`Cu_(2)S` and `M_(2)S` are isomorphous in which percentage of sulphur are `20.14% and 12.94%` respectively. The atomic weight of M is `(Cu=63.5)`

Options

A. 208
B. 108
C. 112
D. 115

Explanation

To solve the problem step by step, we will analyze the given information and perform calculations to find the atomic weight of element M.

### Step 1: Calculate the molecular weight of Cu₂S
The molecular weight of Cu₂S can be calculated as follows:
- Atomic weight of Cu (Copper) = 63.5 g/mol
- Atomic weight of S (Sulfur) = x g/mol (we will find this later)

The molecular weight of Cu₂S = 2 * (Atomic weight of Cu) + (Atomic weight of S)
\[ \text{Molecular weight of Cu}_2\text{S} = 2 \times 63.5 + x = 127 + x \]

### Step 2: Calculate the percentage of sulfur in Cu₂S
The percentage of sulfur in Cu₂S is given by the formula:
\[ \text{Percentage of S} = \left( \frac{\text{Weight of S}}{\text{Molecular weight of Cu}_2\text{S}} \right) \times 100 \]
Substituting the values we have:
\[ 20.14 = \left( \frac{x}{127 + x} \right) \times 100 \]

### Step 3: Solve the equation for x
Rearranging the equation:
\[ 20.14(127 + x) = 100x \]
Expanding and simplifying:
\[ 2557.78 + 20.14x = 100x \]
\[ 2557.78 = 100x - 20.14x \]
\[ 2557.78 = 79.86x \]
\[ x = \frac{2557.78}{79.86} \approx 32 \]
Thus, the atomic weight of sulfur (S) is approximately 32 g/mol.

### Step 4: Calculate the molecular weight of M₂S
Now we will calculate the molecular weight of M₂S:
- Atomic weight of M = y g/mol
- We already know the atomic weight of S = 32 g/mol

The molecular weight of M₂S:
\[ \text{Molecular weight of M}_2\text{S} = 2y + 32 \]

### Step 5: Calculate the percentage of sulfur in M₂S
The percentage of sulfur in M₂S is given as 12.94%:
\[ 12.94 = \left( \frac{32}{2y + 32} \right) \times 100 \]

### Step 6: Solve the equation for y
Rearranging the equation:
\[ 12.94(2y + 32) = 3200 \]
Expanding:
\[ 12.94 \times 2y + 12.94 \times 32 = 3200 \]
Calculating \(12.94 \times 32\):
\[ 12.94 \times 32 = 413.12 \]
So the equation becomes:
\[ 25.88y + 413.12 = 3200 \]
Subtracting 413.12 from both sides:
\[ 25.88y = 3200 - 413.12 \]
\[ 25.88y = 2786.88 \]
Now, solving for y:
\[ y = \frac{2786.88}{25.88} \approx 108 \]

### Conclusion
The atomic weight of M is approximately 108 g/mol.