Problem 24 - Chemistry: Principles and Reactions

Question

A cylindrical piece of pure copper \(\left(d=8.92 \mathrm{~g} / \mathrm{cm}^{3}\right)\) has diameter \(1.15 \mathrm{~cm}\) and height \(4.00\) inches. How many atoms are in that cylinder? (Note: the volume of a right circular cylinder of radius \(r\) and height \(h\) is \(V=\pi r^{2} h .\) )

Step-by-Step Solution

Answer

Answer: The cylindrical piece of pure copper contains approximately \(8.03 \times 10^{23}\) atoms.

Step 1: Calculate the volume of the cylinder

To find the volume of the cylinder, we use the formula \(V = \pi r^2h\), where \(r\) is the radius, and \(h\) is the height. We are given the diameter, which is twice the radius, so \(r = \frac{1.15}{2}\) cm. The height is given as 4.00 inches, which we need to convert to cm. We know that 1 inch is equal to 2.54 cm, so the height in cm is \(4.00 \times 2.54 = 10.16\) cm. Now we can find the volume: \(V = \pi \left(\frac{1.15}{2}\right)^2 (10.16) \approx 9.485 \mathrm{~cm}^{3}\).

Step 2: Calculate the mass of the copper cylinder

Now that we have the volume, we can use the density to find the mass of the copper. The density formula is: \(Density = \frac{Mass}{Volume}\) Rearranging to solve for mass, we get: \(Mass = Density \times Volume\) The density of copper is given as \(8.92 \mathrm{~g} / \mathrm{cm}^{3}\), so the mass of the cylinder is: \(Mass \approx 8.92 \times 9.485 \approx 84.698 \mathrm{~g}\)

Step 3: Determine the number of moles of copper

To find the number of moles of copper in the cylinder, we need to divide the mass by the molar mass of copper. The molar mass of copper is approximately 63.546 g/mol. Therefore, the number of moles in the cylinder is: \(Moles = \frac{84.698}{63.546} \approx 1.333 \mathrm{~mol}\)

Step 4: Calculate the number of atoms using Avogadro's number

Finally, we can use Avogadro's number to find the number of atoms in the copper cylinder. Avogadro's number is approximately \(6.022 \times 10^{23}\) atoms/mol. So, the number of atoms in the cylinder is: \(Atoms \approx 1.333 \times 6.022 \times 10^{23} \approx 8.03 \times 10^{23}\) atoms The cylindrical piece of pure copper contains approximately \(8.03 \times 10^{23}\) atoms.

Key Concepts

Volume of Cylinder

To calculate how much space a cylindrical object occupies, we use the volume formula for a cylinder: \[ V = \pi r^2 h \] Here, \( r \) represents the radius of the cylinder's base, and \( h \) is its height. For our problem, the diameter of the cylinder is given as 1.15 cm, making the radius half of that, or \( r = \frac{1.15\, \text{cm}}{2} = 0.575\, \text{cm} \). The height was provided in inches (4.00 inches), and we first need to convert it to centimeters, using the conversion rate: 1 inch = 2.54 cm. Thus, the height in centimeters is: \[ h = 4.00 \times 2.54 = 10.16\, \text{cm} \] Plugging these values into the volume formula gives us the volume: \[ V = \pi (0.575)^2 (10.16) \approx 9.485\, \text{cm}^3 \] Understanding the volume allows us to proceed with other calculations, like finding mass or number of atoms.

Density of Copper

Density connects the concepts of mass and volume with the formula: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] From this, we can rearrange to find the mass if we have density and volume: \[ \text{Mass} = \text{Density} \times \text{Volume} \]For copper, a common metal, the density is given as 8.92 g/cm³. Multiply this by the volume of our cylinder (9.485 cm³) previously calculated to find the mass: \[ \text{Mass} \approx 8.92 \times 9.485 \approx 84.698\, \text{g} \] This signifies that the copper cylinder weighs about 84.698 grams. Understanding density is essential in determining properties like weight from volume.

Molar Mass of Copper

The molar mass of an element tells us how much one mole of that substance weighs in grams. For copper, the molar mass is about 63.546 g/mol. This measurement is a crucial bridge between the mass of a substance and the amount of substance, expressed in moles.To calculate the number of moles of copper, divide the mass of the copper by its molar mass: \[ \text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}} = \frac{84.698}{63.546} \approx 1.333 \text{ mol} \] This tells us how many "potentially reactive" units (moles) of copper are present in our sample. Calculating moles is a fundamental skill in chemistry, connecting microscopic quantities with macroscopic measurements.

Avogadro's Number

Named after scientist Amedeo Avogadro, Avogadro's number \( 6.022 \times 10^{23} \) represents the number of particles—in this context, atoms—in one mole of a substance. It helps chemists understand quantities at a molecular level. To find the total number of atoms in the copper cylinder, use Avogadro's number in conjunction with the number of moles calculated:\[ \text{Atoms} = \text{Moles} \times \text{Avogadro's Number} = 1.333 \times 6.022 \times 10^{23} \approx 8.03 \times 10^{23} \text{ atoms} \]Thus, the copper cylinder contains approximately \( 8.03 \times 10^{23} \) atoms. This calculation demonstrates how vast the atomic world is even in seemingly small macroscopic quantities.