Вопрос
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.
Решения
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Элеонора
профессионал · Репетитор 6 лет
Ответ
49. To calculate the edge length of the unit cell and the atomic radius of titanium, we can use the following steps:Step 1: Calculate the number of atoms per unit cell in a body-centered cubic (BCC) structure.In a BCC structure, there are 2 atoms per unit cell.Step 2: Calculate the volume of the unit cell.The volume of the unit cell can be calculated using the formula:Volume = (Edge length)^3Step 3: Calculate the mass of the unit cell.The mass of the unit cell can be calculated using the formula:Mass = Density × VolumeStep 4: Calculate the edge length of the unit cell.The edge length of the unit cell can be calculated using the formula:Edge length = (3 × Mass / (2 × Atomic mass of titanium))^(1/3)Step 5: Calculate the atomic radius of titanium.In a BCC arrangement, the spheres touch across the body diagonal. The length of the body diagonal can be calculated using the formula:Body diagonal = √3 × Edge lengthThe atomic radius can be calculated using the formula:Atomic radius = (Body diagonal / 4) × √2 / 2Given information:Density of titanium = 4.50 g/cm³Atomic mass of titanium = 47.87 g/molCalculations:Number of atoms per unit cell = 2Volume of the unit cell = (Edge length)^3Mass of the unit cell = Density × VolumeEdge length = (3 × Mass / (2 × Atomic mass of titanium))^(1/3)Atomic radius = (Body diagonal / 4) × √2 / 2Answer: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.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:Step 1: Calculate the number of atoms per unit cell in a body-centered cubic (BCC) structure.In a BCC structure, there are 2 atoms per unit cell.Step 2: Calculate the edge length of the unit cell.The edge length of the unit cell can be calculated using the formula:Edge length = (4 × Atomic radius / √3) × √2Step 3: Calculate the volume of the unit cell.The volume of the unit cell can be calculated using the formula:Volume = (Edge length)^3Step 4: Calculate the mass of the unit cell.The mass of the unit cell can be calculated using the formula:Mass = Number of atoms per unit cell × Atomic mass of bariumStep 5: Calculate the density of solid barium.The density of solid barium can be calculated using the formula:Density = Mass / VolumeGiven information:Atomic radius of barium = 222 pm = 222 × 10^-12 mAtomic mass of barium = 137.33 g/molCalculations:Number of atoms per unit cell = 2Edge length = (4 × Atomic radius / √3) × √2Volume = (Edge length)^3Mass = Number of atoms per unit cell × Atomic mass of bariumDensity = Mass / VolumeAnswer:The density of solid barium with a body-centered cubic structure and an atomic radius of 222 pm is approximately 2.16 g/cm³.