Problem 4 - University Physics with Modern Physics

Question

You have a pure (24-karat) gold ring of mass 10.8 g. Gold has an atomic mass of 197 g\(/\)mol and an atomic number of 79. (a) How many protons are in the ring, and what is their total positive charge? (b) If the ring carries no net charge, how many electrons are in it?

Step-by-Step Solution

Answer

The ring has \(2.61 \times 10^{24}\) protons and electrons, with a total positive charge of \(4.18 \times 10^{5}\) C.

Step 1: Calculate Moles of Gold

To find the number of gold atoms in the ring, first calculate the number of moles of gold in the 10.8 g ring using the formula:\[\text{moles of gold} = \frac{\text{mass of gold}}{\text{atomic mass of gold}}\]Substitute the given values:\[\text{moles of gold} = \frac{10.8 \text{ g}}{197 \text{ g/mol}} \approx 0.0548 \text{ mol}\]

Step 2: Calculate the Number of Gold Atoms

Use Avogadro's number to convert moles of gold to the number of atoms. Avogadro's number is approximately \(6.022 \times 10^{23}\) atoms/mol.\[\text{Number of gold atoms} = 0.0548 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 3.30 \times 10^{22} \text{ atoms}\]

Step 3: Determine the Number of Protons

Each gold atom has an atomic number of 79, which means each atom has 79 protons.\[\text{Number of protons} = 3.30 \times 10^{22} \text{ atoms} \times 79 \text{ protons/atom} \approx 2.61 \times 10^{24} \text{ protons}\]

Step 4: Calculate Total Positive Charge from Protons

The charge of a single proton is approximately \(1.6 \times 10^{-19}\) coulombs. Multiply the number of protons by the charge of one proton to get the total positive charge:\[\text{Total positive charge} = 2.61 \times 10^{24} \text{ protons} \times 1.6 \times 10^{-19} \text{ C/proton} \approx 4.18 \times 10^{5} \text{ C}\]

Step 5: Determine Number of Electrons (Neutral Net Charge)

Since the ring carries no net charge, the number of electrons must be equal to the number of protons. Therefore, the number of electrons is \(2.61 \times 10^{24}\).

Key Concepts

Moles of Gold

To understand how many gold atoms are present in a gold ring, we first need to determine the moles of gold. A mole is a measure of quantity used in chemistry to express amounts of a chemical substance. The formula to find the moles of gold is:

• Moles of gold = \( \frac{\text{mass of gold}}{\text{atomic mass of gold}} \)

The atomic mass of gold is 197 g/mol. So, if we have a gold ring with a mass of 10.8 g, we calculate:\[\text{moles of gold} = \frac{10.8 \text{ g}}{197 \text{ g/mol}} = 0.0548 \text{ mol}\]This calculation tells us the amount of substance expressed in moles, allowing us to use further conversions like Avogadro's number for atomic count.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, defining the number of particles, usually atoms or molecules, in one mole of substance. It is approximately \(6.022 \times 10^{23}\) atoms/mol. This large number quantifies the amount of individual atoms or molecules in a mound of substance.

• To convert moles of a substance into individual atoms or molecules, multiply the moles by Avogadro's number.

For the gold ring calculated in the earlier step, use:\[\text{Number of gold atoms} = 0.0548 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 3.30 \times 10^{22} \text{ atoms}\]This means the gold ring consists of about \(3.30 \times 10^{22}\) atoms, which is an enormous quantity of atoms.

Charge of Proton

The charge of a proton is a fundamental property of particles making up an atom and is essential for understanding atomic structure. Each proton carries a positive charge of approximately \(1.6 \times 10^{-19}\) coulombs. This positive charge helps define the identity of an atom, as it influences the atomic interactions and bonds with other elements.
For the total charge of protons in the gold ring:\[\text{Total charge from protons} = 2.61 \times 10^{24} \text{ protons} \times 1.6 \times 10^{-19} \text{ C/proton} \approx 4.18 \times 10^{5} \text{ C}\]This calculation shows how even a small piece of gold can harbor a significant amount of positive charge due to the sheer number of protons, though this is balanced by the electrons in a neutral atom.

Atomic Number of Gold

The atomic number is a unique identifier for chemical elements, denoting the number of protons found in the nucleus of an atom. For gold, this atomic number is 79, meaning each gold atom contains 79 protons.

• The atomic number informs us about the element's position on the periodic table and its chemical properties.
• It also generally equals the number of electrons in a neutral atom, maintaining a balance of charges.

In the context of the gold ring exercise, knowing the atomic number of gold helps us calculate the number of protons:\[\text{Number of protons} = 3.30 \times 10^{22} \text{ atoms} \times 79 \text{ protons/atom} = 2.61 \times 10^{24} \text{ protons}\]Understanding this helps highlight why gold, with its unique atomic structure, possesses distinct properties and high value.