Question

Crystalline Structures and Unit Cells Polonium crystallizes with a simple cubic structure. It has a density of 9.3 \(\mathrm{g} / \mathrm{cm}^{3}\) , a radius of \(167 \mathrm{pm},\) and a molar mass of 209 \(\mathrm{g} / \mathrm{mol}\) . Use these data to estimate Avogadro's number (the number of atoms in one mole).

Short answer

Avogadro's number is estimated by taking the molar mass of Polonium and dividing it by the mass of one atom determined from its unit cell.

Step-by-step solution

Determine the Volume of the Unit Cell

Since Polonium crystallizes in a simple cubic structure, each edge of the unit cell is twice the atomic radius. Calculate the volume using the formula for the volume of a cube: \( V = a^3 \), where \( a = 2r \), and \( r = 167 \times 10^{-10} \text{ cm} \).

Calculate the Mass of One Unit Cell

Use the density formula \( \rho = \frac{m}{V} \), where \( \rho \) is the density of Polonium. Solving for mass \( m \), we get \( m = \rho \times V \). Plug in the density, 9.3 \( \text{g/cm}^3 \), and the volume of the unit cell from Step 1.

Determine the Number of Atoms per Unit Cell

In a simple cubic structure, there is one atom per unit cell. So, the mass obtained in Step 2 corresponds to one Polonium atom.

Calculate Avogadro's Number

Using the molar mass of Polonium, which is 209 \( \text{g/mol} \), calculate Avogadro's number by dividing the molar mass by the mass of one atom in the unit cell: \( N_A = \frac{ \text{Molar mass} }{ \text{Mass of one atom} } \).

Key concepts

Crystalline Structures

Materials such as Polonium take form in repeating patterns called crystalline structures, which greatly influence their physical properties. A crystalline structure is composed of atoms, ions, or molecules arranged in a pattern that repeats itself in three dimensions. This orderly arrangement affects how materials interact with light, how they conduct electricity and heat, among other characteristics.

There are several types of crystalline structures, each with its unique geometry and properties. The primary structures include simple cubic, body-centered cubic, face-centered cubic, hexagonal close-packed, and more. Understanding the specific structure of a crystal is crucial for determining various other properties, such as how atoms are packed together, which in turn affects density and strength.

Simple Cubic Structure

Among the crystalline structures is the simple cubic structure, characterized by its simplicity and symmetry. In a simple cubic structure, each corner of a cube is occupied by an atom, and all sides are of equal length. These identical points, known as lattice points, create an overall shape of a cube that is repeated throughout the material.

A key trait of the simple cubic lattice is that each unit cell - the smallest repeating unit in a crystal - contains exactly one atom. This stems from the fact that each atom at the corner is shared by eight adjacent unit cells. As a result, the simple cubic structure is not a particularly dense packing arrangement compared to other structures, which means more space between atoms.

Unit Cell Volume

The volume of a unit cell is a critical measurement when exploring crystalline structures since it dictates the number of atoms per unit volume and contributes to the calculation of density. To calculate the volume of the unit cell, particularly in a simple cubic structure, we can use the formula for the volume of a cube, \( V = a^3 \) where \( a \) is the length of the edge of the cube.

To determine the edge length in a simple cubic crystal, remember that each side is twice the atomic radius: \( a = 2r \). Once the value of \( a \) is known, calculating the volume is straightforward. This volume is integral for determining the mass of the unit cell when multiplied by the material's density, thus connecting volume with a substance's mass and later with Avogadro's number estimation.

Molar Mass Calculation

The molar mass of an element is the mass of one mole of that element's atoms, typically expressed in grams per mole (g/mol). It's a bridge between the macroscopic and atomic scales and is found on the periodic table for each element. In the context of a crystal, molar mass is used to determine the mass of a single atom, which is crucial for estimating Avogadro's number.

To calculate the mass of one atom, you first need the mass of a unit cell which comes from the product of its volume and density. Then, by recognizing that this mass corresponds to a singular atom in a simple cubic structure, you can deduce the mass of a single atom. Finally, dividing the molar mass by the mass of one atom yields Avogadro's number, which represents the number of atoms in one mole of a substance.