What mass of barium is produced when molten \(\mathrm{BaCl}_{2}\) is electrolyzed by a current of \(2.50 \times 10^{5} \mathrm{~A}\) for \(6.00 \mathrm{~h}\) ?

Electrolysis is a chemical process that uses electrical energy to drive a non-spontaneous chemical reaction. Faraday's Laws of Electrolysis provide a quantitative relationship between the amount of substance altered or produced at the electrodes of an electrolytic cell and the amount of electric current passed through the substance.

Faraday's First Law states that the mass of the substance altered at an electrode is directly proportional to the quantity of electricity that passes through the electrolyte. Mathematically, this can be represented as \(m = Z \times Q\), where \(m\) is the mass of the substance, \(Z\) is the electrochemical equivalent (a substance-specific constant), and \(Q\) is the total charge.

Faraday's Second Law says that when the same amount of electric current is passed through different substances, the mass of the substances altered at the respective electrodes is proportional to their chemical equivalent weights. This law allows us to compare the amounts of different substances produced during electrolysis if the charges are equivalent.

In practical applications, such as determining the mass of barium produced from the electrolysis of molten barium chloride (BaCl2), Faraday's laws are essential. To find the total charge transferred (\(Q\)), we multiply the current (\(I\)) by time (\(t\)). The moles of electrons transferred can then be calculated by dividing by Faraday's constant (\(F\)).

The mole concept is fundamental to understanding chemical quantities in reactions. A mole is the SI unit for amount of substance, and it represents approximately \(6.022 \times 10^{23}\) entities, whether they are atoms, ions, electrons, or molecules. This number is known as Avogadro's number.

In electrolysis calculations, the mole concept allows us to relate the physical charge passed through the electrolyte to the chemical reaction happening at the electrodes. By using the mole concept, we can determine the amount of material either produced or consumed at an electrode for every mole of electrons transferred.

For example, in the given exercise involving BaCl2 electrolysis, the mole concept helps calculate the number of moles of barium that can be produced from molten BaCl2. Since each mole of barium requires two moles of electrons (according to the stoichiometry of the reaction), the total moles of electrons (calculated from Faraday's law and the charge transferred) can be divided by 2 to find the moles of barium metal produced.

Stoichiometry is the aspect of chemistry that deals with the relative quantities of reactants and products in chemical reactions. It is based on the conservation of mass and the mole concept. For an electrolysis reaction, stoichiometry explains the ratio in which substances react or are produced based on the balanced chemical equation.

In the example of barium production from molten BaCl2, the stoichiometry of the reaction at the cathode is \(\text{Ba}^{2+} + 2\text{e}^- \rightarrow \text{Ba}\). This indicates that for every mole of barium ions, two moles of electrons are needed. Using stoichiometry, we can determine the moles of barium formed from the moles of electrons transferred during the electrolysis process. Once the moles of barium are known, multiplying by the molar mass of barium (137.33 g/mol) gives the mass of barium produced. This straightforward relationship makes calculations organized and predictable, allowing for clear solutions in quantitative electrolysis problems.