Problem 92 - Chemistry The Central Science

Question

Metallic magnesium can be made by the electrolysis of molten MgCl. (a) What mass of Mg is formed by passing a current of 4.55 A through molten MgCl \(_{2},\) for 4.50 days? (b) How many minutes are needed to plate out 25.00 \(\mathrm{g} \mathrm{Mg}\) from molten \(\mathrm{MgCl}_{2}\) using 3.50 \(\mathrm{A}\) of current?

Step-by-Step Solution

Answer

In part (a), 222.54 g of magnesium is formed when a current of 4.55 A is passed through molten MgCl₂ for 4.50 days. In part (b), 944.20 minutes are needed to plate out 25.00 g of magnesium from molten MgCl₂ using a current of 3.50 A.

Step 1: Part (a) - Mass of Mg formed

Given current = 4.55 A and time = 4.50 days First, we should convert the time to seconds: \(4.50\ days \times 24\frac{hours}{day} \times 60\frac{min}{hour} \times 60\frac{sec}{min} = 388800\ sec\) Now we need to calculate the total charge passed in the electrolysis: \(Charge (Q) = Current (I) \times Time (t)\) \(Q = 4.55\ A \times 388800\ sec\) \(Q = 1769040\ C\) Now we will use Faraday's law: \(Amount\ of\ substance\ (Moles, n) = \frac{Charge\ (Q)}{Faraday\ constant\ (F) \times Valency\ of\ substance\ (z)}\) For magnesium, valency (z) = +2 Faraday constant (F) = 96485 C/mol \(n = \frac{1769040\ C}{96485\frac{C}{mol} \times 2}\) \(n \approx 9.162\ mol\) Now calculate the mass of magnesium produced: \(Mass = Amount\ of\ substance\ (n) \times Molar\ mass\ (M_{Mg})\) \(M_{Mg} = 24.305\ g/mol\) \(Mass = 9.162\ mol \times 24.305\frac{g}{mol}\) \(Mass \approx 222.54\ g\) Thus, 222.54 g of magnesium is formed.

Step 2: Part (b) - Time needed to plate out 25.00 g Mg

Given mass of Mg = 25.00 g, current = 3.50 A We need to find the moles of magnesium in 25.00 g of Mg: \(Amount\ of\ substance\ (n) = \frac{Mass}{Molar\ mass\ (M_{Mg})}\) \(n = \frac{25.00\ g}{24.305\frac{g}{mol}}\) \(n \approx 1.029\ mol\) Now, we need to find the total charge required for this amount: \(Charge\ (Q) = Amount\ of\ substance\ (n) \times Faraday\ constant\ (F) \times Valency\ of\ substance\ (z)\) For magnesium, valency (z) = +2 Faraday constant (F) = 96485 C/mol \(Q = 1.029\ mol \times 96485\frac{C}{mol} \times 2\) \(Q \approx 198281.69\ C\) Now we need to find the time required for this charge and current: \(Time\ (t) = \frac{Charge\ (Q)}{Current\ (I)}\) \(t = \frac{198281.69\ C}{3.50\ A}\) \(t = 56651.91\ sec\) Finally, convert this time to minutes: \(t = 56651.91\ sec \times \frac{1\ min}{60\ sec}\) \(t = 944.20\ min\) Thus, 944.20 minutes are needed to plate out 25.00 g of Mg.

Key Concepts

Understanding Faraday's Law of Electrolysis

Faraday's law of electrolysis is a fundamental principle that quantifies the relationship between the amount of substance produced at an electrode during electrolysis and the quantity of electricity that passes through the electrolyte. It states that the amount of a substance deposited or dissolved at an electrode is directly proportional to the total electric charge passed through the electrolyte.

More formally, we can express Faraday's law mathematically as \( n = \frac{Q}{F \times z} \), where:

• \( n \) is the amount of substance in moles.
• \( Q \) is the total electric charge in coulombs.
• \( F \) is Faraday's constant, approximately 96485 C/mol, representing the charge of one mole of electrons.
• \( z \) is the valency or number of electrons transferred per atom or ion of the substance.

Application in Electrolysis

In the electrolysis of MgCl₂, we calculate the mass of magnesium formed using the electric current and time to determine the total charge passed. Using Faraday’s law, we can then find the number of moles of magnesium that will be deposited at the cathode. The law provides us with a clear relationship between electric current and chemical change in an electrochemical reaction.

Molar Mass and Its Role in Calculations

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is a critical value for converting between the weight of a substance and the number of moles, therefore playing a vital role in stoichiometry and chemical calculations.

The molar mass of an element is numerically equivalent to its atomic mass and is determined by summing the average masses of the isotopes of the element weighted by their relative abundances. For compounds, the molar mass is the sum of the molar masses of all the atoms in the molecule.

• For example, magnesium (Mg) has a molar mass of 24.305 g/mol. This value is fundamental in connecting the electrochemical processing of magnesium to the mass of magnesium deposited through electrolysis.

Calculation in Electrolysis

During electrolysis, after finding the moles of magnesium produced, we multiply this value by the molar mass of magnesium to find the actual mass that precipitates. This conversion step is crucial because it links measurable physical quantities (mass) to abstract chemical quantities (moles), thus allowing us to correlate electrical input with chemical output.

Electrochemical Reactions in Electrolysis

Electrochemical reactions involve the transformation of chemical energy into electrical energy or vice versa. During the electrolysis process, electrical energy is used to provoke a chemical change, specifically, the decomposition of a compound into its elements.

In the case of magnesium chloride (MgCl₂), electrolysis breaks down the ionic compound into magnesium and chlorine gas through redox reactions that occur at the electrodes. Here's what happens at each electrode:

• Cathode (reduction): Magnesium ions (Mg²⁺) gain two electrons and are reduced to form solid magnesium (Mg).
• Anode (oxidation): Chloride ions (Cl^-) lose electrons and are oxidized to form chlorine gas (Cl₂).

This type of reaction is an electrochemical cell that requires an external source of current to occur, opposite to the spontaneous chemical reactions in a galvanic cell.

Calculation Aspect in Electrolysis

In our textbook's exercise, we determine how long we must apply a certain current (in the case of plating out magnesium), or we calculate the mass of magnesium that can be plated out from a given charge. Both scenarios utilize the principles of electrochemistry, combining Faraday's law, molar mass, and the stoichiometry of the electrochemical reactions to reach the final answer.