In a certain electrolysis experiment, \(1.44 \mathrm{~g}\) of \(\mathrm{Ag}\) were deposited in one cell (containing an aqueous \(\mathrm{AgNO}_{3}\) solution), while \(0.120 \mathrm{~g}\) of an unknown metal X was deposited in another cell (containing an aqueous \(\mathrm{XCl}_{3}\) solution) in series with the \(\mathrm{AgNO}_{3}\) cell. Calculate the molar mass of \(\mathrm{X}\).

The first step is calculating the number of moles of silver (Ag) deposited using the given mass and the known molar mass. The molar mass of silver is approximately \(107.868 \, \mathrm{g/mol}\). So, the mole of Ag can be calculated using the formula, \(Moles = \frac{Mass}{Molar \, Mass}\). By substituting the given mass of \(1.44 \, \mathrm{g}\) and molar mass of \(107.868 \, \mathrm{g/mol}\), the result is \(0.0134 \, \mathrm{mol}\).

Using Faraday's first law of electrolysis, it is known that the same amount of electrical charge is responsible for depositing both the silver and the unknown metal. Thus, the moles of metal X deposited is the same as the moles of silver deposited, which is \(0.0134 \, \mathrm{mol}\).

Lastly, the molar mass of the unknown metal can be determined by rearranging the mole calculation formula to solve for molar mass. The rearranged formula is \(Molar \, Mass = \frac{Mass}{Moles}\). By substituting the given mass of \(0.120 \, \mathrm{g}\) and calculated moles of \(0.0134 \, \mathrm{mol}\), the molar mass of metal X is \(8.9552 \, \mathrm{g/mol}\). However, this molar mass does not correspond to any known element. The given problem states that the substance is \(\mathrm{XCl}_{3}\), which indicates that for every mole of X, three moles of electrons are involved. Thus, the molar mass should be \(8.9552 \cdot 3 = 26.87 \, \mathrm{g/mol}\). This molar mass corresponds closely to Aluminum (Al) with a molar mass of \(26.98 \, \mathrm{g/mol}\). Therefore, the unknown metal X is Aluminum.