Problem 75 - Chemistry and Chemical Reactivity

Question

A drop of water has a volume of about \(0.05 \mathrm{mL}\). How many molecules of water are in a drop of water? (Assume water has a density of \(1.00 \mathrm{g} / \mathrm{cm}^{3} .\) )

Step-by-Step Solution

Answer

There are approximately \(1.67 \times 10^{21}\) molecules in a drop of water.

Step 1: Convert volume to mass

First, convert the volume of water from milliliters (mL) to grams (g) using the density of water. Since the density of water is given as \(1.00 \text{ g/cm}^{3}\), the mass of water is equal to its volume in grams:\(\text{mass} = 0.05 \text{ mL} \times 1.00 \text{ g/cm}^{3} = 0.05 \text{ g}\)

Step 2: Convert mass to moles

Now, convert the mass of water to moles. Use the molar mass of water, which is approximately \(18.02 \text{ g/mol}\):\(\text{moles of water} = \frac{0.05 \text{ g}}{18.02 \text{ g/mol}} \approx 0.00278 \text{ mol}\)

Step 3: Convert moles to molecules

Finally, calculate the number of water molecules by using Avogadro's number, which is \(6.022 \times 10^{23} \text{ molecules/mol}\):\(\text{number of molecules} = 0.00278 \text{ mol} \times 6.022 \times 10^{23} \text{ molecules/mol} = 1.67 \times 10^{21} \text{ molecules}\)

Key Concepts

Water Density

Water density is a crucial factor when dealing with the properties of water, such as its mass at a given volume. In the metric system, water density is commonly approximated to be \(1.00 \text{ g/cm}^3\). This means that 1 cubic centimeter (or 1 milliliter) of water has a mass of 1 gram.
This simplification makes it very easy to convert between the volume and mass of water without needing complex calculations. For example, if you have a volume of 0.05 mL of water, like in the exercise, this means you have 0.05 grams of water.
This conversion is particularly handy when dealing with various chemical computations, as it allows you to translate a liquid measurement into a measurable mass amount. This characteristic of water is unique and is tied intimately to its molecular structure and the hydrogen bonding between molecules.

Molar Mass

The concept of molar mass acts as a bridge between the mass of a substance and the amount of substance present.
Molar mass is defined as the mass of one mole of any given substance. For water (\(\text{H}_2\text{O}\)), it is approximately \(18.02 \text{ g/mol}\). This means that one mole of water weighs 18.02 grams.

• The molar mass is calculated from the atomic masses of the atoms that form the compound.
• For water, it is the sum of the masses of 2 hydrogen atoms (approximately 1.01 g/mol each) and 1 oxygen atom (approximately 16.00 g/mol).

Understanding molar mass is critical because it allows us to convert grams into moles, which can then be used to relate to numbers of molecules, especially when using Avogadro's number. This is what we need to solve the given problem.

Avogadro's Number

Avogadro's number is fundamental in chemistry, providing the link between the macroscopic world of grams and moles and the microscopic world of atoms and molecules. It tells us that one mole of any substance contains \(6.022 \times 10^{23}\) entities, be they atoms, molecules, or ions.
This constant allows chemists to convert the amount of a substance in moles to the number of molecules, making it possible to quantitatively describe chemical reactions at the molecular level.

• In the exercise, this number is crucial for determining the number of molecules in a given amount of water (measured in moles).
• Thanks to Avogadro's number, even very small masses can be easily related to an enormous number of molecules, such as \(1.67 \times 10^{21}\) molecules in a single drop of water.

Thus, Avogadro's number is not just a large number but a key unit conversion factor that bridges different scales in chemical equations and reactions.