Problem 44 - Chemistry: The Central Science

Question

At least \(25 \mu \mathrm{g}\) of tetrahydrocannabinol (THC), the active ingredient in marijuana, is required to produce intoxication. The molecular formula of 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?

Step-by-Step Solution

Answer

The number of moles of THC in 25 \(\mu\)g is \(7.96\times 10^{-8}\) mol, and the number of molecules of THC in 25 \(\mu\)g is \(4.79\times 10^{16}\) molecules.

Step 1: Calculate the molar mass of THC

To find the molar mass of THC, we will use the molecular formula \(\mathrm{C}_{21} \mathrm{H}_{30} \mathrm{O}_{2}\) and add the molar masses of all atoms in the formula. The atomic masses of Carbon (C), Hydrogen (H), and Oxygen (O) are approximately 12 g/mol, 1 g/mol, and 16 g/mol, respectively. Molar mass of THC = (21 × 12) + (30 × 1) + (2 × 16) = 252 + 30 + 32 = 314 g/mol

Step 2: Calculate the number of moles

Now, we will use the given mass of THC (25 \(\mu\)g) and its molar mass to calculate the number of moles. To do this, we need to divide the mass by the molar mass: Number of moles = \(\frac{mass}{molar \, mass}\) First, let's convert the given mass to grams: 25 \(\mu g\) = \(25\times 10^{-6} \) g Now, we can find the number of moles: Number of moles = \(\frac{25\times 10^{-6}\, g}{314\, g/mol}\) = \(7.96\times 10^{-8}\) mol

Step 3: Calculate the number of molecules

To find the number of molecules in the given mass of THC, we will use Avogadro's number which is approximately \(6.022 \times 10^{23}\) molecules/mol. Number of molecules = Number of moles × Avogadro's number Number of molecules = \(7.96\times 10^{-8}\, mol \times 6.022 \times 10^{23}\, molecules/mol\) Number of molecules = \(4.79\times 10^{16}\) molecules To summarize: - The number of moles of THC in 25 \(\mu\)g is \(7.96\times 10^{-8}\) mol. - The number of molecules of THC in 25 \(\mu\)g is \(4.79\times 10^{16}\) molecules.