Question

How many water molecules are there in one drop of water (volume \(=0.0018 \mathrm{~mL}\) ) at room temperature? (a) \(4.86 \times 10^{17}\) (b) \(6.023 \times 10^{24}\) (c) \(2.584 \times 10^{19}\) (d) \(6.023 \times 10^{19}\)

Short answer

The correct answer is (d): \(6.023 \times 10^{19}\).

Step-by-step solution

Convert Volume to Liters

The volume of the drop of water is given as \(0.0018\, \text{mL}\). First, we need to convert this volume into liters because the density of water is often given in \(\text{g/mL}\) or \(\text{g/L}\). To convert milliliters to liters, we use the conversion: \(1\, \text{mL} = 0.001\, \text{L}\). Thus, \(0.0018\, \text{mL} = 0.0018 \times 0.001 = 0.0000018\, \text{L}\).

Calculate Mass of Water

Given the density of water is approximately \(1\, \text{g/mL}\) or \(1000\, \text{g/L}\), we can find the mass of the water. Since the volume of the water drop is \(0.0018\, \text{mL}\), the mass is \(0.0018\, \text{g}\).

Molar Mass of Water

The molar mass of water, \(\text{H}_2\text{O}\), is calculated by adding the atomic masses of its constituent elements. The hydrogen atom has an atomic mass of approximately \(1\, \text{g/mol}\) and there are two hydrogens, while oxygen has an atomic mass of approximately \(16\, \text{g/mol}\). So, the molar mass of water is \(2 \times 1 + 16 = 18\, \text{g/mol}\).

Calculate Moles of Water

To find the number of moles of water in the drop, we use the formula: \(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\). Here, \(\text{mass} = 0.0018\, \text{g}\) and \(\text{molar mass} = 18\, \text{g/mol}\). So, \(\text{moles of water} = \frac{0.0018}{18} = 0.0001\, \text{mol}\).

Avogadro's Number for Molecule Count

Avogadro's number \(6.022 \times 10^{23}\) specifies how many molecules are in one mole of a substance. Multiply the moles of water by Avogadro’s number to get the number of molecules. Therefore, the number of water molecules \(= 0.0001 \times 6.022 \times 10^{23} = 6.022 \times 10^{19}\).

Identify Correct Option

Compare the calculated number of molecules, \(6.022 \times 10^{19}\), with the given options. The value matches option (d) \(6.023 \times 10^{19}\).

Key concepts

Molar Mass Calculation

Understanding molar mass is essential to solve any chemistry problem involving conversions from mass to molecular count. For any substance, the molar mass is the mass of 1 mole of its molecules and is expressed in grams per mole (g/mol). Consider water (1C ext{H}_2 ext{O}21D) as an example:
- Hydrogen (16 ext{H}216) has a molar mass of approximately 1 16 ext{g/mol}216. Since there are two hydrogen atoms, this contributes 2 16 ext{g/mol}216. - Oxygen (16 ext{O}216) has a molar mass of 16 16 ext{g/mol}216. - Adding these gives the total molar mass of water as X = 2 + 16 = 18 16 ext{g/mol}216.
This means that one mole of water molecules weighs 18 grams. Knowing the molar mass allows us to convert the mass of a substance into moles, a critical step in determining the number of molecules.

Avogadro's Number

When it comes to counting molecules, using Avogadro's number is key. Avogadro's number (1A6.022 15 10^{23}21A) represents the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. Here is why it's so useful:
- Allows conversion from moles to molecules, or vice versa. - Essential for understanding large-scale chemical reactions.
In our exercise, having calculated the number of moles in the water droplet, we multiply that by Avogadro's number to find the exact count of water molecules in the drop. This step transforms our molar findings into a tangible number of molecules, bridging macroscopic measurements with molecular-level insight.

Density of Water

Density provides a bridge between the volume of a substance and its mass. In the case of water, the density is particularly straightforward: 1 1C ext{g/mL}21D or 1000 1C ext{g/L}21D. How do we use density to move between volume and mass?
- Multiply the volume of water by its density to find mass. - Here, given a water drop of 10.0018 ext{ mL}210, we know water weighs 0.0018 grams.
Understanding density and this simple calculation is crucial as it provides the value necessary for conversion within these exercises. It is this initial conversion that sets the foundation for all subsequent calculations, making it indispensable for solving such problems.

Molecular Count Conversion

Conversion from moles to actual molecules is a matter of simple multiplication when using Avogadro's number. Let’s break down the process:
- First, calculate the number of moles: 16 ext{moles} = rac{ ext{mass}}{ ext{molar mass}}216. - Then, to find how many molecules this accounts for, multiply the moles by Avogadro's number. - For example, from our water drop, we calculate 160.0001 ext{ mol}216, which converts to 166.022 15 10^{19}216 molecules when multiplied by Avogadro's number.
This transformation allows chemists not just to conceptualize but to enumerate microscopic entities based on measurable quantities, like mass or volume, reinforcing the practical application of chemistry concepts in real-world calculations.