Question

Sodium metal (atomic weight \(22.99 \mathrm{~g} / \mathrm{cm}^{3}\) ) adopts a bodycentered cubic structure with a density of \(0.97 \mathrm{~g} / \mathrm{cm}^{3}\). (a) Use this information and Avogadro's number \(\left(N_{\mathrm{A}}=6.022 \times 10^{23}\right)\) to estimate the atomic radius of sodium. (b) If it didn't react so vigorously, sodium could float on water. Use the answer from part (a) to estimate the density of Na if its structure were that of a cubic close-packed metal. Would it still float on water?

Short answer

In summary, we calculated the atomic radius of sodium in its body-centered cubic (BCC) structure using the given density and atomic weight. We then estimated the density of sodium if it had a cubic close-packed (CCP) structure by rearranging atoms and calculating the new unit cell volume. Comparing the density of sodium in the CCP structure to the density of water, we can determine whether sodium would still float on water.

Step-by-step solution

Calculate the volume of the unit cell

First, we determine the volume of a unit cell of sodium. Based on the given density and the atomic weight of sodium, we can calculate the volume of one mole of sodium atoms. Density = mass/volume Rearranging the formula to find the volume, we have: Volume = mass/density Now, we need the volume of one unit cell: Volume_of_unit_cell = Volume / Avogadro's_number \(Volume_{unit\_cell} = \frac{(\frac{22.99 g}{0.97g cm^{-3}})} {6.022\times 10^{23}}\)

Find the edge length of the unit cell

Knowing the volume of a unit cell in a body-centered cubic structure, we can calculate its edge length. In a BCC structure, the volume can be expressed as: \(Volume_{unit\_cell} = a^{3}\) Here, "a" represents the edge length of the unit cell. Solving for "a," we have: \(a = \sqrt[3]{Volume_{unit\_cell}}\)

Determine the atomic radius in sodium's body-centered cubic structure

In a body-centered cubic structure, a diagonal across the unit cell can be expressed as: \(Diagonal = \sqrt{a^{2} + a^{2} + a^{2}}\) And the relationship between the atomic radius (r) and the diagonal in a BCC structure is: \(Diagonal = 4r\) With the edge length calculated in Step 2, we can determine the atomic radius of sodium using the above relationship.

Calculate the density of sodium in a cubic close-packed structure

In a cubic close-packed (CCP) metal structure, there are four atoms per unit cell. To calculate the density of sodium in a CCP structure, we can use the formula: \(Density_{CCP} = \frac{4 * Atomic\_Weight_{Na}}{Volume_{unit\_cell}}\) Now, the value of "Volume_unit_cell" should be changed with respect to the relative arrangement of atoms in a CCP structure. Replace the atomic radius of the current BCC structure into a CCP arrangement to find the volume of the new unit cell.

Determine if sodium would still float on water if it had a cubic close-packed structure

To determine whether Na would still float on water, compare the density of sodium in its CCP structure with the density of water (1 g/cm³). If the density of sodium in its CCP structure is less than that of water, it would still float on water.

Key concepts

Body-Centered Cubic

A body-centered cubic (BCC) structure is a type of crystal lattice where each cube-shaped unit cell contains atoms at each corner of the cube and a single atom at the center. This arrangement results in a total of two atoms per unit cell since each of the corner atoms is shared among eight other unit cells, contributing an equivalent of one whole atom per unit cell, plus the one atom at the center.
- **Key Features**: - Contains 2 atoms per unit cell. - Less tightly packed than some other structures, like FCC. - Common metals that crystallize in a BCC structure include sodium, iron, and chromium.
In a BCC structure, the edge length of the unit cell is mathematically related to its atomic radius. The calculation of properties like atomic radius and density becomes important in determining other characteristics of the metal, such as weight and geometry. Understanding BCC helps you to estimate the physical behavior of metals, like how they might interact with water or other substances.

Cubic Close-Packed

The cubic close-packed (CCP) structure, also known as face-centered cubic (FCC), is a highly efficient way to arrange atoms in a crystal lattice.
- **Key Characteristics**: - Contains 4 atoms per unit cell. - Atoms are located at each of the eight corners of the cube and at the center of each face. - Known for its high packing efficiency of about 74%.
The CCP arrangement allows metals to achieve greater density. This structure is commonly seen in metals such as aluminum, copper, and gold. In a CCP structure, the spheres are arranged in a 3-layer repeating pattern, creating a highly packed structure. This leads to stronger metallic bonds and more durable materials.
Understanding CCP is crucial in material sciences as it affects properties such as strength, ductility, and conductivity.

Atomic Radius

The atomic radius is the distance from the center of an atom to the outer boundary of its electron cloud. In the context of crystal structures:
- **In a BCC structure**: The atomic radius relates to the diagonal of the cubic unit cell. The diagonal can be expressed through the equation: \[ \text{Diagonal} = 4r \] where \(r\) is the atomic radius.
- **In a CCP structure**: The atomic radius defines the touching points between spheres along the diagonal of the face-centered arrangement.
The radius helps to define how tightly atoms are packed within a structure, and is a vital factor in understanding and predicting how materials will respond under different conditions, such as pressure or temperature changes. Accurate calculation of atomic radii assists in constructing a clear mental model of how microscopic structures translate into macroscopic properties of materials.

Density Calculation

Density calculation in crystal structures involves determining how mass is distributed in a given volume. Density is calculated using the formula:
- **Density = Mass/Volume**
To compute the density for a crystal with a known atomic structure like BCC or CCP, you must know:

• The number of atoms per unit cell.
• The atomic mass.
• The volume of the unit cell.

For a BCC structure, the density calculation considers 2 atoms per unit cell, while a CCP structure involves 4 atoms. Altering the structural form changes the arrangement and packing of atoms, thus affecting the density. This principle is handy in predicting how materials behave in different environments or structural states, such as whether sodium can float on water as its density changes when it assumes different crystal structures.