Problem 27 - University Physics with Modern Physics

Question

How many moles are in a \(1.00-\mathrm{kg}\) bottle of water? How many nolecules? The inolar mass of water is 18.0 \(\mathrm{g} / \mathrm{mol}\).

Step-by-Step Solution

Answer

There are approximately 55.56 moles and 3.34 x 10²⁵ molecules in 1.00 kg of water.

Step 1: Convert Mass to Grams

First, we need to convert the mass of the water from kilograms to grams because the molar mass is given in grams per mole. Since there are 1000 grams in a kilogram, we have:\[\text{Mass of water} = 1.00 \text{ kg} \times 1000 \text{ g/kg} = 1000 \text{ g} \]

Step 2: Calculate Moles of Water

Next, use the formula to find the number of moles, which is: \(\text{Number of moles} = \frac{\text{Mass of substance (g)}}{\text{Molar mass (g/mol)}}\). Using the given molar mass of water (18.0 g/mol):\[\text{Number of moles} = \frac{1000 \text{ g}}{18.0 \text{ g/mol}} \approx 55.56 \text{ moles}\]

Step 3: Calculate Number of Molecules

To find the number of molecules, use Avogadro's number, which is approximately \(6.022 \times 10^{23}\) molecules per mole. Multiply the number of moles of water by Avogadro's number:\[\text{Number of molecules} = 55.56 \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mole} \approx 3.34 \times 10^{25} \text{ molecules}\]

Key Concepts

Molar Mass

Molar mass is a key concept in chemistry that helps us understand the amount of matter present in a substance. It is defined as the mass of one mole of a substance, which is usually expressed in grams per mole (g/mol). Think of molar mass as a bridge that connects the mass of a material to the number of moles it contains.
For example, the molar mass of water (H₂O) is 18.0 g/mol. This means that 18 grams of water is equivalent to one mole of water molecules. To put this into perspective:

• Each water molecule consists of 2 hydrogen atoms and 1 oxygen atom.
• It combines the atomic masses of hydrogen (about 1.0 g/mol) and oxygen (about 16.0 g/mol).

Thus, the calculation gives us 2 * 1 + 16 = 18 g/mol for water. Understanding the molar mass allows us to convert between the physical mass and the chemical amount of a substance.

Avogadro's Number

Avogadro's number is an essential constant in chemistry, named after the Italian scientist Amedeo Avogadro. It provides the link between the macroscopic world of grams and liters, and the microscopic world of atoms and molecules.
Avogadro's number is approximately \(6.022 \times 10^{23}\) entities per mole. This colossal number allows chemists to count particles by weighing them. It tells us how many atoms or molecules are present in a mole of a substance.

• This number is used to convert moles of a substance into the number of particles.
• For example, one mole of water contains approximately \(6.022 \times 10^{23}\) molecules of H₂O.

Understanding Avogadro's number is crucial for tasks requiring the counting of molecules or atoms, such as chemical reactions or determining the concentration of solutions.

Molecule Calculation

Calculating the number of molecules in a given mass of substance is a common task in chemistry. This involves understanding the relationships between mass, moles, and Avogadro's number.
To calculate the number of molecules:

• First, determine the number of moles from the given mass using the formula: \(\text{Number of moles} = \frac{\text{Mass}}{\text{Molar mass}}\).
• Then, multiply the number of moles by Avogadro's number to find the total number of molecules.

For instance, if you have a 1 kg bottle of water, which weighs 1000 g, and knowing that the molar mass of water is 18.0 g/mol, you first find there are approximately 55.56 moles of water. By using Avogadro's number, multiplying 55.56 moles by \(6.022 \times 10^{23}\), you can calculate that there are about \(3.34 \times 10^{25}\) water molecules in the bottle. This process highlights the power of chemical calculations to connect macroscopic measurements with microscopic particles.