Question

(a) Find the approximate number of water molecules in \(1.00 \mathrm{L}\) of water. (b) What fraction of the liter's volume is occupied by water nuclei?

Short answer

(a) About \(3.34 \times 10^{25}\) water molecules; (b) around 0.01% of the volume is occupied by nuclei.

Step-by-step solution

Determine Moles of Water in 1 Liter

First, calculate the number of moles of water in 1.00 liter. Since the density of water is approximately 1 g/mL, 1.00 liter of water has a mass of 1000 grams. The molar mass of water (H₂O) is approximately 18.02 g/mol. To find the number of moles, use the formula: \[ n = \frac{mass}{molar \ mass} \]So,\[ n = \frac{1000 \, \text{g}}{18.02 \, \text{g/mol}} \approx 55.5 \, \text{moles} \]

Calculate the Number of Water Molecules

Use Avogadro's number to find the number of molecules in the moles you calculated. Avogadro's number is approximately \( 6.022 \times 10^{23} \) molecules/mol. Use the formula: \[ \text{Number of molecules} = n \times N_A \]Thus:\[ \text{Number of molecules} = 55.5 \, \text{moles} \times 6.022 \times 10^{23} \text{molecules/mol} \]\[ \approx 3.34 \times 10^{25} \text{molecules} \]

Find Volume of Water Nuclei

The water nuclei volume is much smaller compared to the space the water molecules occupy due to the electron cloud. Assume the nucleus of a water molecule roughly takes up about 10,000 times less space than the molecule itself. If the volume of 1 mole of water (18 mL) is entirely filled with water molecules, then the actual nuclear volume will be: \[ \text{Nuclear volume fraction} \approx \frac{Volume}{10,000} \]So:\[ \frac{1}{10,000} \approx 0.0001 \text{ (fraction of total water volume)} \]

Determine Fraction of Total Volume Occupied by Nuclei

Calculate the fraction of the 1-liter volume occupied by the nuclei based on the percentage of the space taken by nuclei in the water molecules. If the entire 1-liter volume was filled with water molecules, the nuclei would occupy approximately 0.01% of that space. Thus:\[ \text{Fraction of total 1L volume} = 1 \text{L} \times 0.0001 \approx 0.0001 \text{ L} \]

Key concepts

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry used to quantify the number of molecules, atoms, or other basic units in a mole of a substance. It is named after the Italian scientist Amedeo Avogadro. This constant is approximately equal to \(6.022 \times 10^{23}\) entities per mole. It provides the bridge between the micro world (individual atoms and molecules) and the macro world (weight and volume measurements we use in labs).

This number is pivotal when calculating the number of molecules in a given sample. For example, if you have a known number of moles of a substance, you can multiply it by Avogadro's number to find out how many molecules or atoms you have. This helps scientists and students alike understand that substances of equal mole quantities have the same number of particles, even if their physical properties differ substantially. Therefore, Avogadro's number enables chemists to quantify substances at the atomic or molecular scale.

Water Molecules

Water is a simple yet fascinating molecule consisting of two hydrogen atoms bonded to one oxygen atom, represented by the chemical formula H₂O. Each water molecule is made up of three atoms, and its unique structure gives it special properties, such as its ability to exist in three states—solid, liquid, and gas—at normal Earth temperatures.

When you calculate the number of water molecules in any sample, such as in the exercise above, you utilize the number of moles of water and Avogadro's number. For instance, if you determine that you have 55.5 moles of water, you multiply this by \(6.022 \times 10^{23}\) to estimate that there are roughly \(3.34 \times 10^{25}\) molecules of water in your sample of 1 liter. This calculation showcases how even a seemingly small amount of water consists of an astronomical number of molecules, underscoring the immense scale of the molecular world.

Understanding water molecules is crucial for diverse scientific fields, from chemistry to biology and environmental science. Water acts as a universal solvent, making it an essential component in biological processes and chemical reactions.

Molar Mass

Molar mass is the mass of one mole of a given substance, usually expressed in grams per mole (g/mol). It is derived from the atomic masses of the constituent elements of the molecule found on the periodic table.

For water, its molar mass is calculated by adding the molar masses of two hydrogen atoms (each approximately 1.01 g/mol) and one oxygen atom (approximately 16.00 g/mol), which equals about 18.02 g/mol. This value is crucial for converting between grams and moles in chemical calculations. If you know the molar mass of a molecule, you can quickly determine how many moles are in a certain mass of a substance, and vice versa.

In the exercise example, determining the molar mass was key to calculating the number of moles in 1 liter of water. Knowing both the mass and the molar mass allows you to determine how many mole (and thus how many molecules) are present in a sample. Molar mass links the macro world of grams to the micro world of moles, harmonizing measurements across different scales.