Question

A cylindrical piece of sodium is \(12.00 \mathrm{cm}\) long and has a diameter of \(4.5 \mathrm{cm} .\) The density of sodium is \(0.971 \mathrm{g} / \mathrm{cm}^{3}\) How many atoms does the piece of sodium contain? (The volume of a cylinder is \(\left.V=\pi \times r^{2} \times \text { length. }\right)\)

Short answer

The cylindrical sodium piece contains approximately \(4.85 \times 10^{24}\) atoms.

Step-by-step solution

Calculate the Volume of the Cylinder

To find the volume of a cylindrical piece of sodium, we use the formula for the volume of a cylinder: \( V = \pi \times r^2 \times \text{height} \). The radius \( r \) is half the diameter, so \( r = \frac{4.5 \text{ cm}}{2} = 2.25 \text{ cm} \). The height of the cylinder is given as \( 12.00 \text{ cm} \). Substitute these values into the formula: \[ V = \pi \times (2.25 \text{ cm})^2 \times 12.00 \text{ cm} \approx 190.85 \text{ cm}^3. \]

Calculate the Mass of the Sodium Cylinder

Using the density formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \), we can rearrange it to find the mass as \( \text{Mass} = \text{Density} \times \text{Volume} \). The density of sodium is given as \( 0.971 \text{ g/cm}^3 \). Using the volume calculated in step 1, the mass is: \[ \text{Mass} = 0.971 \text{ g/cm}^3 \times 190.85 \text{ cm}^3 \approx 185.23 \text{ g}. \]

Convert the Mass to Moles

To find the number of moles of sodium, use the formula: \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \). The molar mass of sodium is approximately \( 22.99 \text{ g/mol} \). Substitute the mass found in step 2: \[ \text{moles} = \frac{185.23 \text{ g}}{22.99 \text{ g/mol}} \approx 8.06 \text{ moles.} \]

Calculate the Number of Atoms using Avogadro's Number

To convert moles to atoms, multiply the number of moles by Avogadro's number, \( 6.022 \times 10^{23} \text{ atoms/mol} \). Using the moles calculated in step 3: \[ \text{Number of atoms} = 8.06 \text{ moles} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 4.85 \times 10^{24} \text{ atoms}. \]

Key concepts

Volume of a Cylinder

To determine the volume of a cylinder, we use the specific formula: \( V = \pi \times r^2 \times \text{height} \).This equation tells us that the volume depends on the radius of the cylinder's base and its height.
In our example, the sodium cylinder has a diameter of 4.5 cm, so its radius is half of that, which is 2.25 cm.
The height of this cylinder is 12.00 cm. When we apply the cylinder's dimensions to the volume formula, we calculate:\[ V = \pi \times (2.25 \, \text{cm})^2 \times 12.00 \,\text{cm} \approx 190.85 \, \text{cm}^3. \]Understanding this volume is crucial because it helps us analyze the physical space the sodium occupies.Whenever you encounter a cylindrical volume problem, remember to square the radius and apply the height to your formula.

Density Calculation

Density is a measure of how much mass is contained in a given volume. It is expressed mathematically as:\( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \).
In our exercise, the sodium has a known density of 0.971 g/cm³.This relationship allows us to find the mass when the volume is known by rearranging the formula to: \( \text{Mass} = \text{Density} \times \text{Volume} \).
So, we substitute our known values in:\[\text{Mass} = 0.971 \, \text{g/cm}^3 \times 190.85 \, \text{cm}^3 \approx 185.23 \, \text{g}.\]This step is essential as it bridges the volume of our substance to its corresponding mass, making it possible to progress further in calculations.Remember, knowing either density or volume can help you find the other with mass in play.

Avogadro's Number

Avogadro’s Number, \(6.022 \times 10^{23}\) atoms/mol, is a fundamental constant in chemistry. It allows us to convert between moles and the actual number of atoms or molecules. Using it, we can determine how many individual particles are present in molar quantities.
After calculating the moles present in sodium, you can find the number of atoms by multiplying by Avogadro's number: \[ \text{Number of atoms} = 8.06 \, \text{moles} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 4.85 \times 10^{24} \, \text{atoms}.\]By applying this principle, we bridge macroscopic measurements (like grams and moles) with the microscopic world of atoms, making quantitative chemical analysis possible.

Molar Mass

Molar mass is the mass of one mole of a substance, usually in grams/mole. Each element has a characteristic molar mass that can be found on the periodic table.
For sodium, the molar mass is approximately 22.99 g/mol.
This value allows us to convert the mass of a substance to moles using the formula:\( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \).
In this example, with a mass of roughly 185.23 g for sodium, the number of moles is:\[ \text{moles} = \frac{185.23 \, \text{g}}{22.99 \, \text{g/mol}} \approx 8.06 \, \text{moles}.\]Understanding molar mass is crucial for translating between mass and the quantity of substance in chemical reactions and other calculations.