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 \(\mathrm{kg}\) ) of \(\mathrm{Cl}_{2}\) is produced?

The electrolysis of molten CaCl₂ consists of the following balanced chemical reaction: \[2\ CaCl_{2}(l) \rightarrow 2\ Ca(s) + Cl_{2}(g)\]

First, we need to convert the given mass of calcium (5.52 kg) into moles using its molar mass: Molar mass of Ca = 40.08 g/mol \[5.52\ kg\ Ca = 5520\ g\ & n = \frac{mass}{molar\ mass}\\ n_(Ca) = \frac{5520\ g}{40.08\ g/mol}= 137.76\ mol\]

In the electrolysis process, two moles of Ca²⁺ ions receive two moles of electrons to become solid calcium as per the equation: \[Ca^{2+} + 2e^- \rightarrow Ca(s)\] From Faraday's first law of electrolysis, the charge (Q) needed for this reaction can be calculated using: \[Q = nze\] where n = moles of Ca produced(137.76 mol), z = number of electrons per mole of Ca (z = 2, since it's a divalent ion), e = elementary charge (1.6 x 10⁻¹⁹ C) \[Q = (137.76\ mol)\ (2)\ (6.022 \times 10^{23}\ mol^{-1})\ (1.6 \times 10^{-19}\ C) = 2.658 \times 10^{5}\ C\] Now we are given electrolysis time (t) is 8 hours, convert that to seconds: \[t = 8\ h \times \frac{3600\ sec}{1\ h} = 28800\ sec\] Now, we can calculate the current (I) needed to produce the given mass of calcium using the equation: \[Q = It\] \[I = \frac{Q}{t} = \frac{2.658 \times 10^{5}\ C}{28800\ sec} = 9.23\ A\] So, the current needed to produce 5.52 kg of calcium is 9.23 A.

Using stoichiometry, we can relate the moles of Ca produced with the moles of Cl₂. From the balanced chemical equation, we have: \[2\ CaCl_{2}(l) \rightarrow 2\ Ca(s) + Cl_{2}(g)\] From the mole ratios, we can see that 2 moles of Ca produced = 1 mole of Cl₂ produced. So, \[n_(Cl_2) = \frac{n_(Ca)}{2} = \frac{137.76\ mol}{2} = 68.88\ mol\] Now, we will convert the moles of Cl₂ to mass using its molar mass (M_Cl₂ = 70.90 g/mol): \[Mass_(Cl_2) = n(Cl_2) \times M_(Cl_2) = (68.88\ mol) \times (70.90\ g/mol) = 4886.32\ g\] So, the mass of Cl₂ produced is 4.88632 kg.