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Chapter 22: Problem 92

Polonium is the only element known to crystallize in the simple cubic form. In this structure, the interatomic distance between a Po atom and each of its six nearest neighbors is \(335 \mathrm{pm}\). Use this description of the crystal structure to estimate the density of polonium.

Short Answer

Expert verified
The estimated density of polonium is approximately \(9.23 \, g/cm^3\).

Step by step solution

01

- Calculation of Unit Cell Volume

The unit cell of a simple cubic structure is a cube, and the length of each side of the cube is equal to the given interatomic distance. Convert this distance from picometers to centimeters:\[335 \mathrm{pm} * \frac{100 \mathrm{cm}}{1 \mathrm{m}} * \frac{1 \mathrm{m}}{10^{12}\mathrm{pm}} = 3.35 \times 10^{-8} \mathrm{cm}\] Now calculate the volume (\(V\)) of the cube using the formula \(V=a^3\), so \(V=(3.35\times 10^{-8})^3=3.76\times 10^{-23}\) cubic centimeters.
02

- Calculation of Density

Density (\(\rho\)) is mass divided by volume. In one unit cell, there's one Po atom, so the mass of one unit cell is the molar mass of Po divided by Avogadro's number. Covert the molar mass of Po from g/mol to g: \[209 \, g/mol * \frac{1 \, mol}{6.022 \times 10^{23} \, atoms} = 3.47 \times 10^{-22} \, g\] Now, use the formula \(\rho=m/V\) to find the density and express the result in g/cm^3.\[\rho = \frac{3.47 \times 10^{-22} \, g}{3.76 \times 10^{-23} \, cm^3} = 9.23 \, g/cm^3\]
03

- Report the Result

After completing the steps above and performing the calculations, report the result for the density of polonium which is found to be around \(9.23 \, g/cm^3\)

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Most popular questions from this chapter

Use Lewis structures and other information to explain the observation that (a) the oxygen-to-oxygen bond lengths in \(\mathrm{O}_{2}, \mathrm{O}_{3}\) and \(\mathrm{H}_{2} \mathrm{O}_{2}\) are \(121,128,\) and \(148 \mathrm{pm},\) respectively. (b) the oxygen-to-oxygen bond length of \(\mathrm{O}_{2}\) is \(121 \mathrm{pm}\) and for \(\mathrm{O}_{2}^{+}\) is \(112 \mathrm{pm}\). Why is the bond length for \(\mathrm{O}_{2}^{+}\) so much shorter than for \(\mathrm{O}_{2} ?\)

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