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Chapter 1: Problem 64

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

Expert verified
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

01

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.
02

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.
03

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...
04

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.

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Most popular questions from this chapter

Do these statements describe chemical or physical properties? (a) Oxygen gas supports combustion. (b) Fertilizers help to increase agricultural production. (c) Water boils below \(100^{\circ} \mathrm{C}\) on top of a mountain. (d) Lead is denser than aluminum. (e) Uranium is a radioactive element.

The price of gold on a certain day in 2004 was \(\$ 315\) per troy ounce. How much did \(1.00 \mathrm{~g}\) of gold cost that day? \((1\) troy ounce \(=31.03 \mathrm{~g} .)\)

How many minutes does it take light from the sun to reach Earth? (The distance from the sun to Earth is 93 million mi; the speed of light \(=3.00 \times\) \(\left.10^{8} \mathrm{~m} / \mathrm{s} .\right)\)

Which of these statements is scientifically correct? "The mass of the student is \(56 \mathrm{~kg}\)." "The weight of the student is \(56 \mathrm{~kg}\)."

(a) Convert the following temperatures to kelvin: (i) \(113^{\circ} \mathrm{C},\) the melting point of sulfur, (ii) \(37^{\circ} \mathrm{C}\), the normal body temperature, (iii) \(357^{\circ} \mathrm{C},\) the boiling point of mercury. (b) Convert the following temperatures to degrees Celsius: (i) \(77 \mathrm{~K}\), the boiling point of liquid nitrogen, (ii) \(4.2 \mathrm{~K},\) the boiling point of liquid helium, (iii) \(601 \mathrm{~K}\), the melting point of lead.
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