Question

The radius of an atom of copper (Cu) is about 140 pm. (a) EXpress this distance in millimeters (mm) and in angstroms \((\AA)\). (b) How many Cu atoms would have to be placed side by side to span a distance of \(5.0 \mathrm{~mm} ?(\mathbf{c})\) If you assume that the Cu atom is a sphere, what is the volume in \(\mathrm{cm}^{3}\) of a single atom?

Short answer

(a) 1.40×10^{-10} mm and 1.4 Å; (b) 1.79×10^{10} atoms; (c) 1.15×10^{-24} cm^3.

Step-by-step solution

Convert Picometers to Millimeters

Given the radius of a copper atom is 140 pm. To convert picometers (pm) to millimeters (mm), use the conversion factor: \[1 \, \text{mm} = 10^{12} \, \text{pm}\]Thus, \[140 \, \text{pm} = \frac{140}{10^{12}} \, \text{mm} = 1.40 \times 10^{-10} \, \text{mm}\]

Convert Picometers to Angstroms

To convert picometers to angstroms (Å), use the conversion factor: \[1 \, \text{Å} = 100 \, \text{pm}\]Thus, \[140 \, \text{pm} = \frac{140}{100} \, \text{Å} = 1.4 \, \text{Å}\]

Calculate Number of Atoms in 5.0 mm

To find out how many copper atoms are needed to span 5.0 mm, first use the radius of one atom in millimeters:\[1.40 \times 10^{-10} \, \text{mm per atom}\]Use the formula:\[\text{Number of atoms} = \frac{\text{Total distance}}{\text{Diameter of one atom}}\]Since the diameter is twice the radius:\[\text{Diameter} = 2 \times 1.40 \times 10^{-10} \, \text{mm} = 2.80 \times 10^{-10} \, \text{mm}\]Thus,\[\text{Number of atoms} = \frac{5.0 \, \text{mm}}{2.80 \times 10^{-10} \, \text{mm}} \approx 1.79 \times 10^{10}\]

Calculate Volume of a Copper Atom

Assuming the copper atom is a sphere with radius 140 pm:To find the volume, use the formula for the volume of a sphere:\[V = \frac{4}{3} \pi r^3\]Convert the radius into centimeters from picometers:\[\text{Radius} = 140 \, \text{pm} = 140 \times 10^{-12} \, \text{m} = 1.4 \times 10^{-10} \, \text{m} = 1.4 \times 10^{-8} \, \text{cm}\]Now, calculate the volume:\[V = \frac{4}{3} \pi (1.4 \times 10^{-8} \text{cm})^3 \approx 1.15 \times 10^{-24} \text{cm}^3\]

Key concepts

Unit Conversion

Understanding unit conversion is crucial when dealing with atomic scale measurements. The radius of a copper atom is given as 140 pm (picometers). To work with a wide range of applications, converting units into more familiar ones like millimeters (mm) or angstroms (Å) is necessary.
- **From Picometers to Millimeters**: Since 1 millimeter is equivalent to \(10^{12}\) picometers, converting 140 pm to mm requires dividing 140 by \(10^{12}\). This results in approximately \(1.4 \times 10^{-10} \text{ mm}\).
- **From Picometers to Angstroms**: With 1 angstrom equal to 100 picometers, dividing 140 by 100 gives us \(1.4 \, \text{Å}\).
Such conversions allow scientists to uniformly communicate values across different scientific disciplines easily. Understanding these conversion factors also ensures accurate recalibrations necessary for precise experimental measurements.

Copper Atom

The copper atom is of great interest in both chemistry and physics. As a fundamental building block of the element copper, it has a specific radius approximately equal to 140 pm. The properties of copper atoms make them a key subject of study.
- **Atomic Radius**: This is the measure from the nucleus of an atom to its outermost electron. In copper, this is 140 pm, which is exceptionally small, demonstrating the scale at which atomic structures exist.
- **Atomic Arrangement**: To understand how atomic structures translate to visible measured quantities, we calculate how many atoms would line up over a certain distance. For instance, if we need a line of copper atoms to reach 5.0 mm, we first find the diameter (twice the radius) and then divide the total length by this diameter.
Understanding the basic structure and calculations involving copper atoms lays the groundwork for future explorations in chemistry and material science.

Spherical Volume Calculation

In the realm of atomic structures, particularly when visualizing atoms, it's often useful to assume spherical shapes for simplicity. Calculating the volume of a spherical object like an atom involves using specialized formulas.
- **Sphere Volume Formula**: The general formula for a sphere's volume is \( V = \frac{4}{3} \pi r^3 \). Here \( r \) refers to the radius, which in the case of copper, is 140 pm.
- **Calculating Atomic Volume**: The first step involves converting the radius into centimeters, since volume is often more conveniently expressed in \( \text{cm}^3 \). Since 1 picometer is \(10^{-12}\) meters, the radius becomes \(1.4 \times 10^{-8} \text{ cm}\). Using this in our volume formula gives an approximate volume of \(1.15 \times 10^{-24} \text{cm}^3\) for a single copper atom.
Understanding how to calculate this helps visualize the minute yet significant mass of individual atoms in atomic-level physics and chemistry.