Caffeine contains 49.5% C, 28.8% N, 16.5% 0 and 5.20% H. Calculate the molecular formula. Given that 0.2 moles of caffeine weigh 38.04 g.

Explanation

To find the molecular formula of caffeine based on its percentage composition and given mass, we can follow these steps:

### Step 1: Convert percentages to grams
Assuming we have 100 g of caffeine, we can directly convert the percentages to grams:
- Carbon (C): 49.5 g
- Hydrogen (H): 5.20 g
- Nitrogen (N): 28.8 g
- Oxygen (O): 16.5 g

### Step 2: Convert grams to moles
Now, we will convert the grams of each element to moles using their atomic masses:
- Moles of Carbon (C) = \( \frac{49.5 \, \text{g}}{12.01 \, \text{g/mol}} \) = 4.125 moles
- Moles of Hydrogen (H) = \( \frac{5.20 \, \text{g}}{1.01 \, \text{g/mol}} \) = 5.15 moles
- Moles of Nitrogen (N) = \( \frac{28.8 \, \text{g}}{14.01 \, \text{g/mol}} \) = 2.056 moles
- Moles of Oxygen (O) = \( \frac{16.5 \, \text{g}}{16.00 \, \text{g/mol}} \) = 1.03 moles

### Step 3: Determine the simplest mole ratio
Next, we will divide each mole value by the smallest number of moles calculated:
- For Carbon: \( \frac{4.125}{1.03} \) ≈ 4
- For Hydrogen: \( \frac{5.15}{1.03} \) ≈ 5
- For Nitrogen: \( \frac{2.056}{1.03} \) ≈ 2
- For Oxygen: \( \frac{1.03}{1.03} \) = 1

### Step 4: Write the empirical formula
From the whole number ratios, we can write the empirical formula:
- Empirical formula = \( \text{C}_4\text{H}_5\text{N}_2\text{O} \)

### Step 5: Calculate the empirical formula mass
Now, we calculate the mass of the empirical formula:
- Empirical formula mass = \( (4 \times 12.01) + (5 \times 1.01) + (2 \times 14.01) + (1 \times 16.00) \)
- Empirical formula mass = 48.04 + 5.05 + 28.02 + 16.00 = 97.11 g/mol

### Step 6: Determine the molar mass of caffeine
Given that 0.2 moles of caffeine weigh 38.04 g, we can find the molar mass:
- Molar mass = \( \frac{38.04 \, \text{g}}{0.2 \, \text{moles}} \) = 190.2 g/mol

### Step 7: Find the value of n
To find the molecular formula, we need to determine the value of n:
- \( n = \frac{\text{Molecular formula mass}}{\text{Empirical formula mass}} \)
- \( n = \frac{190.2}{97.11} \) ≈ 2

### Step 8: Write the molecular formula
Now we can find the molecular formula:
- Molecular formula = \( n \times \text{Empirical formula} = 2 \times \text{C}_4\text{H}_5\text{N}_2\text{O} \)
- Molecular formula = \( \text{C}_8\text{H}_{10}\text{N}_4\text{O}_2 \)

### Final Answer
The molecular formula of caffeine is \( \text{C}_8\text{H}_{10}\text{N}_4\text{O}_2 \).

---