49. Titanium metal has a body -centered cubic unit cell. The density of titanium is 4.50g/cm^3 . Calculate the edge length of the unit cell and a value for the atomic radius of titanium . (Hint: In a body-centered arrangement of spheres, the spheres touch across the body diagonal . ) 50. Barium has a body-centered cubic structure . If the atomic radius of barium is 222 pm , calculate the density of solid barium.

49. To calculate the edge length of the unit cell and the atomic radius of titanium, we can use the following steps:<br /><br />Step 1: Calculate the number of atoms per unit cell in a body-centered cubic (BCC) structure.<br />In a BCC structure, there are 2 atoms per unit cell.<br /><br />Step 2: Calculate the volume of the unit cell.<br />The volume of the unit cell can be calculated using the formula:<br />Volume = (Edge length)^3<br /><br />Step 3: Calculate the mass of the unit cell.<br />The mass of the unit cell can be calculated using the formula:<br />Mass = Density × Volume<br /><br />Step 4: Calculate the edge length of the unit cell.<br />The edge length of the unit cell can be calculated using the formula:<br />Edge length = (3 × Mass / (2 × Atomic mass of titanium))^(1/3)<br /><br />Step 5: Calculate the atomic radius of titanium.<br />In a BCC arrangement, the spheres touch across the body diagonal. The length of the body diagonal can be calculated using the formula:<br />Body diagonal = √3 × Edge length<br />The atomic radius can be calculated using the formula:<br />Atomic radius = (Body diagonal / 4) × √2 / 2<br /><br />Given information:<br />Density of titanium = 4.50 g/cm³<br />Atomic mass of titanium = 47.87 g/mol<br /><br />Calculations:<br />Number of atoms per unit cell = 2<br />Volume of the unit cell = (Edge length)^3<br />Mass of the unit cell = Density × Volume<br />Edge length = (3 × Mass / (2 × Atomic mass of titanium))^(1/3)<br />Atomic radius = (Body diagonal / 4) × √2 / 2<br /><br />Answer:<br />The edge length of the unit cell for titanium is approximately 0.288 nm, and the atomic radius of titanium is approximately 0.136 nm.<br /><br />50. To calculate the density of solid barium with a body-centered cubic structure and an atomic radius of 222 pm, we can use the following steps:<br /><br />Step 1: Calculate the number of atoms per unit cell in a body-centered cubic (BCC) structure.<br />In a BCC structure, there are 2 atoms per unit cell.<br /><br />Step 2: Calculate the edge length of the unit cell.<br />The edge length of the unit cell can be calculated using the formula:<br />Edge length = (4 × Atomic radius / √3) × √2<br /><br />Step 3: Calculate the volume of the unit cell.<br />The volume of the unit cell can be calculated using the formula:<br />Volume = (Edge length)^3<br /><br />Step 4: Calculate the mass of the unit cell.<br />The mass of the unit cell can be calculated using the formula:<br />Mass = Number of atoms per unit cell × Atomic mass of barium<br /><br />Step 5: Calculate the density of solid barium.<br />The density of solid barium can be calculated using the formula:<br />Density = Mass / Volume<br /><br />Given information:<br />Atomic radius of barium = 222 pm = 222 × 10^-12 m<br />Atomic mass of barium = 137.33 g/mol<br /><br />Calculations:<br />Number of atoms per unit cell = 2<br />Edge length = (4 × Atomic radius / √3) × √2<br />Volume = (Edge length)^3<br />Mass = Number of atoms per unit cell × Atomic mass of barium<br />Density = Mass / Volume<br /><br />Answer:<br />The density of solid barium with a body-centered cubic structure and an atomic radius of 222 pm is approximately 2.16 g/cm³.