A compound of carbon, hydrogen, and nitrogen contains the three elements in the respective ratio of `9 : 1 : 3.5` Calculculate the empirical formula. If the molecular weight of the compound is 108, what its molecular formula?

Explanation

To solve the problem step by step, we will first determine the empirical formula from the given mass ratio of carbon, hydrogen, and nitrogen, and then we will find the molecular formula based on the molecular weight provided.

### Step 1: Write down the given ratio
The ratio of carbon, hydrogen, and nitrogen is given as:
- Carbon (C) : Hydrogen (H) : Nitrogen (N) = 9 : 1 : 3.5

### Step 2: Convert the weight ratio to moles
To find the empirical formula, we need to convert the mass ratio into a mole ratio. We do this by dividing the mass of each element by its respective molar mass.

- Molar mass of Carbon (C) = 12 g/mol
- Molar mass of Hydrogen (H) = 1 g/mol
- Molar mass of Nitrogen (N) = 14 g/mol

Calculating the number of moles for each element:
- Moles of Carbon = \( \frac{9 \text{ g}}{12 \text{ g/mol}} = 0.75 \text{ moles} \)
- Moles of Hydrogen = \( \frac{1 \text{ g}}{1 \text{ g/mol}} = 1 \text{ mole} \)
- Moles of Nitrogen = \( \frac{3.5 \text{ g}}{14 \text{ g/mol}} = 0.25 \text{ moles} \)

### Step 3: Determine the simplest mole ratio
Next, we need to find the simplest ratio by dividing each mole value by the smallest number of moles calculated.

The smallest number of moles is 0.25 (for Nitrogen):
- Carbon: \( \frac{0.75}{0.25} = 3 \)
- Hydrogen: \( \frac{1}{0.25} = 4 \)
- Nitrogen: \( \frac{0.25}{0.25} = 1 \)

Thus, the simplest mole ratio is:
- C : H : N = 3 : 4 : 1

### Step 4: Write the empirical formula
From the simplest mole ratio, we can write the empirical formula:
- Empirical formula = \( C_3H_4N \)

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

### Step 6: Calculate the molecular formula
Given that the molecular weight of the compound is 108 g/mol, we can find the factor \( n \) to determine the molecular formula:
- \( n = \frac{\text{Molecular weight}}{\text{Empirical formula weight}} = \frac{108}{54} = 2 \)

### Step 7: Write the molecular formula
The molecular formula is obtained by multiplying the subscripts in the empirical formula by \( n \):
- Molecular formula = \( n \times C_3H_4N = 2 \times C_3H_4N = C_6H_8N_2 \)

### Final Answer
The empirical formula is \( C_3H_4N \) and the molecular formula is \( C_6H_8N_2 \).

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