Textbook Question
Titanium metal has a density of 4.506 g>cm3 and an atomic radius of 144.8 pm. In what cubic unit cell does titanium crystallize?
Ch.12 - Solids and Solid-State Materials
Chapter 12, Problem 46
If a protein can be induced to crystallize, its molecular structure can be determined by X-ray crystallography. Protein crystals, though solid, contain a large amount of water molecules along with the protein. The protein chicken egg-white lysozyme, for instance, crystallizes with a unit cell having angles of 90° and with edge lengths of 7.9 * 103 pm, 7.9 * 103 pm, and 3.8 * 103 pm. There are eight molecules in the unit cell. If the lysozyme molecule has a molecular weight of 1.44 * 104 and a density of 1.35 g>cm3, what percent of the unit cell is occupied by the protein?
Verified step by step guidance1
Calculate the volume of the unit cell using the formula for the volume of a cuboid, which is given by the product of its edge lengths. In this case, the formula is V = a imes b imes c, where a, b, and c are the edge lengths of the unit cell.
Convert the volume from picometers cubed (pm^3) to cubic centimeters (cm^3) for compatibility with the density units. Note that 1 pm = 10^{-12} meters and 1 meter = 100 cm.
Calculate the mass of the protein in the unit cell by multiplying the number of molecules in the unit cell by the molecular weight of each molecule.
Calculate the theoretical mass of the unit cell using its volume and the given density, using the formula mass = density imes volume.
Determine the percent occupancy of the protein in the unit cell by dividing the mass of the protein by the theoretical mass of the unit cell and then multiplying by 100 to get the percentage.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Unit Cell and Molecular Weight
A unit cell is the smallest repeating unit in a crystal lattice that defines the structure of the entire crystal. The molecular weight of a substance, in this case, chicken egg-white lysozyme, is the mass of one mole of its molecules, expressed in grams per mole. Understanding these concepts is crucial for calculating the number of molecules in a unit cell and their contribution to the overall mass.
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Simple Cubic Unit Cell
Density and Volume Calculation
Density is defined as mass per unit volume and is a key property in determining how much space a substance occupies. To find the volume of the unit cell, one can use the edge lengths provided. The density of the protein allows for the calculation of the total mass of the protein in the unit cell, which is essential for determining the percentage of the unit cell occupied by the protein.
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Percent Occupancy Calculation
Percent occupancy refers to the fraction of the total volume of a unit cell that is occupied by a specific component, in this case, the protein. This is calculated by dividing the total mass of the protein in the unit cell by the total volume of the unit cell, then multiplying by 100 to express it as a percentage. This concept is vital for understanding how much of the crystal structure is made up of the protein versus other components, such as water.
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Related Practice
Textbook Question
The atomic radius of Pb is 175 pm, and the density is 11.34 g>cm3. Does lead have a primitive cubic structure or a face-centered cubic structure?
Textbook Question
The density of a sample of metal was measured to be 6.84 g>cm3. An X-ray diffraction experiment measures the edge of a face-centered cubic cell as 350.7 pm. What is the atomic weight, atomic radius, and identity of the metal?
Textbook Question
Iron crystallizes in a body-centered cubic unit cell with an edge length of 287 pm. Iron metal has a density of 7.86 g>cm3 and a molar mass of 55.85 g. Calculate a value for Avogadro's number.
Textbook Question
Sodium hydride, NaH, crystallizes in a face-centered cubic unit cell similar to that of NaCl (Figure 12.11). How many Na+ ions touch each H- ion, and how many H- ions touch each Na+ ion?
Textbook Question
Cesium chloride crystallizes in a cubic unit cell with Cl- ions at the corners and a Cs+ ion in the center. Count the numbers of + and - charges, and show that the unit cell is electrically neutral.