Problem 6 - Physics

Question

Water has a mass per mole of \(18.0 \mathrm{~g} / \mathrm{mol}\), and each water molecule \(\left(\mathrm{H}_{2} \mathrm{O}\right)\) has 10 electrons. (a) How many electrons are there in one liter \(\left(1.00 \times 10^{-3} \mathrm{~m}^{3}\right)\) of water? (b) What is the net charge of all these electrons?

Step-by-Step Solution

Answer

(a) There are approximately \(3.34 \times 10^{26}\) electrons. (b) The net charge is approximately \(-5.35 \times 10^{7} \, \text{C}\).

Step 1: Calculate the number of moles in the water

To find the number of electrons, we first need to calculate the number of moles in a liter of water. We know that water has a density of approximately \(1000 \, \text{kg/m}^3\), so 1 liter (\(1 \, \text{L} = 1 \, \text{kg}\) of water has a mass of 1000 grams. Using the molar mass of water (18.0 g/mol), we find the number of moles by dividing the mass of the water by its molar mass: \[\text{Number of moles} = \frac{1000 \, \text{g}}{18.0 \, \text{g/mol}} = 55.56 \, \text{mol}\]

Step 2: Calculate the number of molecules

Now, we calculate the number of water molecules in these moles using Avogadro's number, which is approximately \(6.022 \times 10^{23} \, \text{molecules/mol}\). Thus, the number of water molecules is:\[\text{Number of molecules} = 55.56 \, \text{mol} \times 6.022 \times 10^{23} \, \text{molecules/mol} \approx 3.34 \times 10^{25} \, \text{molecules}\]

Step 3: Calculate the number of electrons

Each water molecule (H₂O) contains 10 electrons. Therefore, the total number of electrons is given by multiplying the number of molecules by the number of electrons per molecule:\[\text{Number of electrons} = 3.34 \times 10^{25} \, \text{molecules} \times 10 \, \text{electrons/molecule} = 3.34 \times 10^{26} \, \text{electrons}\]

Step 4: Calculate the net charge of the electrons

Electrons have a negative charge of \(-1.602 \times 10^{-19} \, \text{C}\) (Coulombs) each. To find the total charge, multiply the number of electrons by the charge of a single electron:\[\text{Net charge} = 3.34 \times 10^{26} \, \text{electrons} \times (-1.602 \times 10^{-19} \, \text{C/electron}) \approx -5.35 \times 10^{7} \, \text{C}\]

Key Concepts

Molecular Calculations

Understanding molecular calculations is crucial in many chemistry and physics problems. It allows us to quantify substances at a molecular level. When you have a substance like water, you can easily calculate its properties knowing its molecular weight and density. For example, water has a molar mass of 18 grams per mole, which indicates that one mole of water weighs 18 grams. Calculating the amount of a substance in moles provides a clearer picture of its molecular composition.

In the exercise provided, we first determine how many moles of water are present in one liter. Given that the density of water is 1000 kg/m³, one liter (equivalent to one kilogram) contains water with a mass of 1000 grams. By dividing this mass by the molar mass of water (18 g/mol), we derive that there are approximately 55.56 moles of water in that one liter.

These calculations are foundational for further molecular analysis, such as determining the number of specific particles (like electrons) within a substance. This approach of converting mass to moles is commonly utilized across various scientific disciplines.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, representing the number of units (usually atoms or molecules) per mole of a substance: approximately \(6.022 \times 10^{23}\). This large number is pivotal because it provides a bridge between the macroscopic world, where we measure quantities in grams, and the microscopic world of individual atoms and molecules.

In the exercise, after determining the number of moles of water, Avogadro's number helped us calculate the total number of water molecules. Multiplying the moles of water (55.56 mol) by Avogadro's number gives approximately \(3.34 \times 10^{25}\) water molecules. Understanding Avogadro's number and its application allows us to transition smoothly from moles to actual molecular or atomic counts, which is critical for comprehensively analyzing chemical reactions and physical processes at a particle level.

Net Charge Calculation

In physics and chemistry, calculating the net charge is essential, especially when dealing with ions and charged particles. Each electron carries a fundamental charge of approximately \(-1.602 \times 10^{-19}\) coulombs. To compute the total charge exerted by a group of electrons, you multiply the number of electrons by this charge.

In the scenario from the exercise, once we have the total number of electrons in water, determined from knowing each molecule has 10 electrons, we can find the net charge by multiplying the number of electrons \( (3.34 \times 10^{26}) \) by the charge per electron. The resulting net charge is approximately \(-5.35 \times 10^{7}\) coulombs. It's negative because electrons carry a negative charge, influencing how substances interact at both atomic and macroscopic scales.

Calculating net charge is not just a theoretical exercise; it's crucial in many practical applications, such as in electrical engineering, to understand how materials interact and to design systems that utilize or manage electric charges effectively.