Question

How long will it take to produce \(1.00 \times 10^{3} \mathrm{kg}\) of magnesium metal by the electrolysis of molten magnesium chloride using a current of \(5.00 \times 10^{4} \mathrm{A} ?\)

Short answer

The electrolysis of molten magnesium chloride will take approximately 44.2 hours to produce \(1.00 \times 10^3 \mathrm{kg}\) of magnesium metal using a current of \(5.00 \times 10^4 \mathrm{A}\).

Step-by-step solution

Identify the relevant information in the problem

The relevant information given in the problem is 1. Mass of magnesium to be produced: \(1.00 \times 10^3 \mathrm{kg}\) 2. Current for electrolysis: \(5.00 \times 10^4 \mathrm{A}\) 3. Molar mass of magnesium: 24.31 g/mol (Note: Not given in the problem, but can be found in the periodic table.) 4. Charge of a magnesium ion: +2 (Note: Not given in the problem, but can be deduced from its position in the periodic table.)

Identify the relevant equation for electrolysis

Faraday's law of electrolysis states that: \(Q = n \times F\) where Q represents the charge in coulombs (C), n is the number of moles of electrons exchanged, F is Faraday's constant, which is equal to \(96485 \mathrm{C/mol}\).

Calculate the moles of magnesium and the required charge

To find the moles of magnesium produced, use the given mass and its molar mass: Moles of magnesium = \(\frac{1.00 \times 10^6 \mathrm{g}}{24.31 \mathrm{g/mol}} = 4.12 \times 10^4 \mathrm{mol}\) Each magnesium ion requires two moles of electrons to complete the reaction: Moles of electrons needed = \(2 \times 4.12 \times 10^4 \mathrm{mol} = 8.24 \times 10^4 \mathrm{mol}\) Using Faraday's law, calculate the required charge: \(Q = n \times F = 8.24 \times 10^4 \mathrm{mol} \times 96485 \mathrm{C/mol} = 7.95 \times 10^9 \mathrm {C}\)

Calculate the time required

Now, use the given current to find the time required for the electrolysis process. The relationship between charge, current, and time is given by: \(Q = I \times t\) where Q is the charge in coulombs (C), I is the current in amperes (A), and t is the time in seconds (s). Rearrange the equation to find the time: \(t =\frac{Q}{I} = \frac{7.95 \times 10^9 \mathrm{C}}{5.00 \times 10^4 \mathrm{A}} = 1.59 \times 10^5 \mathrm{s}\)

Convert the time to a more suitable unit

Convert the time from seconds to hours for easier interpretation: \(t = 1.59 \times 10^5 \mathrm{s} \times \frac{1 \mathrm{hour}}{3600 \mathrm{s}} = 44.2 \mathrm{hours}\) The electrolysis of molten magnesium chloride will take approximately 44.2 hours to produce 1.00 x 10^3 kg of magnesium metal using a current of 5.00 x 10^4 A.

Key concepts

Faraday's Law of Electrolysis

Faraday's Law of Electrolysis is vital for understanding the intricacies of how electrolysis works. Michael Faraday, a pioneering chemist and physicist, laid the groundwork for this law which revolutionized our understanding of electrochemistry. The law states that the amount of substance produced or consumed at an electrode during electrolysis is directly proportional to the quantity of electricity that passes through the electrolyte. This quantity is quantified by the charge in coulombs (C).

The law is usually articulated in mathematical form as: \(Q = n \times F\) where \(Q\) represents the total charge passed through the solution in coulombs, \(n\) is the number of moles of electrons transferred, and \(F\) is Faraday's constant, approximately \(96485 \mathrm{C/mol}\). This relationship demonstrates how the moles of electrons are correlated to the mass of substance altered at the electrode.
Understanding this law enables us to predict and calculate the outcomes of the electrolysis process, such as the mass of a substance produced given a certain amount of current passed over a period.

Practical Application in Electrolysis

When applying Faraday's Law to practical electrolysis, it's important to consider the valence of the ion being deposited or dissolved. For magnesium, with a valency of +2, every magnesium ion requires the transfer of two electrons to be deposited as a solid metal. This relationship is crucial when determining the total charge necessary to produce a desired amount of magnesium during electrolysis.

Molar Mass of Magnesium

The molar mass of a substance is one of the most fundamental concepts in chemistry, and it is particularly relevant when performing calculations related to electrolysis. Molar mass signifies the mass of one mole of a substance, and it's expressed in grams per mole (g/mol). For magnesium, the molar mass is approximately 24.31 g/mol. This value is crucial as it serves as a conversion factor between the mass of magnesium and the number of moles.

When engaged in calculations for electrolysis, knowing the molar mass allows for the conversion from the mass of magnesium produced (or desired) to moles, which can then be used in Faraday’s law calculations.

Link to Electrolysis Calculations

To lay it out more practically, suppose you are calculating how much magnesium you can produce with a given amount of charge—knowing the molar mass allows you to determine this amount by linking moles of electrons to the physical mass of magnesium deposited.

Calculating Electrolysis Time

Time is a significant factor in the process of electrolysis; specifically, calculating the time required to produce a certain amount of substance is a common task. This is where a comprehensive understanding of the relationship between current, charge, and time comes into play.

To calculate the time needed for magnesium to deposit through electrolysis, we manipulate the charge equation: \(Q = I \times t\). By rearranging the equation, time \(t\) can be expressed as \(t = \frac{Q}{I}\), with \(Q\) being the charge and \(I\) the current. Once you have calculated the total charge needed to produce the desired amount of magnesium, divide that by the current to find the time required.

Time Conversion

Often, it's beneficial to convert this time into hours or even days to gain a practical sense of the duration of the electrolysis process. This helps in planning and can impact the cost-efficiency and feasibility of the electrolysis setup.

Charge and Current Relationship

Charge and current have a direct and linear relationship in the context of electrolysis. Charge, measured in coulombs (C), is the total quantity of electricity conveyed, while current, measured in amperes (A), is the rate at which charge is being transferred. In essence, current is the flow of electric charge, and the relationship between the two is a cornerstone of understanding electrochemical processes.

The equation that describes this relationship is \(Q = I \times t\), where \(Q\) is the charge, \(I\) is the current, and \(t\) is time in seconds. This equation suggests that if you know the amount of charge required to produce a certain mass of a substance (as determined through Faraday's Law), you can adjust the current and time to meet the specific production requirements.

Adjusting Electrolysis Parameters

For practical applications, this means that by altering the current, you can affect the time it takes to complete the electrolysis process. A higher current will reduce the time needed; conversely, a lower current will increase it. Correctly balancing these elements is crucial for optimizing electrolysis in industrial and laboratory settings.