Question

The diameter of a copper (Cu) atom is roughly \(1.3 \times\) \(10^{-10} \mathrm{~m} .\) How many times can you divide evenly a piece of 10 -cm 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

The 10-cm copper wire can be divided evenly approximately 36 times until it is reduced to two separate copper atoms.

Step-by-step solution

Convert wire's length to meters

We first convert the length of the wire from centimeters to meters because the diameter of the copper atom is also given in meters. Given that 1 meter = 100 cm, the length of a 10-cm copper wire is \(10/100 = 0.1\) m.

Calculate the number of copper atoms in the wire

Then, we calculate the number of copper atoms that could be laid end-to-end along the length of the wire. This is done by dividing the total length of the wire (0.1 m) by the diameter of a single copper atom (\(1.3 \times 10^{-10}\) m). This results in approximately \(0.1/ (1.3 \times 10^{-10}) \)which equals to \(7.69 \times 10^{10}\) copper atoms.

Calculate the number of times the wire could be divided evenly

The problem asks for the number of divisions required until the wire is reduced to just two individual copper atoms. Each division effectively halves the number of atoms. Thus, we are looking for the number of times we can divide \(7.69 \times 10^{10}\) by 2 until we get to 2. This is a logarithmic operation with base 2. The formula to calculate the number of divisions is \(log_{2}(7.69 \times 10^{10})\). This calculation gives us 36.22...

Round off to an Integer

Since it's impossible to make a fraction of a division, we round the value we obtained in the previous step to the nearest whole number. In this case, we round 36.22 to 36.

Key concepts

Atomic Diameter

The atomic diameter refers to the width across an atom, measured from one side of its electronic cloud to the other. For a copper (Cu) atom, this measurement is approximately \(1.3 \times 10^{-10}\) meters. This tiny dimension provides insight into the scale at which atoms interact with each other.
Atoms are the building blocks of all matter, and understanding their size helps us comprehend concepts in material science and chemistry. When comparing atomic diameters, they are often expressed in meters or nanometers for precision. An atom's size affects properties like electrical conductivity and heat transfer, making this measurement crucial in scientific applications.

Division Process

Dividing a piece of copper wire at the atomic level involves understanding the concept of sequential halving. Starting with a 10-cm copper wire, which converts to 0.1 meters, we ascertain how many copper atoms lay aligned in this length by dividing it by the diameter of a single copper atom.
Once we know the number of atoms, we use a division process to count how many times the wire can be evenly split. Each halving action reduces the number of atoms in the wire until only two atoms are left. This thought experiment helps visualize the scale and number of atoms making up everyday objects.

Logarithms

Logarithms are incredibly useful when solving problems involving repeated division or multiplication, such as finding how many halving steps result in two atoms. A logarithm, with a base of 2 in this context, helps calculate the repeated divisions needed.
To determine how many times the original number of atoms can be divided to reach two atoms, we take the logarithm base 2 of the initial number of atoms, \(7.69 \times 10^{10}\). The formula \(log_{2}(7.69 \times 10^{10})\) simplifies this problem, giving us an approximate number of 36 divisions. It's an efficient mathematical tool for problems involving exponential changes or reductions.

Measurement Conversion

When dealing with measurements across different units, conversions are essential. In this exercise, converting the length of copper wire from centimeters to meters was a critical first step. Knowing that 1 meter equals 100 centimeters, we find that 10 cm is equivalent to 0.1 meters.
Such conversions ensure consistency, especially when comparing quantities like atomic diameters in scientific calculations. Always remember to verify units are compatible; this checks accuracy and aids in clear communication of scientific ideas. Measurement conversion is fundamental for anyone working in fields where precision is mandatory.