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 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 25 μg of THC represents 7.95 × 10⁻⁸ moles and approximately 4.79 × 10¹⁶ THC molecules.

Step-by-step solution

Find the molar mass of THC

To find the molar mass of THC, we will use the atomic masses of its constituent elements: Carbon (C), Hydrogen (H), and Oxygen (O). The molecular formula of THC is C₂₁H₃₀O₂, which means it contains 21 carbon atoms, 30 hydrogen atoms, and 2 oxygen atoms. The atomic mass of Carbon (C) = 12.01 g/mol The atomic mass of Hydrogen (H) = 1.01 g/mol The atomic mass of Oxygen (O) = 16.00 g/mol Now, we will calculate the molar mass of THC: Molar mass of THC = (21 × Atomic mass of C) + (30 × Atomic mass of H) + (2 × Atomic mass of O) Molar mass of THC = (21 × 12.01 g/mol) + (30 × 1.01 g/mol) + (2 × 16.00 g/mol) Molar mass of THC = 252.21 g/mol + 30.30 g/mol + 32.00 g/mol Molar mass of THC = 314.51 g/mol

Calculate the number of moles of THC

Now, we will use the given mass of THC (25 μg) and its molar mass (314.51 g/mol) to calculate the number of moles of THC. First, we need to convert the mass from micrograms to grams. Mass of THC = 25 μg = 0.000025 g Next, we will use the formula: Number of moles = (Mass of the substance) / (Molar mass) Number of moles of THC = (0.000025 g) / (314.51 g/mol) Number of moles of THC = 7.95 × 10⁻⁸ mol

Calculate the number of THC molecules

Finally, we will use Avogadro's number to calculate the number of THC molecules. Avogadro's number is approximately 6.022 × 10²³ particles/mol. Number of THC molecules = (Number of moles of THC) × (Avogadro's number) Number of THC molecules = (7.95 × 10⁻⁸ mol) × (6.022 × 10²³ particles/mol) Number of THC molecules = 4.79 × 10¹⁶ particles So, 25 μg of THC represents 7.95 × 10⁻⁸ moles and approximately 4.79 × 10¹⁶ THC molecules.

Key concepts

Molar Mass Calculation

Understanding molar mass calculation is essential for converting between mass and moles of a substance, which is a common practice in chemistry. Molar mass represents the mass of one mole of a substance and is measured in grams per mole (g/mol).

To calculate the molar mass of a compound, such as tetrahydrocannabinol (THC), you need to know the chemical formula and the atomic masses of each element within that compound. For THC, with the formula \(\mathrm{C}_{21} \mathrm{H}_{30} \mathrm{O}_{2}\), you calculate the molar mass by multiplying the number of atoms of each element by their respective atomic masses and adding these values together. You can find atomic masses on the periodic table, and the ones needed for THC are 12.01 g/mol for carbon, 1.01 g/mol for hydrogen, and 16.00 g/mol for oxygen.

To simplify, the molar mass calculation in our example would look like this:

• Carbon: 21 atoms × 12.01 g/mol
• Hydrogen: 30 atoms × 1.01 g/mol
• Oxygen: 2 atoms × 16.00 g/mol

Then sum these values for the total molar mass of THC. Remember, precision in these numbers is critical as it directly affects the accuracy of all subsequent calculations.

Avogadro's Number

When dealing with atoms and molecules at the micro scale, Avogadro's number becomes a fundamental concept in chemistry. Avogadro's number, approximately \(6.022 \times 10^{23}\) entities per mole, represents the number of particles found in one mole of a substance. This constant enables chemists to count particles by weighing them.

In quantitative chemistry, to convert moles to the number of discrete particles - atoms, molecules, or ions - you multiply the moles by Avogadro’s number. In the example with THC, calculating the number of THC molecules from the moles determined involves this simple multiplication: \(\text{{Number of molecules}} = \text{{Number of moles}} \times \text{{Avogadro's number}}\). This provides an astonishingly large number, reflecting the incredibly small size and mass of individual molecules.

Chemical Formula Interpretation

The chemical formula interpretation is an invaluable skill for identifying the composition and proportions of atoms in a chemical compound. It provides an immediate understanding of the molecule's structure and is key for various chemical calculations, including determination of molar mass. The molecular formula, such as \(\mathrm{C}_{21} \mathrm{H}_{30} \mathrm{O}_{2}\) for THC, tells us that each molecule consists of 21 carbon atoms, 30 hydrogen atoms, and 2 oxygen atoms.

To interpret this effectively, think of the subscript numbers as a simple count for each type of atom in the molecule. If no subscript is present, it implies only one atom of that element. By knowing the atomic composition from the chemical formula, scientists can predict chemical properties, reaction behaviors, and how the molecule will interact with others. This information is foundational to tasks from balancing chemical equations to calculating reactions yields and is central to understanding chemistry.