Question

Many companies are working to develop cups, straws, and even nail polish that will be able to detect a variety of drugs that may impair a person's ability to make good decisions. One of these drugs is rohypnol, also known as the date rape drug. This molecule is similar to the one found in Valium, a drug prescribed to treat anxiety disorders, seizures, and other medical conditions. If a nail polish will change color when it comes in contact with \(50.0\) mg of rohypnol \(\left(\mathrm{C}_{16} \mathrm{H}_{12} \mathrm{FN}_{3} \mathrm{O}_{3}\right)\), how many molecules of rohypnol are required to change the color of the nail polish?

Short answer

9.60 x 10^{19} molecules

Step-by-step solution

Determine the molar mass of rohypnol

Calculate the molar mass of rohypnol \ \ \ \ \ \[\text{C}_{16} \text{H}_{12} \text{FN}_{3} \text{O}_{3}\text{ as follows:} \ \ (16 \times 12.01) + (12 \times 1.01) + 19.00 + (3 \times 14.01) + (3 \times 16.00) \ \ = 192.16 + 12.12 + 19.00 + 42.03 + 48.00 \ \ = 313.31 \text{ g/mol} \ \]

Convert milligrams to grams

Convert the given mass of rohypnol from milligrams to grams: \ \ \ \ \[\text{Given:} 50.0 \text{ mg} = 0.050 \text{ g} \ \]

Calculate moles of rohypnol

Use the molar mass calculated in Step 1 to find the number of moles: \ \ \ \ \[\text{Moles of rohypnol} = \frac{\text{Mass in grams}}{\text{Molar mass}} \ \ = \frac{0.050 \text{ g}}{313.31 \text{ g/mol}} \ \ = 0.0001596 \text{ moles} \ \]

Calculate number of molecules

Convert the moles from Step 3 to molecules using Avogadro's number \ \ \ \ \[\text{Number of molecules} = \text{Moles} \times 6.022 \times 10^{23} \ \ = 0.0001596 \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mole} \ \ = 9.60 \times 10^{19} \text{ molecules} \ \]

Key concepts

Molar Mass

In chemistry, the term 'molar mass' refers to the mass of one mole of a given substance. Molar mass is typically expressed in grams per mole (g/mol). This concept is fundamental because it allows us to convert between mass and the number of entities (atoms, molecules) in a substance.
For example, the molar mass of rohypnol \(\text{C}_{16} \text{H}_{12} \text{FN}_{3} \text{O}_{3} \) can be calculated by adding up the atomic masses of all its constituent atoms:

• Carbon (C): 16 atoms at 12.01 g/mol each
• Hydrogen (H): 12 atoms at 1.01 g/mol each
• Fluorine (F): 1 atom at 19.00 g/mol
• Nitrogen (N): 3 atoms at 14.01 g/mol each
• Oxygen (O): 3 atoms at 16.00 g/mol each

Summing these values results in a molar mass of 313.31 g/mol for rohypnol.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry that defines the number of particles (atoms, molecules, ions) in one mole of substance. The value is approximately 6.022 \(\times 10^{23}\) entities per mole. This large number is essential for understanding the relationship between the macroscopic (grams) and microscopic (atoms/molecules) worlds.
For instance, if you have 1 mole of rohypnol, you would have approximately 6.022 \(\times 10^{23}\) molecules of it. This means when we know the moles of a substance, we can directly use Avogadro's number to determine the total number of individual molecules or atoms present.

Moles to Molecules Conversion

The conversion from moles to molecules is a crucial step in many chemical calculations. It involves using Avogadro's number to switch between these two units.
In our example, once we calculate the number of moles of rohypnol (Step 3), we can find the number of molecules using the formula:
\[ \text{Number of molecules} = \text{moles} \times 6.022 \times 10^{23} \text{ molecules/mole} \]
Using this approach, the number of molecules in 0.0001596 moles of rohypnol is calculated as follows: \[ 0.0001596 \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mole} \] resulting in approximately 9.60 \(\times 10^{19}\) molecules.

Unit Conversion

Unit conversion is a fundamental skill in chemistry that allows you to switch between different measurement systems. Accurate unit conversion is crucial for performing precise calculations.
In our problem, we first converted milligrams to grams for the calculation:

• 1 milligram (mg) = 0.001 grams (g)

So, 50.0 mg of rohypnol is equivalent to 0.050 g.
Unit conversions can also be more complex, involving multiple steps and dimensional analysis techniques to ensure proper units are used throughout a calculation. These techniques are essential for clear and accurate chemical calculations and analyses.