All the alkali metals crystallize with bec structures. (a) Find a general equation relating the metallic radius to the density of a bcc solid of an element in terms of its molar mass and use it to deduce the atomic radius of each of the following elements, given the density of each \(\left(\right.\) in \(\left.\mathrm{g} \cdot \mathrm{cm}^{-3}\right): \mathrm{Li}, 0.53 ; \mathrm{Na}, 0.97 ; \mathrm{K}, 0.86 ; \mathrm{Rb}, 1.53\); Cs, 1.87. (b) Find a factor for converting the density of a bcc element into the density that it would have if it crystallized in a ccp structure. (c) Calculate what the densities of the alkali metals would be if they were ccp. (d) Which, if any, would float on water?

The body-centered cubic (bcc) structure is one of the ways that atoms can be arranged in a solid. In a bcc lattice, each unit cell—a small, repeating entity that creates the structure of the crystal—consists of atoms at each of the eight corners of a cube and one atom at the very center of the cube. This central atom is equidistant from all corner atoms, hence the term 'body-centered'.

Understanding this structure is important for calculating the density and other properties of a metal. In the bcc arrangement, there is a specific geometric relationship between the atomic radius and the edge length of the unit cell. The edge length, denoted as 'a', can be related to the atomic radius 'r' using the equation \( a = \frac{4r}{\sqrt{3}} \). Since each unit cell in a bcc structure contains two atoms, this relationship is crucial in deriving formulas for the atomic radius based on the density of the metal.

The relationship between density and atomic radius in a solid is significant as it allows us to determine one property from the other. Density, \( \rho \), is defined as the mass of atoms within a unit volume of the solid. By understanding the unit cell geometry, we can derive an equation that links the density of a bcc solid to the atomic radius.

The equation derived from this relationship is \( \rho = \frac{2M}{N_A a^3} = \frac{2M}{N_A (4r/\sqrt{3})^3} \), where \( M \) is the molar mass, \( N_A \) is Avogadro's number, \( a \) is the edge length of the cube, and \( r \) is the atomic radius. With this formula, given the density and molar mass of a bcc metal, one can calculate its atomic radius, which is fundamental for understanding the metal's physical characteristics.

When comparing different crystalline structures, it's often useful to be able to convert the density from one structure to another. The body-centered cubic (bcc) and face-centered cubic (ccp or cubic close-packed) structures have different packing efficiencies, meaning that atoms are packed more tightly in one structure compared to the other.

The packing efficiency of a bcc structure is approximately 68%, while that of a ccp structure is around 74%. To convert the density from bcc to ccp, we use the ratio of these packing efficiencies:\[ \text{Conversion Factor} = \frac{74}{68} \]Multiplying the density of a metal in its bcc form by this conversion factor will give an estimation of what its density would be if the metal crystallized in a ccp structure.

Alkali metals are a group of elements located in group 1 of the periodic table. They share similar properties due to their electronic configuration, which includes a single electron in their outermost shell. Some defining properties of alkali metals include:

• High reactivity, especially with water.
• Low melting and boiling points relative to other metals.
• Softness, often being able to be cut with a knife.
• Low densities compared to other metals.

These properties influence how alkali metals interact with their environment, including how they crystallize. Typically, alkali metals crystallize in a bcc structure, which affects their calculated densities. By applying the general equation relating metallic radius to the density of a bcc solid, we can deduce their atomic radii. This is essential for predicting how these metals will behave under different physical conditions, including whether they will float when placed in water - an aspect that depends on their densities relative to the density of water (1 g/cm^3).