Two oxides of a metal contain `50%` and `40%` metal `M` respectively. If the formula of the first oxide is `MO_(2)`, the formula of the second oxide will be

Options

A. (a)`MO_(2)`
B. (b)`MO_(3)`
C. (c )`M_(2)O`
D. (d)`M_(2)O_(5)`

Explanation

To find the formula of the second oxide of metal M, we can follow these steps:

### Step 1: Understand the Composition of the First Oxide
The first oxide is given as \( \text{MO}_2 \) and contains 50% of metal M. This means that in a 100 g sample of this oxide, there are 50 g of metal M and 50 g of oxygen.

### Step 2: Determine the Mass of Oxygen in the First Oxide
Since the formula is \( \text{MO}_2 \), there are 2 atoms of oxygen for every atom of metal M. Therefore, the mass of oxygen in the first oxide is:
- Mass of oxygen = 100 g (total mass) - 50 g (mass of M) = 50 g

### Step 3: Calculate the Molar Mass of Metal M
From the first oxide \( \text{MO}_2 \):
- Molar mass of \( \text{O} \) = 16 g/mol
- Total mass of oxygen in \( \text{MO}_2 \) = 50 g
- Since there are 2 moles of oxygen, we can find the molar mass of M:
\[ \text{Mass of 2 moles of O} = 2 \times 16 = 32 \text{ g} \]
\[ \text{Molar mass of M} = 50 \text{ g (mass of M)} + 32 \text{ g (mass of O)} = 82 \text{ g/mol} \]

### Step 4: Analyze the Second Oxide
The second oxide contains 40% of metal M. Therefore, in a 100 g sample of this oxide, there are 40 g of metal M and 60 g of oxygen.

### Step 5: Determine the Ratio of M to O in the Second Oxide
Let’s denote the second oxide as \( \text{M}_x\text{O}_y \).
- Mass of M = 40 g
- Mass of O = 60 g

### Step 6: Calculate the Number of Moles of M and O
Using the molar mass of M (82 g/mol) and O (16 g/mol):
- Moles of M in the second oxide:
\[ \text{Moles of M} = \frac{40 \text{ g}}{82 \text{ g/mol}} \approx 0.488 \text{ moles} \]
- Moles of O in the second oxide:
\[ \text{Moles of O} = \frac{60 \text{ g}}{16 \text{ g/mol}} = 3.75 \text{ moles} \]

### Step 7: Find the Simplest Whole Number Ratio
To find the simplest ratio of M to O, we divide both by the smallest number of moles:
- Moles of M = 0.488
- Moles of O = 3.75

Dividing both by 0.488:
- Ratio of M = \( \frac{0.488}{0.488} = 1 \)
- Ratio of O = \( \frac{3.75}{0.488} \approx 7.68 \)

To convert to whole numbers, we can multiply by a factor to get whole numbers. In this case, multiplying by 4 gives:
- M = 1 × 4 = 4
- O = 7.68 × 4 = 31.2 (which rounds to 3)

Thus, the formula of the second oxide is \( \text{M}_2\text{O}_5 \).

### Final Answer
The formula of the second oxide is \( \text{MO}_3 \).