Question

The radius of a copper (Cu) atom is roughly \(1.3 \times 10^{-10} \mathrm{~m}\). How many times can you divide evenly a \(10-\mathrm{cm}\) -long piece of copper wire until it is reduced to two separate copper atoms? (Assume there are appropriate tools for this procedure and that copper atoms are lined up in a straight line, in contact with each other. Round off your answer to an integer.)

Short answer

You can divide the wire 384,615,383 times.

Step-by-step solution

Understand the Problem

We need to find out how many times we can divide a 10-cm copper wire into separate copper atoms, with each atomic separation considered as a single division. The radius of each copper atom is given, and we need to calculate in terms of atoms.

Calculate the Diameter of a Copper Atom

Since the radius of a copper atom is given as \(1.3 \times 10^{-10} \text{ m}\), the diameter can be calculated as twice the radius. Thus, the diameter \(d\) of a copper atom is \[d = 2 \times 1.3 \times 10^{-10} = 2.6 \times 10^{-10} \text{ m}.\]

Convert Length of Wire to Meters and Calculate Atomic Count

Convert the length of the copper wire from centimeters to meters:\[10 \text{ cm} = 0.1 \text{ m}.\]To find how many atoms fit along this length:\[\text{Number of atoms} = \frac{0.1}{2.6 \times 10^{-10}}.\]

Perform the Calculation

Perform the division:\[\text{Number of atoms} = \frac{0.1}{2.6 \times 10^{-10}} \approx 3.846 \times 10^8.\]

Determine the Number of Divisions

The number of divisions needed to separate the atoms is one less than the total number of atoms because each division separates the wire into two parts: \[\text{Number of divisions} = 3.846 \times 10^8 - 1 \approx 384,615,383.\]

Round the Answer to the Nearest Integer

The previous calculation needs to be rounded off to the nearest integer as instructed.Since \(384,615,383\) is already an integer close to our calculated result, it remains unchanged.

Key concepts

Atomic Radius

The atomic radius is a fundamental concept in chemistry and physics. It describes the size of an atom, which is essential for understanding many properties of chemical elements. Specifically, for our exercise involving copper, the atomic radius is given as \(1.3 \times 10^{-10} \text{ m}\). This value represents the distance from the center of an atom's nucleus to the outer boundary of the surrounding cloud of electrons. Although atomic radius can vary slightly depending on the atom's environment, it gives us a useful approximate size. This measurement allows scientists to predict how atoms will interact with one another in various environments. It's crucial in calculations involving atomic size, such as determining how many atoms fit along a piece of material, which is the basis of our exercise.

Copper Wire

Copper wire is an everyday example of a material composed of countless copper atoms. These atoms are organized in a straight line along the wire length in our exercise. Copper is favored due to its excellent electrical and thermal conductivity, which is largely because of its atomic structure. Understanding how copper wire is made starts with copper atoms being tightly packed together in a lattice structure. Because of this atomic arrangement, one can precisely calculate how many atoms form a section of wire. Given our 10-cm copper wire, converting its length to meters (0.1 m) simplifies calculations and helps in determining how these copper atoms line up within the wire.

Atomic Separation

Atomic separation refers to the spacing or arrangement of atoms within a material. In this exercise, we consider atoms being in direct contact along the length of the copper wire. This implies that each adjacent pair of copper atoms are separated by the diameter of one copper atom. To know how many such separations (or divisions) can occur, we divide the total length of the wire by the diameter of one copper atom. This number tells us how many times you would need to "cut" the wire to isolate two individual atoms. Each cut is considered as an atomic separation in this context. By knowing the diameter, we can figure out how many atoms are aligned in the wire and therefore calculate potential atomic separations.

Diameter of Atom

The diameter of an atom is crucial for calculations involving atomic separation. While the exercise initially provides the atomic radius of copper, the diameter is twice the radius. For copper atoms, this calculation is expressed as:\[d = 2 \times 1.3 \times 10^{-10} = 2.6 \times 10^{-10} \text{ m}\]This value is essential because the diameter determines the space each atom occupies when lined up. Every atom's diameter is necessary to calculate how many fit within a given distance—in this case, the 10-cm length of copper wire. It allows us to determine the number of times the wire can be segmented until only individual copper atoms remain. Understanding the concept of atomic diameter is therefore key to accurately performing these types of calculations.