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Chapter 2: Problem 20

The radius of an atom of copper (Cu) is about $140 \mathrm{pm} .(\mathbf{a}) \mathrm{Ex}-\( press this distance in millimeters \)(\mathrm{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

Expert verified
The radius of a copper atom is \(140 \times 10^{-9} mm\) and \(140 \times 10^{-2} Å\). To span a distance of 5.0 mm, \(\frac{5.0}{2 \times 140 \times 10^{-9}}\) copper atoms are needed. The volume of a single copper atom, assuming it is a sphere, is \(\frac{4}{3}\pi (140 \times 10^{-12}m)^3 \times 10^{6}~cm^3\).

Step by step solution

01

Convert the radius from picometers to millimeters and angstroms

Given the radius of a copper atom is 140 pm. We need to convert this distance into millimeters (mm) and angstroms (Å). To do this, we will use the following conversion factors: 1 pm = \(10^{-12}\) m 1 mm = \(10^{-3}\) m 1 Å = \(10^{-10}\) m First, convert the radius to meters: \(140/pm \times (1/10^{-12}/m) = 140 \times 10^{-12}m\) Now, convert the radius from meters to millimeters: \(140 \times 10^{-12}m \times (10^{3}/mm) = 140 \times 10^{-9} mm\) And finally, convert the radius from meters to angstroms: \(140 \times 10^{-12}m \times (10^{10}/Å) = 140 \times 10^{-2} Å\)
02

Calculate the number of copper atoms required to span 5.0 mm

We are now given that the total distance we want to span with copper atoms placed side by side is 5.0 mm. To find out how many copper atoms are required, we can use the radius found in step 1 (in mm) and the formula: Number of copper atoms = Total distance / Diameter of one atom Since the diameter of an atom is twice its radius, the diameter of a copper atom is: \(2 \times 140 \times 10^{-9} mm\) Now, calculate the number of copper atoms required: Number of copper atoms = \(\frac{5.0/mm}{2 \times 140 \times 10^{-9}/mm} = \frac{5.0}{2 \times 140 \times 10^{-9}}\)
03

Calculate the volume of a single copper atom, assuming it is a sphere

Now, we need to find the volume of a single copper atom, assuming it's a sphere. To do this, we can use the formula for the volume of a sphere: Volume = \(\frac{4}{3}\pi r^3\) Since we have the radius (in meters) from step 1, we can easily calculate the volume: Volume = \(\frac{4}{3}\pi (140 \times 10^{-12}m)^3\) We are asked for the volume in \(cm^3\), so we need to convert the volume from \(m^3\) to \(cm^3\). We can use the following conversion factor: 1 \(m^3\) = \(10^{6}\) \(cm^3\) So, the volume in \(cm^3\) is: \(\frac{4}{3}\pi (140 \times 10^{-12}m)^3 \times 10^{6}~cm^3\)

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Most popular questions from this chapter

Millikan determined the charge on the electron by studying the static charges on oil drops falling in an electric field (Figure 2.5\()\). A student carried out this experiment using several oil drops for her measurements and calculated the charges on the drops. She obtained the following data: $$ \begin{array}{lc} \hline \text { Droplet } & \text { Calculated Charge (C) } \\ \hline \text { A } & 1.60 \times 10^{-19} \\ \text {B } & 3.15 \times 10^{-19} \\ \text {C } & 4.81 \times 10^{-19} \\ \text {D } & 6.31 \times 10^{-19} \\ \hline \end{array} $$ (a) What is the significance of the fact that the droplets carried different charges? (b) What conclusion can the student draw from these data regarding the charge of the electron? (c) What value (and to how many significant figures) should she report for the electronic charge?

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