Question

Argon has a cubic closest packed structure as a solid. Assuming that argon has a radius of \(190 . \mathrm{pm}\), calculate the density of solid argon.

Short answer

The density of solid argon is approximately 1.72 g/ cm³.

Step-by-step solution

Calculate the edge length of the unit cell

Since argon has a cubic closest packed (ccp) structure, its unit cell is face-centered cubic (fcc). In a face-centered cubic structure, the edge length 'a' and the radius 'r' are related by the formula: \(a = 2\sqrt{2}r\). The radius of argon is given as 190 pm. Plug in the value of r to find the edge length of the unit cell: \[a = 2\sqrt{2}(190\,\text{pm})\] \[a \approx 536.66\,\text{pm}\]

Calculate the volume of the unit cell

The volume V of a cubic unit cell can be calculated using the edge length 'a'. The formula for the volume is: \(V = a^3\). Plug in the value of a obtained in the previous step to calculate the volume of the unit cell: \[V = (536.66\,\text{pm})^3\] \[V \approx 1.54\times 10^8\,\text{pm}^3\]

Calculate mass of argon atoms in the unit cell

In an fcc unit cell, there are four atoms per unit cell. The molar mass of argon is 39.95 g/mol. We can calculate the mass of one argon atom by dividing the molar mass by Avogadro's number (\(6.022\times10^{23}\,\text{atoms/mol}\)): \[mass_{Ar} = \frac{39.95\,\text{g/mol}}{6.022\times10^{23}\,\text{atoms/mol}} \approx 6.64\times10^{-23}\,\text{g/atom}\] To find the mass of four argon atoms in the unit cell, multiply the mass of one argon atom by 4: \[mass_{unit\ cell} = 4 (6.64\times10^{-23}\,\text{g/atom}) = 2.656\times10^{-22}\,\text{g}\]

Convert the volume of the unit cell to cm³

To calculate the density, we need to convert the volume of the unit cell from \(\text{pm}^3\) to cm³. We can use the conversion factor of \(1\,\text{cm} = 10^{12}\,\text{pm}\): \[V\,(\text{cm}^3) = \frac{1.54\times10^8\,\text{pm}^3}{(10^{12}\,\text{pm/cm})^3} = 1.54\times10^{-16}\,\text{cm}^{3}\]

Calculate the density of solid argon

Finally, we can calculate the density of solid argon using the formula \(\rho = \frac{mass}{volume}\): \[\rho_{Ar} = \frac{2.656\times10^{-22}\,\text{g}}{1.54\times10^{-16}\,\text{cm}^3} \approx 1.72\,\text{g/cm}^3\] The density of solid argon is approximately 1.72 g/ cm³.