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Chapter 10: Problem 53

Calcium has a cubic closest packed structure as a solid. Assuming that calcium has an atomic radius of \(197 \mathrm{pm},\) calculate the density of solid calcium.

Short Answer

Expert verified
The density of solid calcium, with a cubic closest packed structure and an atomic radius of \(197 \, pm\), can be calculated as approximately \(1.219 \times 10^{7} \ g/m^3\).

Step by step solution

01

Determine the number of atoms in an FCC unit cell

In an FCC lattice, each corner atom is shared by eight adjacent unit cells and each face-centered atom is shared by two adjacent unit cells. There are 8 corner atoms and 6 face-centered atoms within each unit cell. Thus, the total number of atoms in an FCC unit cell can be calculated as: \[\frac{8}{8} (Corner \ atoms) + \frac{6}{2}(Face \ centered \ atoms) = 1 + 3 = 4 \ atoms\]
02

Mass of one unit cell

Now that we know there are 4 calcium atoms in each unit cell, we need to find the mass of these atoms within the unit cell. The atomic mass of calcium is 40.08 atomic mass units (amu) which is equivalent to 40.08 g/mol. Therefore, the mass of one calcium atom can be found using Avogadro's number, \(N_{A} = 6.022 \times 10^{23} \ mol^{-1}\). Mass of one calcium atom: \[\frac{40.08 \ g/mol}{6.022 \times 10^{23} \ atoms/mol} = 6.66 \times 10^{-23} \ g\] To find the mass of one unit cell, we multiply the mass of one atom by the number of atoms in the unit cell (which is 4): Mass of unit cell: \(4 \times 6.66 \times 10^{-23} \ g = 2.664 \times 10^{-22} \ g\)
03

Find the length of the unit cell

We are given the atomic radius of calcium is \(197 \, pm = 1.97 \times 10^{-10} \ m\). In an FCC structure, the diagonal of the face connecting the opposite face-centered atoms is equal to \(4 \times radius\). Using the Pythagorean theorem and considering that the face diagonal forms a right-angled triangle with the two sides and the length of the unit cell, or edge, as 'a', we get: \( (a)^{2} + (a)^{2} = (4 \times 1.97 \times 10^{-10})^{2} \) We can use this relation to find the length of the unit cell.
04

Calculate the edge length and volume of the unit cell

Rewriting the previous equation and solving for 'a' to find the edge length of the unit cell: \( 2a^{2} = (4 \times 1.97 \times 10^{-10})^{2} \) \( a^{2} = \frac{(4 \times 1.97 \times 10^{-10})^{2}}{2} \) \( a = \sqrt{\frac{(4 \times 1.97 \times 10^{-10})^{2}}{2}} = 2.794 \times 10^{-10} \ m \) Now, we can find the volume of the unit cell by cubing the edge length: Volume of the unit cell: \( V = a^3 = (2.794 \times 10^{-10})^3 = 2.186 \times 10^{-29} \ m^3 \)
05

Calculate the density of solid calcium

With the mass and volume of the unit cell, we can now calculate the density of solid calcium: Density: \( \rho = \frac{Mass}{Volume} = \frac{2.664 \times 10^{-22} \ g}{2.186 \times 10^{-29} \ m^3} = 1.219 \times 10^{7} \ g/m^3 \) Finally, we have the density of solid calcium in g/m鲁: \( \rho \approx 1.219 \times 10^{7} \ g/m^3 \)

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Most popular questions from this chapter

The melting point of a fictional substance \(X\) is \(225^{\circ} \mathrm{C}\) at 10.0 atm. If the density of the solid phase of \(\mathrm{X}\) is 2.67 \(\mathrm{g} / \mathrm{cm}^{3}\) and the density of the liquid phase is 2.78 \(\mathrm{g} / \mathrm{cm}^{3}\) at 10.0 atm, predict whether the normal melting point of \(\mathrm{X}\) will be less than, equal to, or greater than \(225^{\circ} \mathrm{C}\) . Explain.

You and a friend each synthesize a compound with the formula \(\mathrm{XeCl}_{2} \mathrm{F}_{2} .\) Your compound is a liquid and your friend's compound is a gas (at the same conditions of temperature and pressure). Explain how the two compounds with the same formulas can exist in different phases at the same conditions of pressure and temperature.

Some water is placed in a sealed glass container connected to a vacuum pump (a device used to pump gases from a container), and the pump is turned on. The water appears to boil and then freezes. Explain these changes using the phase diagram for water. What would happen to the ice if the vacuum pump was left on indefinitely?

The structure of the compound \(\mathrm{K}_{2} \mathrm{O}\) is best described as a cubic closest packed array of oxide ions with the potassium ions in tetrahedral holes. What percent of the tetrahedral holes are occupied in this solid?

What is an alloy? Explain the differences in structure between substitutional and interstitial alloys. Give an example of each type.
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