Textbook Question
Platinum crystallizes with the face-centered cubic unit cell. The radius of a platinum atom is 139 pm. Calculate the edge length of the unit cell and the density of platinum in g/cm3.
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Ch.12 - Solids and Modern Material
Chapter 12, Problem 34
Molybdenum crystallizes with the body-centered unit cell. The radius of a molybdenum atom is 136 pm. Calculate the edge length of the unit cell and the density of molybdenum
Verified step by step guidance1
Step 1: Understand that in a body-centered cubic (bcc) unit cell, atoms at the corners and the center of the cell touch each other along the body diagonal. The length of the body diagonal is equal to 4 times the radius of the atom (4r).
Step 2: The edge length (a) of the unit cell can be found using the Pythagorean theorem in three dimensions, which states that the square of the body diagonal is equal to the sum of the squares of the edge lengths. Since all edges are of equal length in a cubic cell, this simplifies to √3a = 4r. Solve this equation for a to find the edge length of the unit cell.
Step 3: To find the density of molybdenum, first calculate the volume of the unit cell using the formula for the volume of a cube, V = a^3.
Step 4: In a bcc unit cell, there are two atoms per unit cell. So, calculate the mass of two molybdenum atoms. Remember that the molar mass of molybdenum is 95.94 g/mol and Avogadro's number is 6.022 x 10^23 atoms/mol.
Step 5: Finally, calculate the density of molybdenum using the formula density = mass/volume. The mass is the mass of two molybdenum atoms and the volume is the volume of the unit cell calculated in step 3.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Body-Centered Cubic (BCC) Structure
The body-centered cubic (BCC) structure is a type of crystal lattice where atoms are located at each corner of a cube and a single atom is positioned at the center of the cube. In this arrangement, each unit cell contains two atoms, as the corner atoms contribute only a fraction of their volume to the unit cell. Understanding this structure is essential for calculating properties like edge length and density.
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Body Centered Cubic Example
Unit Cell Edge Length Calculation
The edge length of a unit cell in a BCC structure can be calculated using the relationship between the atomic radius and the geometry of the unit cell. For BCC, the edge length 'a' is related to the atomic radius 'r' by the formula a = 4r/√3. This relationship is crucial for determining the dimensions of the unit cell based on the given atomic radius.
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Density Calculation
Density is defined as mass per unit volume and can be calculated for a crystalline solid using the formula density = (mass of atoms in unit cell) / (volume of unit cell). For a BCC structure, the mass can be determined from the number of atoms in the unit cell and the molar mass of the element, while the volume is derived from the cube of the edge length. This concept is vital for understanding the material properties of molybdenum.
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Related Practice
Textbook Question
Rhodium has a density of 12.41 g/cm3 and crystallizes with the face-centered cubic unit cell. Calculate the radius of a rhodium atom.
Textbook Question
Barium has a density of 3.59 g/cm3 and crystallizes with the body-centered cubic unit cell. Calculate the radius of a barium atom.
Textbook Question
Plonium crystallizes with a simple cubic structure. It has a density of 9.3 g/cm3, a radius of 167 pm, and a molar mass of 209 g/mol. Use these data to calculate Avogadro's number (the number of atoms in one mole).