Problem 4 - University Physics with Modern Physics

Question

Particles in a Gold Ring. You have a pure (24 karat) gold ring with mass 17.7 \(\mathrm{g}\) . Gold has atomic mass of 197 \(\mathrm{g} / \mathrm{mol}\) 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 approximately 4.27 × 10^24 protons, with a total charge of approximately 6.83 × 10^5 C. It also has the same number of electrons if it carries no net charge.

Step 1: Understand the Atomic Properties of Gold

Gold (\(\text{Au}\)) has an atomic mass of \(197 \, \text{g/mol}\) and each gold atom has \(79\) protons, as indicated by its atomic number. These protons reside in the nucleus and have a positive charge each.

Step 2: Calculate the Number of Gold Atoms in the Ring

To find the number of atoms in the ring, calculate the moles of gold using the formula:\[\text{moles of Au} = \frac{\text{mass of ring}}{\text{atomic mass}} = \frac{17.7 \, \text{g}}{197 \, \text{g/mol}} \approx 0.08984 \, \text{mol}\]Then, use Avogadro's number (\(6.022 \times 10^{23} \, \text{atoms/mol}\)) to find the number of atoms:\[\text{number of atoms} = 0.08984 \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 5.41 \times 10^{22} \, \text{atoms}\]

Step 3: Calculate the Number of Protons

Each gold atom has \(79\) protons. Thus, the total number of protons in the ring is:\[\text{total protons} = 5.41 \times 10^{22} \, \text{atoms} \times 79 \, \text{protons/atom} \approx 4.27 \times 10^{24} \, \text{protons}\]

Step 4: Calculate the Total Positive Charge

Each proton has a charge of \(1.602 \times 10^{-19} \, \text{C}\). Therefore, the total positive charge is:\[\text{total charge} = 4.27 \times 10^{24} \, \text{protons} \times 1.602 \times 10^{-19} \, \text{C/proton} \approx 6.83 \times 10^{5} \, \text{C}\]

Step 5: Determine the Number of Electrons

If the gold ring carries no net charge, the number of electrons equals the number of protons. Given that we calculated there are \(4.27 \times 10^{24} \, \text{protons}\), there are also \(4.27 \times 10^{24} \, \text{electrons}\) in the ring.

Key Concepts

Atomic Number

The atomic number is a fundamental characteristic of any element. It represents the number of protons found in the nucleus of an atom of that element. For gold, the atomic number is 79, meaning that each gold atom contains 79 protons. This also implies that each neutral atom of gold will have 79 electrons. The atomic number also determines the position of an element in the periodic table and is crucial for identifying the element and predicting its chemical behavior.

Atomic Mass

Atomic mass is the mass of an atom, usually expressed in atomic mass units (amu) or grams per mole (g/mol). It accounts for the mass of protons, neutrons, and electrons, though the electrons contribute negligibly. Gold has an atomic mass of 197 g/mol, which indicates that one mole of gold atoms weighs 197 grams. This value is essential for converting between grams and moles when conducting calculations in chemistry.

Avogadro's Number

Avogadro's number is a constant that defines the number of atoms, molecules, or particles in one mole of a substance. It is approximately \(6.022 \times 10^{23} \) particles per mole. This large number is crucial when working with macroscopic quantities of materials. In the context of a gold ring, once you determine the moles of gold, Avogadro’s number helps calculate the total number of individual gold atoms in the ring.

Protons and Electrons

Protons are positively charged particles found in the nucleus of an atom. Electrons are negatively charged particles orbiting the nucleus. Both are essential in determining an atom's chemical properties and neutrality. For gold, the number of protons is equal to the atomic number, which is 79. Therefore, in a neutral gold atom, there are also 79 electrons, balancing the positive charge of the protons and rendering the atom electrically neutral.

Moles Calculation

Calculating moles involves using the mass of a substance and its atomic mass. For the gold ring problem, the mass of the ring was divided by the atomic mass of gold to determine the number of moles. The formula for this is:

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

Inserting the values:\( \frac{17.7 \, \text{g}}{197 \, \text{g/mol}} \approx 0.08984 \, \text{mol} \).After finding the moles, Avogadro's number is used to determine the total number of atoms: \(0.08984 \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 5.41 \times 10^{22} \, \text{atoms}\). This allows for further calculations like finding the total number of protons.