Question

Receptor proteins that allow us to sense heat have been isolated as well as molecules that cause us to perceive heat such as hot pepper and wasabi. If such proteins are able to detect as little as \(20.0 \mu \mathrm{g}\) of capsaicin, the chemical responsible for the burn in peppers, how many molecules of capsaicin \(\left(\mathrm{C}_{18} \mathrm{H}_{27} \mathrm{NO}_{3}\right)\) are required before the tongue can detect it?

Short answer

Approximately \( 3.94 \times 10^{16} \) molecules of capsaicin are required.

Step-by-step solution

Convert Micrograms to Grams

First, convert the mass of capsaicin from micrograms to grams. Recall that \(1 \text{ gram} = 10^6 \text{ micrograms}\). Therefore, \( 20.0 \text{ μg} = 20.0 \times 10^{-6} \text{ g} = 2.0 \times 10^{-5} \text{ g} \).

Calculate Molar Mass of Capsaicin

Next, calculate the molar mass of capsaicin using its chemical formula, \( \text{C}_{18} \text{H}_{27} \text{NO}_3 \). Calculate the total molar mass by summing the molar masses of each element:\begin{align*} \text{C}: & \, 18 \times 12.01 \text{ g/mol} = 216.18 \text{ g/mol} \ \text{H}: & \, 27 \times 1.01 \text{ g/mol} = 27.27 \text{ g/mol} \ \text{N}: & \, 1 \times 14.01 \text{ g/mol} = 14.01 \text{ g/mol} \ \text{O}: & \, 3 \times 16.00 \text{ g/mol} = 48.00 \text{ g/mol} \ \text{Total Molar Mass}: & \, 216.18 + 27.27 + 14.01 + 48.00 = 305.46 \text{ g/mol} \ \text{Molar Mass of } \text{C}_{18} \text{H}_{27} \text{NO}_{3} & \, = 305.46 \text{ g/mol} \ \text{ \, } \ \text{ \, } \ \text{ \, } \ \text{ \, } \ \text{ \, } \ \text{ \, } \ \text{ \, } \ \text{ \, } \ \text{ \, } \ \text{ \, } \ \text{ \, } \.

Calculate Moles of Capsaicin

Determine the number of moles of capsaicin present in \( 2.0 \times 10^{-5} \text{ g} \), using the molar mass calculated in Step 2: \( \text{Moles of Capsaicin} = \frac{2.0 \times 10^{-5} \text{ g}}{305.46 \text{ g/mol}} = 6.55 \times 10^{-8} \text{ mol} \).

Convert Moles to Molecules

Finally, convert the moles of capsaicin to the number of molecules by using Avogadro's number, which is \( 6.022 \times 10^{23} \text{ molecules/mol} \).Therefore, \( 6.55 \times 10^{-8} \text{ mol} \times 6.022 \times 10^{23} \text{ molecules/mol} = 3.94 \times 10^{16} \text{ molecules} \).

Key concepts

Chemical Calculations

Chemical calculations are vital for determining quantities and properties of substances in chemistry. These calculations often involve steps such as converting units, calculating molar masses, and using fundamental constants like Avogadro's number. Breaking down complex problems into manageable steps is essential for precise results. For instance, if we need to find the number of molecules in a given mass of a substance, we first convert the mass into moles and then use Avogadro's number to find the required quantity.

Molar Mass

The molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It is calculated by summing the atomic masses of all the atoms in a molecule. For example, the molar mass of capsaicin \(\text{C}_{18} \text{H}_{27} \text{NO}_3\) can be found by adding up the contributions from carbon, hydrogen, nitrogen, and oxygen as follows:

• The atomic mass of Carbon (C) is 12.01 g/mol.
• Hydrogen (H) is 1.01 g/mol.
• Nitrogen (N) is 14.01 g/mol.
• Oxygen (O) is 16.00 g/mol.

By calculating the sum:
18 \( \times \) 12.01 + 27 \( \times \) 1.01 + 1 \( \times \) 14.01 + 3 \( \times \) 16.00, we get the total molar mass of capsaicin, which is approximately 305.46 g/mol. This value is crucial for converting grams to moles.

Avogadro's Number

Avogadro's number \(6.022 \times 10^{23}\) is the number of units (atoms, molecules, etc.) in one mole of any substance. This constant allows chemists to count particles by weighing them.
For example, if you know the number of moles of a substance, you can use Avogadro's number to determine the number of molecules. In our capsaicin example, once we calculated the moles, \(6.55 \times 10^{-8} \, mol\), we multiplied by Avogadro's number to find the number of molecules:
\(6.55 \times 10^{-8} \, mol \) \( \times \) \(6.022 \times 10^{23} \, molecules/mol \) = \(3.94 \times 10^{16} \, molecules\).

Unit Conversion

Unit conversion is a fundamental step in solving many chemistry problems. It involves converting one unit of measurement to another to ensure consistency and accuracy.
For instance, in the capsaicin example, we needed to convert 20.0 micrograms to grams. Knowing that 1 gram equals \(10^6\) micrograms, we converted 20.0 micrograms to 2.0 \( \times \) \(10^{-5} \, grams\).
This step ensured we could use the molar mass in grams per mole for further calculations. Proper unit conversion is essential for obtaining correct and meaningful results in chemical calculations.