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?

Short 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-by-step solution

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

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 = $\frac{mass}{molarmass}$ 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 = $\frac{25\times {10}^{-6}g}{314g/mol}$ = $7.96\times {10}^{-8}$ mol

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.