Question

At least \(25 \mu \mathrm{g}\) of tetrahydrocannabinol \((\mathrm{THC}),\) the active ingredient in marijuana, is required to produce intoxication. The molecular formula of \(\mathrm{THC}\) is \(\mathrm{C}_{21} \mathrm{H}_{30} \mathrm{O}_{2} .\) How many moles of THC does this \(25 \mu \mathrm{g}\) represent? How many molecules?

Short answer

25 μg of THC represents \(7.95 \times 10^{-8}\) moles and \(4.79 \times 10^{16}\) molecules of THC.

Step-by-step solution

Convert the mass of THC into grams

To make the calculations easier, we need to first convert the mass of THC from micrograms (μg) to grams (g). To do this, use the following conversion factor: 1 g = 1,000,000 μg So, 25 μg = 25 × (1/1,000,000) g = 25 × 10⁻⁶ g

Calculate the molar mass of THC

We have the molecular formula for THC as C₂₁H₃₀O₂. So, let's find the molar mass for each element and sum them up. Molar mass of Carbon (C): 12.01 g/mol Molar mass of Hydrogen (H): 1.01 g/mol Molar mass of Oxygen (O): 16.00 g/mol Molar mass of C₂₁H₃₀O₂: = 21 × (12.01 g/mol) + 30 × (1.01 g/mol) + 2 × (16.00 g/mol) = 252.21 g/mol + 30.3 g/mol + 32 g/mol = 314.51 g/mol

Calculate the moles of THC

To calculate the number of moles, divide the mass of THC by its molar mass. Moles of THC = (Mass of THC)/(Molar mass of THC) = (25 × 10⁻⁶ g)/(314.51 g/mol) = 7.95 × 10⁻⁸ mol

Calculate the number of molecules of THC

Now, use Avogadro's constant to find the number of molecules in this amount of THC. Recall that Avogadro's constant is approximately 6.022 × 10²³ mol⁻¹. Number of molecules = (Moles of THC) × (Avogadro's constant) = (7.95 × 10⁻⁸ mol) × (6.022 × 10²³ mol⁻¹) = 4.79 × 10¹⁶ molecules So, 25 μg of THC represents 7.95 × 10⁻⁸ moles and 4.79 × 10¹⁶ molecules of THC.

Key concepts

Molecular Formula

Understanding the molecular formula of a compound like THC (tetrahydrocannabinol) is crucial in chemistry. The molecular formula represents the number and type of atoms present in a molecule of a substance. For instance, the molecular formula for THC is \(\mathrm{C}_{21}\mathrm{H}_{30}\mathrm{O}_{2}\). This tells us that one molecule of THC consists of:

• 21 carbon atoms (C)
• 30 hydrogen atoms (H)
• 2 oxygen atoms (O)

Each element contributes differently to the properties and the mass of the molecule. The subscript numbers (e.g., 21 in \(\mathrm{C}_{21}\)) indicate the number of atoms of each element in a single molecule of THC. Knowing the molecular formula allows chemists to calculate other properties such as the molar mass, which is essential for further environmental and medical applications of the compound.

Molar Mass Calculation

The molar mass calculaton is a process used to determine the mass of one mole of a given substance. This is important as it allows us to convert between mass and moles, which is often necessary in chemical reactions and formulations. Here's how we calculate the molar mass of THC, which has the formula \(\mathrm{C}_{21}\mathrm{H}_{30}\mathrm{O}_{2}\):
We first find the atomic masses of its constituent elements:

• Carbon (C) has an atomic mass of 12.01 g/mol.
• Hydrogen (H) has an atomic mass of 1.01 g/mol.
• Oxygen (O) has an atomic mass of 16.00 g/mol.

Next, we multiply these atomic masses by the number of each type of atom in the molecule and sum the results:
\[(21 \times 12.01) + (30 \times 1.01) + (2 \times 16.00) = 252.21 + 30.3 + 32 = 314.51 \, \text{g/mol}\]
This gives us a total molar mass of 314.51 g/mol for THC. With this information, we can calculate how many moles a specific mass of THC represents.

Avogadro's Constant

Avogadro's constant, denoted as \(6.022 \times 10^{23}\, \text{mol}^{-1}\), is a fundamental concept in chemistry. It connects the macroscopic scale of materials with the microscopic scale of atoms and molecules. Essentially, Avogadro's constant tells us the number of atoms or molecules in one mole of a substance.
For THC, if we know the number of moles, we can easily calculate the number of molecules using this constant. For example, if we have \(7.95 \times 10^{-8}\, \text{mol}\) of THC, we determine the number of molecules by multiplying by Avogadro's constant:
\[7.95 \times 10^{-8} \text{ mol} \times 6.022 \times 10^{23} \text{ mol}^{-1} = 4.79 \times 10^{16} \text{ molecules}\]
This calculation is vital in understanding how much of a substance is present at the molecular level, which is essential for dosage calculations in medicine or determining reactivity in chemical processes.