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

### Step 1: Determine the Molar Mass of Tetrahydrocannabinol
We know that the number of moles in 1 gram of tetrahydrocannabinol is 0.00318 moles. Using the formula:

\[ \text{Molar Mass} = \frac{\text{mass}}{\text{number of moles}} \]

Substituting the values:

\[ \text{Molar Mass} = \frac{1 \text{ g}}{0.00318 \text{ moles}} \approx 314.46 \text{ g/mol} \]

### Step 2: Set Up the Relationships Between Atoms
Let:
- \( x \) = number of oxygen atoms
- The number of hydrogen atoms = \( 15x \) (since there are 15 times as many hydrogen atoms as oxygen atoms)
- The number of carbon atoms = \( 0.7 \times (15x) = \frac{10.5x}{1} \) (since there are 70% as many carbon atoms as hydrogen atoms)

### Step 3: Express the Molar Mass in Terms of x
The molar mass can be expressed as:

\[ \text{Molar Mass} = (12 \times \text{number of C}) + (1 \times \text{number of H}) + (16 \times \text{number of O}) \]

Substituting the expressions for the number of atoms:

\[ 314.46 = \left(12 \times \frac{10.5x}{1}\right) + \left(1 \times 15x\right) + \left(16 \times x\right) \]

### Step 4: Simplify the Equation
Simplifying the equation gives:

\[ 314.46 = 126x + 15x + 16x \]
\[ 314.46 = 157x \]

### Step 5: Solve for x
Now, solve for \( x \):

\[ x = \frac{314.46}{157} \approx 2 \]

### Step 6: Calculate the Number of Each Atom
Now that we have \( x \):

- Number of oxygen atoms = \( x = 2 \)
- Number of hydrogen atoms = \( 15x = 15 \times 2 = 30 \)
- Number of carbon atoms = \( 10.5x = 10.5 \times 2 = 21 \)

### Step 7: Write the Molecular Formula
The molecular formula can now be written as:

\[ C_{21}H_{30}O_{2} \]

### Conclusion
The molecular formula of tetrahydrocannabinol is \( C_{21}H_{30}O_{2} \).

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