A \(2.077-\mathrm{g}\) sample of an element, which has an atomic mass between 40 and 55, reacts with oxygen to form \(3.708 \mathrm{~g}\) of an oxide. Determine the formula of the oxide (and identify the element).

To find the moles of the element, we can first find the mass of the element in the oxide sample. We are given the total mass of the oxide, 3.708 g. Subtract the mass of the element (2.077 g) from the total mass, which will give us the mass of oxygen. Mass of Oxygen = Mass of Oxide - Mass of Element Mass of Oxygen = 3.708 g - 2.077 g Mass of Oxygen = 1.631 g Now we can find the moles of the element and moles of oxygen using their atomic masses. Since the atomic mass of the element is between 40 and 55, we can assume a median value (e.g., 47.5) to find the moles: Moles of Element = Mass of Element / Atomic Mass of Element Moles of Element = 2.077 g / 47.5 g/mol ≈ 0.0437 mol For oxygen, the atomic mass is 16 g/mol. Moles of Oxygen = Mass of Oxygen / Atomic Mass of Oxygen Moles of Oxygen = 1.631 g / 16 g/mol ≈ 0.102 mol

To find the mole ratio, divide the moles of each element by the smallest value that we found in the previous step. Mole Ratio of Element = Moles of Element / Smallest Value of Moles Mole Ratio of Element ≈ 0.0437 mol / 0.0437 mol = 1 Mole Ratio of Oxygen = Moles of Oxygen / Smallest Value of Moles Mole Ratio of Oxygen ≈ 0.102 mol / 0.0437 mol ≈ 2.3 ≈ 2 Notice that the mole ratio of oxygen is approximately 2, so the empirical formula of the oxide is likely X₂O, where X is the unknown element.

From the empirical formula (X₂O), we can calculate the molar mass of the oxide using the molar mass of oxygen and the unknown element. Molar Mass of X₂O ≈ (2 * Molar Mass of X) + Molar Mass of O 3.708 g / 0.102 mol ≈ (2 * Molar Mass of X) + 16 g/mol Solve for the molar mass of the unknown element: Molar Mass of X ≈ [(3.708 g / 0.102 mol) - 16 g/mol] / 2 Molar Mass of X ≈ 45.7 g/mol Considering the atomic mass of the element lies between 40 and 55, the closest integer value is 46 g/mol, which corresponds to the atomic mass of palladium (Pd). Therefore, the element is identified as palladium and the empirical formula of the oxide is Pd₂O.