Question

Polonium, the Period 6 member of Group \(6 \mathrm{~A}(16),\) is a rare radioactive metal that is the only element with a crystal structure based on the simple cubic unit cell. If its density is \(9.142 \mathrm{~g} / \mathrm{cm}^{3}\), calculate an approximate atomic radius for polonium.

Short answer

Atomic radius of polonium is calculated using its density and the simple cubic unit cell properties.

Step-by-step solution

- Understand the Unit Cell

Polonium has a simple cubic unit cell, which means there is one atom per unit cell. The side length of the unit cell will be denoted as 'a'.

- Calculate the Volume of the Unit Cell

The volume of the simple cubic unit cell is found using the formula \[ V = a^3 \].

- Use Density to Find the Mass of the Unit Cell

Density (\( \rho \)) is given by mass per unit volume. Rearrange the formula to find the mass:\[ \rho = \frac{m}{V} \] so, \[ m = \rho \times a^3 \].

- Calculate the Mass per Atom

Since there is one atom per unit cell in a simple cubic structure, the mass of one polonium atom is the mass of the unit cell, \[ m_{atom} = m = \rho \times a^3 \].

- Convert Density to Correct Units

Convert the given density \(9.142 \; \mathrm{g}/ \mathrm{cm}^3\) into \( \mathrm{kg}/ \mathrm{m}^3\) since SI units are more accurate for calculations. \(9.142 \; \mathrm{g}/ \mathrm{cm}^3\) equals \(9142 \; \mathrm{kg}/ \mathrm{m}^3\).

- Use Avogadro's Number

We need the number of atoms in a mole (\(6.022 \times 10^{23}\)) and the molar mass of polonium which is approximately 209 g/mol.

- Determine the Volume of One Atom

Using the density equation rearranged to solve for volume and knowing the number of atoms, calculate the volume of one unit cell containing one atom:\[ V = \frac{M}{N \cdot \rho} \] where \(M\) is the molar mass, \(N\) is Avogadro's number, and \(\rho\) is the density.

- Solve for the Side Length of the Unit Cell

Since \(V = a^3\), the side length is found by \[ a = V^{1/3} \].

- Calculate the Atomic Radius

For a simple cubic unit cell, the atomic radius \(r\) is given by \( r = \frac{a}{2} \). Use this to find the atomic radius.

Key concepts

Simple Cubic Unit Cell

A simple cubic unit cell is one of the basic structures in crystallography. It is composed of points that are arranged at the corners of an imaginary cube. Each corner point represents an atom. The simple cubic unit cell is unique in that it has only one atom per unit cell. This is because each corner atom is shared among eight neighboring cells in a 3D grid. Therefore, the total contribution of each corner atom to the unit cell is \(\frac{1}{8}\) per corner. This simple arrangement often makes understanding and calculations straightforward, but less common than other more efficient packing structures.

Density

Density is a key factor in determining various properties of materials. It is defined as mass per unit volume and is expressed with the formula \(\rho = \frac{m}{V}\). For solids, density tells us how much mass is packed into a given volume. In the context of the simple cubic unit cell of polonium, knowing its density helps in calculating the mass of the unit cell. For instance, with a given density of 9.142 \(\text{g/cm}^{3}\) or 9142 \(\text{kg/m}^{3}\), we can easily find the mass of a unit cell once we determine its volume. Density is essential for understanding the physical context of the crystal structure.

Crystal Structure

The crystal structure of a material determines many of its physical properties. A crystal structure refers to the arrangement of atoms in a crystal. For polonium, the simple cubic unit cell is the fundamental building block of its crystal structure. Polonium is unique as it is the only element naturally occurring in this type of structure. The arrangement of atoms in a crystal directly affects its density, stability, and how it interacts with other substances. Crystallographers often study these arrangements to predict and explain the behavior of materials. In a simple cubic structure, understanding how atoms fit together helps in calculating attributes like atomic radius.

Avogadro's Number

Avogadro's Number is a constant that is fundamental in chemistry and physics. It is the number of atoms, molecules, or particles in one mole of a substance, approximately 6.022 \(\times 10^{23}\). This number allows us to convert between atomic scale measurements and macroscopic quantities. In terms of the crystal structure of polonium, Avogadro's Number enables us to relate the mass of the unit cell to the molar mass of polonium. By knowing the molar mass and using Avogadro's Number, we can find out the volume occupied by one mole worth of atoms, and eventually, determine the size of individual atoms. This is critical for nuclear and atomic physics, as well as in various applications in material science.