Ch.12 - Solids and Modern Materials

Problem 57a

Chapter 12, Problem 57a

Tausonite, a mineral composed of Sr, O, and Ti, has the cubic unit cell shown in the drawing. (a) What is the empirical formula of this mineral?

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Identify the positions of each type of atom in the cubic unit cell. Barium (red) atoms are at the corners, Oxygen (green) atoms are at the face centers, and Titanium (yellow) atom is at the body center.

Determine the contribution of each type of atom to the unit cell. Each corner atom is shared by 8 unit cells, each face-centered atom is shared by 2 unit cells, and the body-centered atom is entirely within the unit cell.

Calculate the number of each type of atom in the unit cell. For Barium: 8 corners * 1/8 = 1 atom. For Oxygen: 6 faces * 1/2 = 3 atoms. For Titanium: 1 body-centered atom = 1 atom.

Combine the number of each type of atom to form the empirical formula. The unit cell contains 1 Barium, 3 Oxygen, and 1 Titanium atom.

Write the empirical formula based on the ratio of atoms: BaTiO3.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Empirical Formula

The empirical formula of a compound represents the simplest whole-number ratio of the elements present in that compound. It is derived from the molecular formula by reducing the subscripts to their smallest values. For example, if a mineral contains Sr, O, and Ti in a ratio of 1:2:1, the empirical formula would be SrO2Ti.

Cubic Unit Cell

A cubic unit cell is the smallest repeating unit in a crystal lattice that reflects the symmetry and structure of the entire crystal. In the context of the question, the face-centered cubic (FCC) structure indicates that atoms are located at each corner and the centers of all the cube faces, which influences the arrangement and ratio of the constituent elements in the mineral.

Stoichiometry in Minerals

Stoichiometry in minerals involves understanding the quantitative relationships between the different elements in a mineral's composition. By analyzing the arrangement of atoms in the unit cell, one can determine how many of each type of atom are present, which is essential for calculating the empirical formula of the mineral.

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Textbook Question

Which of the following statements does not follow from the fact that the alkali metals have relatively weak metal–metal bonding? (a) The alkali metals are less dense than other metals. (b) The alkali metals are soft enough to be cut with a knife. (c) The alkali metals are more reactive than other metals. (d) The alkali metals have higher melting points than other metals. (e) The alkali metals have low ionization energies.

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Open Question

Arrange the following metals in increasing order of expected melting points: Mo, Zr, Y, Nb. Explain this trend in melting points.

Textbook Question

For each of the following groups, which metal would you expect to have the highest melting point: (b) rubidium, molybdenum, or indium?

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Textbook Question

The unit cell of a compound containing Co and O has a unit cell shown below. The Co atoms are on the corners, and the O atoms are completely within the unit cell. What is the empirical formula of this compound? What is the oxidation state of the metal?

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Open Question

Alabandite is a mineral composed of manganese(II) sulfide (MnS). The mineral adopts the rock salt structure. The length of an edge of the MnS unit cell is 5.223 Å at 25 °C. Determine the density of MnS in g/cm³.

Textbook Question

A particular form of cinnabar (HgS) adopts the zinc blende structure. The length of the unit cell edge is 5.852 Å. (a) Calculate the density of HgS in this form. (c) Which of the two substances has the higher density? How do you account for the difference in densities?

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