Le titanate de calcium CaTiO3 cristallise, à température...

Le titanate de calcium $\mathrm{CaTiO}_3$ cristallise, à température ambiante, dans le système cubique (structure perovskite); la maille élémentaire de $\mathrm{CaTiO}_3$ est présentée ci-dessous : a) Décrire la maille élémentaire présentée ci-dessus. Faire une projection de cette maille (maille 1) sur le plan de base ( $\vec{a}, \vec{b})$. b) À l'aide cette projection, définir la maille élémentaire ayant pour origine les ions calcium (maille 2). Faire une projection de cette nouvelle maille sur le plan de base $(\vec{a}, \vec{b})$. c) Dessiner une vue en perspective de la maille élémentaire 2 . d) Quel est le polyèdre de coordination formé par les ions $\mathrm{O}^{2-}$ autour de l'ion titane? En déduire la coordinence du titane par rapport à l'oxygène.

Le titanate de calcium $\mathrm{CaTiO}_3$ cristallise, à température ambiante, dans le système cubique (structure perovskite); la maille élémentaire de $\mathrm{CaTiO}_3$ est présentée ci-dessous :
a) Décrire la maille élémentaire présentée ci-dessus. Faire une projection de cette maille (maille 1) sur le plan de base ( $\vec{a}, \vec{b})$.
b) À l'aide cette projection, définir la maille élémentaire ayant pour origine les ions calcium (maille 2). Faire une projection de cette nouvelle maille sur le plan de base $(\vec{a}, \vec{b})$.
c) Dessiner une vue en perspective de la maille élémentaire 2 .
d) Quel est le polyèdre de coordination formé par les ions $\mathrm{O}^{2-}$ autour de l'ion titane? En déduire la coordinence du titane par rapport à l'oxygène.

Key Concepts

Coordination Polyhedron and Coordination Number

In crystallography, a coordination polyhedron is formed by connecting the nearest neighbor atoms or ions surrounding a central atom or ion. The coordination number is derived from this polyhedron and indicates how many ions or atoms are directly bonded to the central one. This concept is vital in understanding the local bonding environment and the physical properties of the material.

Crystallographic Projection

This concept involves projecting a three-dimensional arrangement onto a two-dimensional plane, such as the (a, b) plane. Projections are used to simplify and analyze complex crystal structures by visualizing the relative positions of atoms or ions in specific layers or directions. These projections assist in identifying symmetries and repeating patterns within the crystal.

Perovskite Structure

This refers to a specific type of crystal structure originally found in the mineral perovskite. It is characterized by a cubic lattice arrangement where a large cation exists at the corners of the cube, a smaller cation is centrally located, and the oxygen anions are found at the face centers. The perovskite structure is fundamental in materials science and solid state chemistry due to its versatile electronic, magnetic, and optical properties.

Unit Cell

The unit cell is the smallest repeating unit in a crystal lattice from which the entire structure can be built through translation. Descriptions of the unit cell typically include the lattice parameters and the positions of the ions or atoms within the cell. It is essential to understand the geometry and symmetry of the unit cell to comprehend the overall crystallographic structure.

Transcript

00:02 So this question wants us to determine the edge length of perobskite, as well as to determine the radius of the titanium ion.

00:12 So to determine the edge length, we first need to see how many atoms are inside this perovskite unit cell.

00:21 So there are eight calcium ions at the corner, which each contribute one eighth of an atom for a total of one calcium atom.

00:27 There are six oxide ions in the middle of the cell faces, which each contribute half an atom for a total of three oxyme.

00:32 Oxygen atoms.

00:33 And there is one titanium atom at the center of the unit cell, which contributes one titanium atom.

00:40 So perovskite is composed of a total of five atoms.

00:45 And we can use this information to calculate the mass of one perovskite unit cell.

00:50 So we need the molar mass of perovskite, given its formula.

00:57 And we can determine that based on the periodic table by summing the weight of the atomic weights of the components of this molecule based on the formula.

01:08 And because a perovskite unit cell only consists of these five atoms, the calcium, the titanium, and the three oxides, we also need to take that into account when doing the mass calculation, such that the mass is only for the weight of these five atoms.

01:28 And we should find that the weight is equal to two point.

01:32 257 times 10 to the negative 22 grams.

01:37 And we can then use the mass of perovskite to calculate the unit cell volume, as the density was given in the question to be 4 .1 grams per cubic centimeter.

01:47 After we determine the volume, we can then find the edge length.

01:53 So if we rearrange the density equation, we can find volume by dividing mass by the density.

01:59 And when we do that, we find that the volume of the parovskite unit cell is 5 .506 times 10 to the negative 23 cubic centimeters.

02:11 And if we cube root, the volume, we can determine the edge length.

02:18 And we should find the edge length is equal to, once we convert it to picometers, 380 .4 picometers.

02:28 Next, the question wants us to calculate the radius of the titanium ion at the center of this unit cell.

02:35 So the titanium atom at the center of the unit cell is bound to every oxide in an octahedral manner.

02:41 So we can assume that the oxide ions at the top of the unit cell and the oxide ions in the middle of the faces are touching...

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