Two oxides of a metal contain 36.4% and 53.4% of oxygen by mass respectively. If the formula of the first oxide is `M_(2)O`, then that of the second is

Options

A. `M_(2)O_(2)`
B. MO
C. `MO_(2)`
D. `M_(2)O_(5)`

Explanation

To solve the problem, we need to determine the formula of the second oxide of the metal given the percentages of oxygen by mass in both oxides.

### Step 1: Calculate the molar mass of the metal (M) from the first oxide (M2O)
We know that the first oxide has a formula of M2O and contains 36.4% oxygen by mass.

1. Let the molar mass of the metal be \( M \).
2. The molar mass of oxygen (O) is approximately 16 g/mol.
3. The molar mass of M2O can be calculated as:
\[ \text{Molar mass of } M2O = 2M + 16 \]
4. The percentage of oxygen in M2O is given by:
\[ \text{Percentage of O} = \frac{\text{mass of O}}{\text{mass of M2O}} \times 100 = \frac{16}{2M + 16} \times 100 = 36.4 \]
5. Setting up the equation:
\[ \frac{16}{2M + 16} \times 100 = 36.4 \]
6. Simplifying this:
\[ 16 \times 100 = 36.4 \times (2M + 16) \]
\[ 1600 = 72.8M + 582.4 \]
\[ 1600 - 582.4 = 72.8M \]
\[ 1017.6 = 72.8M \]
\[ M = \frac{1017.6}{72.8} \approx 13.95 \text{ g/mol} \]

### Step 2: Determine the mass of the metal in the second oxide
The second oxide has 53.4% oxygen by mass. Therefore, the mass percentage of the metal in the second oxide is:
\[ 100 - 53.4 = 46.6\% \]

### Step 3: Set up the ratio of the metal to oxygen in the second oxide
1. Let’s assume we have 100 g of the second oxide.
2. The mass of oxygen in this oxide is 53.4 g, and the mass of the metal is 46.6 g.
3. The number of moles of oxygen in the second oxide is:
\[ \text{Moles of O} = \frac{53.4}{16} \approx 3.34 \text{ moles} \]
4. The number of moles of the metal is:
\[ \text{Moles of M} = \frac{46.6}{13.95} \approx 3.34 \text{ moles} \]

### Step 4: Determine the empirical formula of the second oxide
1. The ratio of moles of metal (M) to moles of oxygen (O) is:
\[ \text{Ratio} = \frac{3.34}{3.34} = 1 \]
2. This indicates that the formula of the second oxide is:
\[ \text{M}O \]

### Final Answer
The formula of the second oxide is \( \text{MO} \).

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