolyte. This means that more charge will result in a greater amount of substance deposited.
Faraday's Law is expressed through the formula:

• \[ m = \frac{Q}{F} \times \frac{M}{n} \]

Here, \(m\) is the mass of the substance deposited, \(Q\) is the total electric charge, \(F\) is Faraday's constant (\(96485\ \mathrm{C/mol}\)), \(M\) is the molar mass of the substance, and \(n\) is the number of electrons required to deposit one atom/ion of the substance.
Using this law, we can calculate the mass of cadmium deposited given the electric current and the time it was passed through the solution. By first determining the total charge (\(Q\)) with the formula \(Q = I \times t\), we convert the amount of charge into moles of electrons and then use stoichiometry to find the mass of cadmium deposited.

Molar Mass Calculation

Molar mass is a critical concept in chemistry that allows us to link the mass of a given substance to the number of moles it contains. It's expressed in grams per mole (g/mol). Knowing the molar mass enables us to perform calculations that involve converting between the quantities of mass and moles.
For example, cadmium in its elemental form has a molar mass of approximately \(112.41\ \text{g/mol}\). This value is essential when calculating the mass of cadmium deposited during electrolysis. Once we determine the number of moles of cadmium, we simply multiply by the molar mass to find the mass in grams.

• In practice, always check the periodic table for accurate molar masses as they might vary slightly.

The purpose of using molar mass in such calculations is to translate the moles of a substance (which are derived using Faraday's Law) into a measurable mass, which is often the form needed for practical applications.

Stoichiometry

Stoichiometry encompasses the quantitative relationships used to determine the masses, moles, and molecules of substances involved in a chemical reaction. This concept is crucial in translating chemical reactions into the language of mathematics, allowing predictions about the quantities of reactants and products.

• To understand stoichiometry in electrolysis, consider the reaction where cadmium ions \(\mathrm{Cd}^{2+}\) gain electrons to form cadmium metal: \[\mathrm{Cd}^{2+} + 2\mathrm{e}^{-} \rightarrow \mathrm{Cd}\]

Here, stoichiometry tells us that two moles of electrons are required to reduce one mole of cadmium ions to one mole of solid cadmium. This relationship is crucial when using Faraday's Law of Electrolysis to calculate the amount of cadmium deposited. By knowing the number of electrons involved, we can determine the equivalent amount of cadmium produced.
Stoichiometry thereby acts as the bridge between theoretical calculations and real-world chemical quantities, making it indispensable for tasks like determining the mass of elements deposited in electrochemical reactions.