Problem 44 - Chemistry The Central Science In Si Units

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?

Step-by-Step Solution

Answer

25 \(\mu\mathrm{g}\) of THC represents approximately \(7.95 \times 10^{-8}\) moles or \(4.79 \times 10^{16}\) molecules.

Step 1: Convert Micrograms to Grams

The mass of THC given is in micrograms, so the first step is to convert this to grams, since molar mass is typically given in grams per mole. We know that \(1 \mu\mathrm{g} = 10^{-6}\) g. Therefore, the mass in grams is: \(25 \mu\mathrm{g} = 25 \times 10^{-6}\) g.

Step 2: Find the Molar Mass of THC

The molecular formula for THC is \(\mathrm{C}_{21}\mathrm{H}_{30}\mathrm{O}_{2}\). To find the molar mass, sum the atomic masses of all the atoms:- Carbon: \(21 \times 12.01 = 252.21\) g/mol- Hydrogen: \(30 \times 1.01 = 30.30\) g/mol- Oxygen: \(2 \times 16.00 = 32.00\) g/molAdding these together, the molar mass of THC is \(252.21 + 30.30 + 32.00 = 314.51\) g/mol.

Step 3: Calculate Moles of THC

Using the formula \(\text{moles} = \frac{\text{mass in grams}}{\text{molar mass}}\), calculate the moles of THC. Substitute in the values:\[\text{moles of THC} = \frac{25 \times 10^{-6}}{314.51} \approx 7.95 \times 10^{-8}\] moles.

Step 4: Calculate Number of Molecules

Use Avogadro's number, which is \(6.022 \times 10^{23}\) molecules/mol, to find the number of molecules:\[\text{number of molecules} = 7.95 \times 10^{-8} \times 6.022 \times 10^{23} \approx 4.79 \times 10^{16}\] molecules.

Key Concepts

Molecular Mass

Molecular mass is a crucial concept in chemistry that refers to the mass of a single molecule of a chemical compound. It is measured in atomic mass units (amu) and is calculated by summing up the atomic masses of all the atoms present in the molecule. Each element has a specific atomic mass, which you can find on the periodic table. For a compound like tetrahydrocannabinol (THC), with the formula \(\mathrm{C}_{21}\mathrm{H}_{30}\mathrm{O}_{2}\), you calculate it as follows:

• Carbon (\(\mathrm{C}\)): 21 atoms with an atomic mass of 12.01 amu each, summing up to 252.21 amu.
• Hydrogen (\(\mathrm{H}\)): 30 atoms with an atomic mass of 1.01 amu each, summing up to 30.30 amu.
• Oxygen (\(\mathrm{O}\)): 2 atoms with an atomic mass of 16.00 amu each, summing up to 32.00 amu.

By adding these values, the molecular mass of THC is 314.51 amu. Understanding molecular mass is essential for determining the number of moles of a substance in reactions, which, in turn helps to translate between mass and quantity of molecules involved.

Avogadro's Number

Avogadro's number is a constant that helps chemists understand the quantity of particles, such as atoms or molecules, in a sample. It is defined as \(6.022 \times 10^{23}\) particles per mole. This number allows you to convert between the number of moles and the actual number of molecules or atoms.
Suppose you have 1 mole of a substance, like THC. You can confidently say it contains exactly \(6.022 \times 10^{23}\) molecules due to Avogadro's number.
When given a small sample, like the 25 micrograms of THC, the conversion of its mass to moles allows you to use Avogadro's number to find out how many molecules are present in that sample. In the example provided, there were approximately \(4.79 \times 10^{16}\) molecules in 25 micrograms of THC. This relationship is crucial for scientists when quantifying chemical reactions and understanding how much of each substance they are working with.

Micrograms to Grams Conversion

In many scientific calculations, converting units is an essential skill, especially in chemistry where standard measurements are crucial for accuracy. One common conversion is from micrograms to grams. The prefix "micro" in the metric system indicates \(10^{-6}\), meaning that 1 microgram is equivalent to \(1 \times 10^{-6}\) grams.
This knowledge is essential when dealing with small quantities. For instance, the problem initially provides the mass of THC as 25 micrograms. Before performing calculations related to molecular mass or moles, you must convert this to grams:

• 25 micrograms = \(25 \times 10^{-6}\) grams.

This conversion allows chemists to standardize their calculations using grams as the mass unit. This is important because molecular mass is typically measured in grams per mole, ensuring consistency and precision throughout chemical calculations.