Problem 7 - College Physics

Question

\(\bullet\) \(\bullet\) Particles in a gold ring. You have a pure \((24-\) karat) gold ring with mass 17.7 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 car- ries no net charge, how many electrons are in it?

Step-by-Step Solution

Answer

The ring has 4.73 x 10^22 protons with a total positive charge of 7.57 x 10^3 C and contains the same number of electrons.

Step 1: Calculate Moles of Gold

Find the number of moles of gold using the formula: \( \text{number of moles} = \frac{\text{mass}}{\text{atomic mass}} \). Here mass is 17.7 g, and the atomic mass of gold is 197 g/mol.

Step 2: Find Number of Atoms in Gold Ring

Multiply the number of moles of gold by Avogadro's number \( (6.022 \times 10^{23} \text{ atoms/mol}) \) to find the number of gold atoms in the ring.

Step 3: Determine Number of Protons

Every gold atom has 79 protons (atomic number of gold). Multiply the total number of gold atoms by 79 to find the total number of protons in the ring.

Step 4: Calculate Total Positive Charge

Since each proton has a charge of \(1.6 \times 10^{-19} \text{ C} \), multiply the total number of protons by this charge to find the total positive charge in coulombs.

Step 5: Determine Number of Electrons

For a neutral gold ring, the total number of electrons equals the total number of protons. Therefore, the number of electrons is the same as the number of protons.

Key Concepts

Gold Atomic Mass

The atomic mass of an element is the average mass of its atoms, including its isotopic compositions, measured in atomic mass units (amu) or grams per mole (g/mol). For gold, the atomic mass is 197 g/mol. This means that one mole of gold atoms weighs 197 grams on average. The atomic mass helps in calculating the amount of substance present without counting individual atoms. By knowing the mass of a gold object, like a ring, and the atomic mass of gold, one can ascertain the number of moles using the formula:\[ \text{moles of gold} = \frac{\text{mass of gold}}{\text{atomic mass}}. \]. By doing so, we could simply divide the mass of the ring, which is 17.7 grams, by the atomic mass of 197 g/mol to obtain the moles of gold, which can be further used in calculations.

Protons and Electrons

Gold has an atomic number of 79, meaning each gold atom contains 79 protons in its nucleus. Protons are positively charged components that define the element and its properties. In a neutral atom, the number of protons is equal to the number of electrons. Electrons, which carry a negative charge, orbit the nucleus and balance out the positive charge of the protons. To find the total number of protons in a gold ring, we multiply the total number of atoms by 79. To understand the neutrality, realize that for a gold ring with no net charge, the number of electrons must also be 79 times the number of gold atoms. Consequently, both the protons and electrons balance each other out, leading to no net charge.

Avogadro's Number

Avogadro's number, which is \(6.022 \times 10^{23}\) atoms/mol, is a constant that indicates the number of constituent particles, usually atoms or molecules, in one mole of a given substance. It provides a bridge between the microscopic world of atoms and the macroscopic quantities of material that we can measure. To find how many atoms are in a gold ring, multiply the moles of gold, calculated from its mass and atomic mass, by Avogadro's number. This results in an enormous number of atoms even in a tiny gold ring. Calculating the number of atoms is key to determining quantities such as the total number of protons, and thereby calculating properties like charge.

Charge Calculation

The charge of an atom is usually measured in coulombs, where one proton has a charge of \(1.6 \times 10^{-19}\) coulombs. To calculate the total positive charge in a gold ring, multiply the total number of protons by the charge of a single proton. This gives the total positive charge brought by the protons in coulombs. Since a neutral gold ring has the same number of protons and electrons, understanding these calculations provides insight into how charges balance in elements, leading to stable substances with no net charge. The mastery of such calculations aids in deeper insights into the atomic structure and electronic configuration of materials.