Understanding the Problem
To find the atomic mass of a metal in a trivalent oxide, we must understand the chemical stoichiometry of the compound. A 'trivalent' metal (let's call it $M$) has a valency of +3. Oxygen typically has a valency of -2.
Using the criss-cross rule for chemical formulas:
- Metal $M$ has valency 3
- Oxygen $O$ has valency 2
- The formula becomes $M_2O_3$
Step-by-Step Solution
1. Identify the composition
Given: Oxygen content = 31.6% by mass. Therefore, the Metal content = $100\% - 31.6\% = 68.4\%$.
2. Set up the mass ratio
In the compound $M_2O_3$:
- The mass of 3 atoms of Oxygen = $3 \times 16 = 48$ g/mol
- The mass of 2 atoms of Metal = $2 \times A$, where $A$ is the atomic mass of the metal.
Since 31.6% corresponds to the oxygen component and 68.4% to the metal component, we can establish the following ratio:
$$\frac{Mass \, of \, 2M}{Mass \, of \, 3O} = \frac{68.4}{31.6}$$
$$\frac{2A}{48} = \frac{68.4}{31.6}$$
3. Solve for A
$$2A = \frac{68.4 \times 48}{31.6}$$ $$2A = \frac{3283.2}{31.6}$$ $$2A \approx 103.9$$ $$A \approx 51.95 \, \text{g/mol}$$
Conclusion: The atomic mass of the metal is approximately 51.95 u, which suggests the metal is Chromium (Cr).
Intuition
By treating the percentage as a weight ratio, we can compare the actual known mass of oxygen in the formula against the unknown mass of the metal. Since we know the molar ratio (2 metal atoms per 3 oxygen atoms), we can solve the algebraic equation for the atomic weight of a single atom of the metal.