Atoms of solid polonium are packed in a repeating array that can be represented by a cubic unit cell, which is the simplest three-dimensional cross-section that repeats throughout the solid. If the atomic radius r is 1.67 × 10⁻⁸ cm and the mass of one polonium atom is 3.47 × 10⁻²² g, which expression correctly calculates the density of a pure sample of solid polonium?

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Quizplus · Accepted Answer

11ecba2d_123c_02ba_a3bf_7fdedb52a8f8_MD0008_00 The density ρ of a substance is defined as the mass m it contains per unit of volume V that it occupies. This physical property is expressed mathematically as: 11ecba2d_123c_02bb_a3bf_135865e4453d_MD0008_00 Because materials tend to expand when heated and contract when cooled, the volume that a substance occupies can undergo minute changes with temperature. As a result, the exact density of a substance varies slightly with temperature. On the most fundamental level, the density of a substance is the result of its atomic packing. Atoms that pack together more tightly achieve a greater density than molecules with a less space-efficient packing arrangement. Accordingly, the density of an element can be calculated based on its atomic radius and the atomic packing within its unit cell. Polonium atoms pack together in a simple cubic unit cell. As such, the length of one side S of the unit cell is equal to twice the atomic radius (2r). When this length is used, the volume of the cube (unit cell) is: 11ecba2d_123c_29cc_a3bf_777768a75f28_MD0008_00 Substituting this volume expression into the density equation gives: 11ecba2d_123c_29cd_a3bf_c5ff13fd1c93_MD0008_00 Because each of the eight corners of the unit cell hold one-eighth of a polonium atom, the unit cell contains the equivalent mass of one polonium atom. Substituting the values for mass and the atomic radius into the equation above gives the density: 11ecba2d_123c_29ce_a3bf_513c86212f23_MD0008_00 (Choice A) The length of one side S of the unit cell is equal to twice the atomic radius (2r). (Choice B) Although the unit cell cuts through eight atoms, the interior of the unit cell contains the equivalent mass of only one atom (one-eighth of each of the eight atoms). (Choice D) The volume of a sphere is found using 4πr³/3, but this does not account for the volume between the spheres. The volume of the unit cell cube must be used instead. Educational objective: The density of a substance varies slightly with temperature and is defined as the mass of the substance per unit of volume that it occupies. The density of an element can be calculated based on its atomic radius and the atomic packing within its unit cell.

Atoms of Solid Polonium Are Packed in a Repeating Array