Problem 75 - Chemistry The Molecular Science

Question

If you have a ring that contains \(1.94 \mathrm{~g}\) gold, calculate how many gold atoms are in the ring.

Step-by-Step Solution

Answer

The ring contains approximately \(5.93 \times 10^{21}\) gold atoms.

Step 1: Identify the known values

To solve the problem, we first need to list the known values:- The mass of gold: 1.94 g- Molar mass of gold (Au): 197 g/mol. This is the standard atomic weight of gold.- Avogadro's number: \(6.022 \times 10^{23}\) atoms/mol.

Step 2: Calculate moles of gold

Use the molar mass of gold to convert the given mass into moles. The formula to calculate moles from mass is: \[\text{moles} = \frac{\text{mass in grams}}{\text{molar mass in g/mol}}\]Substitute the known values:\[\text{moles of gold} = \frac{1.94 \, \text{g}}{197 \, \text{g/mol}} \approx 0.00985 \, \text{mol}\]

Step 3: Calculate the number of gold atoms

To find the number of atoms, multiply the moles of gold by Avogadro's number. The formula is:\[\text{Number of atoms} = \text{moles} \times \text{Avogadro's number}\]Substitute the values we have:\[\text{Number of gold atoms} = 0.00985 \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 5.93 \times 10^{21} \, \text{atoms}\]

Key Concepts

Molar Mass

Understanding molar mass helps us relate mass and moles. The molar mass of an element or compound is the mass of one mole of its particles. For atoms like gold (Au), the molar mass matches its standard atomic weight measured in grams per mole (g/mol).
For gold, the molar mass is approximately 197 g/mol, reflecting the average mass of a mole of gold atoms.

• This value accounts for isotopic distribution of gold but is often rounded when used in calculations.

When calculating the mass-to-moles conversion, we use the formula:

• \( \text{moles} = \frac{\text{mass in grams}}{\text{molar mass in g/mol}} \)

For example, using this formula, we determine that 1.94 grams of gold equals approximately 0.00985 moles. Understanding this conversion is key for further calculations involving the number of atoms.

Avogadro's Number

Avogadro's number, a fundamental constant in chemistry, allows us to scale atomic scale quantities to macroscopic levels. It is the number of particles, usually atoms or molecules, in one mole of a substance.

• Numerically, Avogadro's number is \( 6.022 \times 10^{23} \) particles per mole.

Avogadro's number was conceptualized to provide a bridge between the atomic scale and our everyday experiences with matter. It's how chemists count chemical entities by weighing them.

• For ease of calculation, consider that one mole of gold atoms equals \( 6.022 \times 10^{23} \) gold atoms.

Using Avogadro's number in calculations, such as determining the number of gold atoms in a given mass, involves multiplying the number of moles by this constant, which expands our understanding from single atoms to an actual, observable quantity.

Mole Concept

The mole is a central concept in chemistry, serving as a bridge between the atomic and macroscopic worlds. It lets us understand how the tiny particles that make up matter translate into quantities we can see and measure.
Key aspects of the mole concept include:

• One mole is defined as \( 6.022 \times 10^{23} \) particles, also known as Avogadro’s number.
• It represents a definite number of atoms, molecules, or ions present in a substance.

When using moles in calculations:

• You can calculate moles from mass using the formula \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \).
• The number of molecules or atoms can be found by multiplying the number of moles by Avogadro’s number.

In our example of gold, where 1.94 grams translates into 0.00985 moles, we then multiply by Avogadro’s number to find the total count of atoms, helping us quantify the microscopic world.