Problem 92 - Chemistry: The Central Science

Question

Elemental calcium is produced by the electrolysis of molten \(\mathrm{CaCl}_{2}\). (a) What mass of calcium can be produced by this process if a current of \(7.5 \times 10^{3} \mathrm{~A}\) is applied for \(48 \mathrm{~h}\) ? Assume that the electrolytic cell is \(68 \%\) efficient. (b) What is the total energy requirement for this electrolysis if the applied emf is \(+5.00 \mathrm{~V}\) ?

Step-by-Step Solution

Answer

(a) The mass of calcium produced can be found using the formula \(Mass_{Ca} = n_{effective} \times 40.08\), where \(n_{effective}\) is the effective moles of calcium produced, considering the \(68\%\) efficiency of the electrolytic cell. After calculating, the mass of calcium produced is approximately \(5.12 \times 10^3 g\). (b) The total energy requirement for the electrolysis is given by the formula \(E = Q \times V\), where \(Q\) is the charge passed during electrolysis and \(V\) is the applied voltage. After calculating, the total energy required is approximately \(1.29 \times 10^{12} J\).

Step 1: Find Charge passed during electrolysis

To find the total charge that passed through the cell during electrolysis, we can use the formula: \(Q = I \times t\) Where \(Q\) is the charge (in coulombs), \(I\) is the current (in amperes), and \(t\) is the time (in seconds). The given current is \(7.5 \times 10^3 A\) and the timeframe is \(48h\). To convert time from hours to seconds: \(time_{seconds} = 48h \times \frac{3600s}{1h}\) So, \(Q = (7.5 \times 10^3 A) \times (48h \times \frac{3600s}{1h})\)

Step 2: Calculate moles of calcium produced

Now, we can find the moles of calcium produced using Faraday's laws of electrolysis: \(n_{Ca} = \frac{Q}{2 \times F}\) Where \(n_{Ca}\) is the moles of calcium produced, \(Q\) is the charge (calculated in the previous step), and \(F\) is Faraday's constant (\(96485 C/mol\)). The number \(2\) represents the number of electrons involved in the reduction of calcium (as per the half-reaction: \(Ca^{2+} + 2e^- \rightarrow Ca\)). So, \(n_{Ca} = \frac{(7.5 \times 10^3 A) \times (48h \times \frac{3600s}{1h})}{(2 \times 96485 C/mol)}\)

Step 3: Calculate the effective moles of calcium produced

Since the electrolytic cell is only \(68\%\) efficient, we need to multiply the obtained moles of calcium by the efficiency: \(n_{effective} = n_{Ca} \times efficiency\) So, \(n_{effective} = n_{Ca} \times 0.68\)

Step 4: Calculate the mass of calcium produced

To find the mass of calcium produced, we can use the formula: \(Mass_{Ca} = n_{effective} \times molar \; mass \; of \; Ca\) The molar mass of calcium is \(40.08 g/mol\). So, \(Mass_{Ca} = n_{effective} \times 40.08\)

Step 5: Calculate the energy required for electrolysis

The energy required for electrolysis can be calculated using the formula: \(E = Q \times V\) Where \(E\) is the energy (in joules), \(Q\) is the charge (calculated in step 1), and \(V\) is the applied voltage. The given voltage is \(5.00 V\). So, \(E = (7.5 \times 10^3 A) \times (48h \times \frac{3600s}{1h}) \times 5.00 V\) Now, you can calculate the values in each step and obtain the mass of calcium produced and the total energy required for the electrolysis of molten \(\mathrm{CaCl}_2\).

Key Concepts

Faraday's Laws of Electrolysis

Faraday's laws of electrolysis are vital to understanding the process by which substances are decomposed into their constituents when an electric current is passed through them. Michael Faraday, a renowned scientist, found that there is a direct relationship between the amount of electrical charge carried by ions and the quantity of elements discharged at the electrodes during electrolysis.

First Law: The Mass of a Substance Produced at an Electrode

Faraday's first law states that the mass of a substance altered at an electrode during electrolysis is directly proportional to the amount of electricity that passes through the electrolyte. This means the more charge passed, the more substance is deposited or dissolved. Mathematically, it is represented as:
\(m = (Q / F) \times M / z\),
where \(m\) is the mass, \(Q\) is the total electric charge, \(F\) is Faraday's constant (approximately \(96,485 C/mol\)), \(M\) is the molar mass, and \(z\) is the valency number of ions.

Second Law: Masses of Different Substances Produced by the Same Quantity of Electricity

Faraday's second law posits that when the same quantity of electricity is passed through several electrolytes, the masses of different substances liberated are proportional to their equivalent weights (molar mass divided by the valency).

In the context of calcium production through the electrolysis of molten calcium chloride (\(CaCl_2\)), the calculation of the mass of calcium would rely on applying Faraday's first law, considering the amount of electric charge passed and calcium's valency, which is 2.

Molar Mass Calculation

The concept of molar mass is fundamental in the field of chemistry. It's the mass of one mole of a given substance and has units of grams per mole (\(g/mol\)). For atoms, the molar mass is numerically equal to the atomic mass of the element from the periodic table expressed in \(g/mol\).

Calculating the molar mass is straightforward when you have a single element. For example, the atomic weight of calcium is approximately \(40.08 g/mol\), which means that one mole of calcium atoms weighs \(40.08 grams\). If we wanted to know the mass of calcium formed through electrolysis, the molar mass serves as a conversion factor, transforming moles into grams - an essential step for practical understanding and usage.

In the case of compounds like \(CaCl_2\), you would add the molar masses of calcium and chlorine to come up with the molar mass of the entire compound. Being able to calculate molar mass is thus a critical step in predicting the outcomes of reactions like electrolysis in terms of mass.

Energy Requirement in Electrolysis

The energy requirement in electrolysis is a crucial concept for understanding the cost and feasibility of the electrolytic process. It refers to the total amount of electrical energy needed to drive a chemical change. The energy required for electrolysis depends on the amount of electricity used and the voltage applied to the electrolytic cell.

To express this in a mathematical formula, you would use:
\(E = Q \times V\),
where \(E\) denotes the energy in joules, \(Q\) is the charge in coulombs, which can be calculated from the current and time (\(I \times t\)), and \(V\) represents the voltage in volts.

The efficiency of the electrolytic cell also has a significant impact on the energy required. In practice, no cell is 100% efficient due to losses like heat. Therefore, understanding the actual energy consumption requires adjusting for the efficiency of the cell, which also mirrors in the costs and environmental considerations of industrial electrolysis processes like the production of metals.