Question

Molten \(\mathrm{CaCl}_{2}\) is electrolyzed for 8.00 \(\mathrm{h}\) to produce \(\mathrm{Ca}(s)\) and \(\mathrm{Cl}_{2}(g) .\) a. What current is needed to produce 5.52 \(\mathrm{kg}\) of calcium metal? b. If 5.52 \(\mathrm{kg}\) calcium metal is produced, what mass (in kg) of \(\mathrm{Cl}_{2}\) is produced?

Short answer

The current needed to produce 5.52 kg of Calcium metal is 923.6 A, and when 5.52 kg of Calcium metal is produced, 9.765 kg of \(\mathrm{Cl}_{2}\) is also produced.

Step-by-step solution

Convert mass to moles

Since we are given 5.52 kg of Calcium metal, we need to convert it into moles. The molar mass of \(\mathrm{Ca}\) is 40.08g/mol. Number of moles = \(\frac{5.52 kg * 1000g/kg}{40.08g/mol}\) = 137.7 mol Step 2: Calculate the moles of electrons transferred

Determine the moles of electrons

The balanced chemical equation is given below: \(\mathrm{CaCl}_{2}(l) \rightarrow \mathrm{Ca}(s) + \mathrm{Cl}_{2}(g)\) From the reaction, for every mole of \(\mathrm{Ca}\) formed, 2 moles of electrons are transferred. Thus, we can calculate the moles of electrons transferred as: Moles of electrons = 2 * moles of \(\mathrm{Ca}\) = 2 * 137.7 mol = 275.4 mol Step 3: Calculate the total charge transferred

Use Faraday's Law

Faraday's Law states that the charge transferred is equal to the product of the moles of electrons transferred and the Faraday's constant (F). Total charge (Q) = moles of electrons * F F = 96485 C/mol Q = 275.4 mol * 96485 C/mol = \(2.66 * 10^7\) C Step 4: Calculate the current

Determine the current

We are given that the electrolysis process took 8 hours. So, we can find the current needed (I) using the formula: Current (I) = \(\frac{Total\:Charge}{time}\) Convert the time into seconds: 8 hours * 60 min/hour * 60 s/min = 28,800 s I = \(\frac{2.66 * 10^7 C}{28800 s} = 923.6 A\) Thus, the current needed to produce 5.52 kg of Calcium metal is 923.6 A. Step 5: Calculate the mass of \(\mathrm{Cl}_{2}\) produced

Find the mass of \(\mathrm{Cl}_{2}\) produced

From the balanced chemical equation, for every mole of \(\mathrm{Ca}\) formed, 1 mole of \(\mathrm{Cl}_{2}\) is produced. Therefore, the moles of \(\mathrm{Cl}_{2}\) produced is equal to the moles of \(\mathrm{Ca}\) produced. Moles of \(\mathrm{Cl}_{2}\) = moles of \(\mathrm{Ca}\) = 137.7 mol Now, convert moles of \(\mathrm{Cl}_{2}\) to mass. The molar mass of \(\mathrm{Cl}_{2}\) is 2 * 35.45 g/mol = 70.9 g/mol. Mass of \(\mathrm{Cl}_{2}\) = moles of $\mathrm{Cl}_{2} * 70.9 g/mol = 137.7 mol * 70.9 g/mol = 9765 g Convert the mass from grams to kilograms: 9765 g * 1 kg/1000 g = 9.765 kg Thus, when 5.52 kg of Calcium metal is produced, 9.765 kg of \(\mathrm{Cl}_{2}\) is also produced.

Key concepts

Faraday's Law

Faraday's Law plays a crucial role in understanding electrolysis. It connects the amount of electric charge used with the substance that is electrolyzed. This law states that the amount of substance deposited or released at an electrode is proportional to the total electric charge passed. It's expressed as:

• Total charge (Q) = moles of electrons × Faraday's constant (F)

The Faraday's constant is approximately 96485 C/mol, representing the charge of one mole of electrons. In our scenario, knowing the moles of electrons helps us determine the charge needed. If we have 275.4 moles of electrons, the total charge is calculated by multiplying by the Faraday's constant.

Chemical Equations

Chemical equations balance the reactants and products in a chemical reaction. For the electrolysis of \(CaCl_{2} \), the equation is: \(CaCl_{2}(l) \rightarrow Ca(s) + Cl_{2}(g) \)

• Each mole of calcium formed requires transferring 2 moles of electrons.
• For every mole of calcium, one mole of chlorine gas is produced.

Balancing these chemical equations is vital because it ensures the conservation of mass and charge, providing a map of how substances react and form new products. Here, understanding the electron transfer is essential for using Faraday’s Law.

Moles and Molar Mass

The concept of moles and molar mass helps in converting between grams and moles, a critical step in chemical calculations.

• The molar mass of calcium, Ca, is 40.08 g/mol.
• Molar mass of chlorine gas, Cl\(_2\), is 70.9 g/mol.

To find the moles of calcium from mass:\[\text{Number of moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}\]For instance, 5.52 kg of calcium is converted to moles as follows:\[\frac{5520 \text{ g}}{40.08 \text{ g/mol}} = 137.7 \text{ mol}\]This process helps in determining how much of each reactant or product is involved in a reaction.

Current Calculation

Current calculation in electrolysis links time, charge, and current. Here's how:

• First, find the total charge using the number of moles of electrons and the Faraday's constant.
• Then, calculate the current using the formula:

\[I (\text{current}) = \frac{Q (\text{total charge})}{t (\text{time})}\]Time must be in seconds, so convert hours to seconds by multiplying by 3600.If the process takes 8 hours, that equates to 28800 seconds. With total charge \(Q = 2.66 \times 10^7\) C, the current is:\[\frac{2.66 \times 10^7 \text{ C}}{28800 \text{ s}} = 923.6 \text{ A}\]This value is the amount of electrical current needed to produce 5.52 kg of calcium.