In a compound `C, H, N` atoms are present in `9:1:3.5` by weight. Molecular weight of compound is `108`. Its molecular formula is:

Options

A. `C_(6)H_(8)N_(2)`
B. `C_(7)H_(10)N`
C. `C_(5)H_(6)N_(3)`
D. `C_(4)H_(18)N_(3)`

Answer

To find the molecular formula of the compound with the given atomic ratios and molecular weight, we can follow these steps:

### Step 1: Determine the mass of each element based on the given ratio
The ratio of the masses of Carbon (C), Hydrogen (H), and Nitrogen (N) is given as 9:1:3.5. Let's assume we have a total mass of 100 grams for simplicity.

- Mass of Carbon = 9 parts = \( \frac{9}{9 + 1 + 3.5} \times 100 = \frac{9}{13.5} \times 100 = 66.67 \, \text{g} \)
- Mass of Hydrogen = 1 part = \( \frac{1}{13.5} \times 100 = 7.41 \, \text{g} \)
- Mass of Nitrogen = 3.5 parts = \( \frac{3.5}{13.5} \times 100 = 25.93 \, \text{g} \)

### Step 2: Calculate the number of moles of each element
Using the atomic weights:
- Atomic weight of Carbon (C) = 12 g/mol
- Atomic weight of Hydrogen (H) = 1 g/mol
- Atomic weight of Nitrogen (N) = 14 g/mol

Now, we can calculate the number of moles for each element:
- Moles of Carbon = \( \frac{66.67 \, \text{g}}{12 \, \text{g/mol}} = 5.56 \, \text{mol} \)
- Moles of Hydrogen = \( \frac{7.41 \, \text{g}}{1 \, \text{g/mol}} = 7.41 \, \text{mol} \)
- Moles of Nitrogen = \( \frac{25.93 \, \text{g}}{14 \, \text{g/mol}} = 1.85 \, \text{mol} \)

### Step 3: Find the simplest ratio of moles
To find the simplest whole number ratio, we divide each mole value by the smallest number of moles calculated:
- For Carbon: \( \frac{5.56}{1.85} \approx 3 \)
- For Hydrogen: \( \frac{7.41}{1.85} \approx 4 \)
- For Nitrogen: \( \frac{1.85}{1.85} = 1 \)

Thus, the simplest ratio of C:H:N is 3:4:1.

### Step 4: Write the empirical formula
From the simplest ratio, the empirical formula is \( C_3H_4N \).

### Step 5: Calculate the molar mass of the empirical formula
Now, we calculate the molar mass of the empirical formula:
- Molar mass of \( C_3H_4N = (3 \times 12) + (4 \times 1) + (1 \times 14) = 36 + 4 + 14 = 54 \, \text{g/mol} \)

### Step 6: Determine the molecular formula
Given that the molecular weight of the compound is 108 g/mol, we can find the ratio of the molecular weight to the empirical formula weight:
- Ratio = \( \frac{108}{54} = 2 \)

Thus, the molecular formula is obtained by multiplying the subscripts in the empirical formula by this ratio:
- Molecular formula = \( C_{3 \times 2}H_{4 \times 2}N_{1 \times 2} = C_6H_8N_2 \)

### Final Answer
The molecular formula of the compound is \( C_6H_8N_2 \).

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