A drug marijuana owes its activity to tetrahydrocannabinol, which contains 70% as many carbon atoms as hydrogen atoms and 15 times as many hydrogen atoms as oxygen atoms. If the number of moles in 1 gm of tetrahydrocannabinol are 0.00318` moles then its molecular formula will be:

Options

A. `C_(21) H_(30) O_2`
B. `C_(21) H_(15) O`
C. `C_(31) H_(15) O`
D. `C_(31) H_(15) O_2`

Answer

To find the molecular formula of tetrahydrocannabinol based on the given information, we can follow these steps:

### Step 1: Define Variables
Let:
- \( x \) = number of oxygen atoms

From the problem statement:
- The number of hydrogen atoms = \( 15x \)
- The number of carbon atoms = \( 0.7 \times (15x) = 10.5x \)

### Step 2: Express the Ratios
We have the following ratios based on our definitions:
- Carbon (C) = \( 10.5x \)
- Hydrogen (H) = \( 15x \)
- Oxygen (O) = \( x \)

### Step 3: Eliminate the Decimal
Since the number of atoms must be whole numbers, we can multiply all coefficients by 2 to eliminate the decimal:
- Carbon (C) = \( 10.5x \times 2 = 21x \)
- Hydrogen (H) = \( 15x \times 2 = 30x \)
- Oxygen (O) = \( x \times 2 = 2x \)

### Step 4: Determine the Ratio of Atoms
Now we have:
- C = 21
- H = 30
- O = 2

Thus, the ratio of atoms in the molecular formula is:
- C : H : O = 21 : 30 : 2

### Step 5: Write the Empirical Formula
The empirical formula can be written as:
\[ \text{C}_{21}\text{H}_{30}\text{O}_{2} \]

### Step 6: Calculate the Molar Mass
Next, we need to calculate the molar mass of the empirical formula:
- Molar mass of C = 12 g/mol
- Molar mass of H = 1 g/mol
- Molar mass of O = 16 g/mol

Calculating the molar mass:
\[ \text{Molar mass} = (21 \times 12) + (30 \times 1) + (2 \times 16) = 252 + 30 + 32 = 314 \text{ g/mol} \]

### Step 7: Find the Molecular Formula
Given that the number of moles in 1 g of tetrahydrocannabinol is 0.00318 moles, we can find the molar mass:
\[ \text{Molar mass} = \frac{\text{mass}}{\text{moles}} = \frac{1 \text{ g}}{0.00318 \text{ moles}} \approx 314.5 \text{ g/mol} \]

Since the calculated molar mass (314 g/mol) is approximately equal to the empirical formula mass (314 g/mol), the molecular formula is the same as the empirical formula.

### Conclusion
Thus, the molecular formula of tetrahydrocannabinol is:
\[ \text{C}_{21}\text{H}_{30}\text{O}_{2} \]

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