Question

A \(25.0 \mathrm{~mm} \times 40: 0 \mathrm{~mm}\) piece of gold foil is \(0.25 \mathrm{~mm}\) thick. The density of gold is \(19.32 \mathrm{~g} / \mathrm{cm}^{3}\). How many gold atoms are in the sheet? (Atomic weight : \(\mathrm{Au}=197.0\) ) (a) \(7.7 \times 10^{23}\) (b) \(1.5 \times 10^{23}\) (c) \(4.3 \times 10^{21}\) (d) \(1.47 \times 10^{22}\)

Short answer

The number of gold atoms in the sheet is approximately 7.7 x 10^23.

Step-by-step solution

Calculate the volume of the gold foil

Firstly, convert the dimensions of the gold foil from millimeters to centimeters and then calculate its volume. Dimensions in centimeters are: 2.50 cm by 4.00 cm by 0.025 cm. The volume is volume = length * width * height = 2.50 cm * 4.00 cm * 0.025 cm.

Calculate the mass of the gold foil

Using the density formula (mass = density * volume), multiply the volume from Step 1 by the density of gold (19.32 g/cm^3) to calculate the mass of the gold foil.

Convert the mass to grams

Ensure that the mass calculated in Step 2 is in grams because the atomic weight is given in grams per mole.

Calculate moles of gold

Use the atomic weight of gold (197.0 g/mol) and the mass of the gold foil to calculate the number of moles of gold (moles = mass / atomic weight).

Calculate the number of gold atoms

Use Avogadro's number (6.022 x 10^23 atoms/mol) to convert the number of moles of gold to the number of atoms (atoms = moles * Avogadro's number).

Check against the options

Match your calculated number of atoms to one of the options provided in the question.

Key concepts

Density and Volume Calculations

Understanding density and volume is crucial when determining the amount of substance in a given space. Density is defined as mass per unit volume (\frac{g}{cm^3}). To solve problems involving density, it is often necessary to calculate the volume of the object first, as was the case with the gold foil in our exercise.

To calculate the volume, you need the three spatial dimensions of the object. In the context of the foil, its length, width, and thickness were provided in millimeters, but it's important to convert these measurements into centimeters since density is given in grams per cubic centimeter (\frac{g}{cm^3}). Once the dimensions are in the correct units, the volume can be calculated using the formula: \[ \text{volume} = \text{length} \times \text{width} \times \text{height} \].

After obtaining the volume, you can find the mass of the gold foil by using the formula: \[ \text{mass} = \text{density} \times \text{volume} \]. It is essential to ensure the units are consistent to avoid any calculation errors.

Stoichiometry and Moles

Stoichiometry is a section of chemistry that deals with the quantitative relationships between the substances involved in chemical reactions. More commonly, stoichiometry is the process used to calculate the amounts of substances consumed and produced in a chemical reaction. In this scenario, stoichiometry refers to the conversion between mass and moles using the molar mass.

A mole is a standard scientific unit for measuring large quantities of very small entities such as atoms, molecules, or other particles. One mole is Avogadro's number (6.022 \times 10^{23}) of particles. The molar mass is the mass of one mole of a substance and has units of grams per mole (\frac{g}{mol}).

To calculate the number of moles from mass, the formula used is: \[ \text{moles} = \frac{\text{mass}}{\text{molar mass}} \]. In the gold foil example, the molar mass of gold (Au) is 197.0 g/mol. By dividing the mass of the gold foil by this molar mass, you find the number of moles of gold in the foil.

Avogadro's Number Application

Avogadro's number is a fundamental constant in chemistry, representing the number of atoms, ions, or molecules in one mole of a substance. It is defined as exactly 6.02214076 × 10^23 particles per mole.

Applying Avogadro's number is often the final step in conversions from grams to particles (atoms, molecules, etc.). After calculating the number of moles of a substance, you multiply this value by Avogadro's number to find out how many individual particles you have. For the exercise focusing on gold foil, once you have calculated the number of moles of gold, you determine the number of gold atoms by multiplying the moles by Avogadro's number, according to the formula:\[ \text{atoms} = \text{moles} \times \text{Avogadro's number} \].

This application of Avogadro's number allows us to relate a quantity of matter that is easy to weigh with one that involves countable particles, bridging the macroscopic world of grams with the microscopic world of atoms.