Chapter 10, Problem 88E

The structures of another class of ceramic, high-temperature superconductors are shown in figures a-d.

a. Determine the formula of each of these four superconductors.

b. One of the structural features that appears to be essential for high-temperature superconductivity is the presence of planar sheets of copper and oxygen atoms. As the number of sheets in each unit cell increases, the temperature for the onset of superconductivity increases. Order the four structures from lowest to the highest superconducting temperature.

c. Assign oxidation states to Cu in each structure assuming Tl exists as Tl³⁺. The oxidation states of Ca, Ba, and O are assumed to be + 2, + 2, and −2, respectively.

d. It also appears that copper must display a mixture of oxidation states for a material to exhibit superconductivity. Explain how this occurs in these materi.als as well as in the superconductor in Exercise 87.

Interpretation Introduction

Interpretation:

The formula, order of thermal conductivity and oxidation states of elements of given high-temperature superconductors are should be explained.

Concept Introduction:

• In packing of atoms/ions in a crystal structure, they are imagined as spheres. The two major types of close packing of the spheres in the crystal are hexagonal close packing and cubic close packing.

• In edge-centered cubic unit cell, the eight corners is occupied by single atom and shared by 8 unite calls and the 12 edges are having 12 atoms and shared by 4 unites center of the cube is occupied by one atom.

• In the In face-centered cubic unit cell, the eight corners is occupied by single atom and shared by 8 unite calls and the 6 faces are having 6 atoms and shared by 2 unites, center of the cube is occupied by one atom.

Explanation of Solution

(a)

To calculate the empirical formulas given high-temperature superconductors a, b, c, d

• Two Barium are present in centers of cubic unite.

$\begin{array}{l}\text{The}\text{number}\text{of}\text{Tl}\text{atoms}\text{in}\text{corners}\text{is,}\\ \\ \text{8}\text{corners×}\frac{\text{1/8}\text{Tl}}{\text{corner}}\text{=}\text{1}\text{Tl}\text{atom}\\ \\ \text{The}\text{number}\text{of}\text{Cu}\text{atoms}\text{in}\text{edges}\text{is,}\\ \\ \text{4}\text{edges}\text{×}\frac{\text{1/4}\text{Cu}}{\text{edges}}\text{=}\text{1}\text{Cu}\text{atom}\\ \\ \text{The}\text{number}\text{of}\text{O}\text{atoms}\text{in}\text{faces}\text{is,}\\ \\ \text{6}\text{faces}\text{×}\frac{\text{1/2}\text{Cu}}{\text{faces}}\text{=}\text{3}\text{O}\text{atoms}\\ \\ \text{8}\text{edges}\text{×}\frac{\text{1/4}\text{O}}{\text{edges}}\text{=}\text{2}\text{O}\text{atom}\\ \\ \text{The total O }\text{atoms }\text{=}\text{5}\end{array}$

The formula of given high-temperature superconductors (a) is ${\text{TlBa}}_{\text{2}}{\text{CuO}}_{\text{5}}$.

• Thallium and Barium are the same as in structure (a).

•  The one Calcium atom present in the center of the cubic unit than,

• $\begin{array}{l}\text{The}\text{number}\text{of}\text{Cu}\text{atoms}\text{in}\text{edges}\text{is,}\\ \\ \text{8}\text{edges×}\frac{\text{1/4 Cu}}{\text{edge}}\text{=}\text{2}\text{Cu}\text{atoms}\\ \\ \text{The}\text{number}\text{of}\text{O}\text{atoms}\text{in}\text{faces}\text{is,}\\ \\ \text{10}\text{faces}\text{×}\frac{\text{10/2}\text{O}}{\text{face}}\text{=}\text{5}\text{O}\text{atoms}\\ \\ \text{8}\text{edges×}\frac{\text{1/4 O}}{\text{edge}}\text{=}\text{2}\text{O}\text{atoms}\\ \\ \text{The total O }\text{atoms }\text{=}\text{7}\end{array}$

The empirical formula of given high-temperature superconductors (b) is ${\text{TlBa}}_{\text{2}}{\text{CaCu}}_{\text{2}}{\text{O}}_{\text{7}}$.

• Thallium and Barium are the same as in structure (a).

• The two Calcium atoms present in the center of the cubic unit than,

• $\begin{array}{l}\text{The}\text{number}\text{of}\text{Cu}\text{atoms}\text{in}\text{edges}\text{is,}\\ \\ 12\text{edges×}\frac{\text{1/4 Cu}}{\text{edge}}\text{=}3\text{Cu}\text{atoms}\\ \\ \text{The}\text{number}\text{of}\text{O}\text{atoms}\text{in}\text{faces}\text{is,}\end{array}$

• $\begin{array}{l}\text{14faces}\text{×}\frac{\text{14/2}\text{O}}{\text{face}}\text{=}7\text{O}\text{atoms}\\ \\ \text{8}\text{edges×}\frac{\text{1/4 O}}{\text{edge}}\text{=}2\text{O}\text{atoms}\\ \\ \text{The total O }\text{atoms }\text{=}9\end{array}$

The empirical formula of given high-temperature superconductors (c) is ${\text{TlBa}}_{\text{2}}{\text{Ca}}_{\text{2}}{\text{Cu}}_{\text{3}}{\text{O}}_{\text{9}}$.

• Thallium and Barium are the same as in structure (a).

• The three Calcium atom present in the center of the cubic unit than,

$\begin{array}{l}\text{The}\text{number}\text{of}\text{Cu}\text{atoms}\text{in}\text{corners}\text{is,}\\ \\ 32\text{corner}\text{×}\frac{\text{1/8 Cu}}{\text{corner}}\text{=}4\text{Cu}\text{atoms}\\ \\ \text{The}\text{number}\text{of}\text{O}\text{atoms}\text{in}\text{faces}\text{is,}\\ \\ \text{18faces}\text{×}\frac{\text{18/2}\text{O}}{\text{face}}\text{=}\text{9}\text{O}\text{atoms}\\ \\ \text{8}\text{edges×}\frac{\text{1/4 O}}{\text{edge}}\text{=}\text{2}\text{O}\text{atoms}\\ \\ \text{The total O }\text{atoms }\text{=}\text{11}\end{array}$

The empirical formula of given high-temperature superconductors (d) is ${\text{TlBa}}_{\text{2}}{\text{Ca}}_3{\text{Cu}}_4{\text{O}}_{11}$.

• The number of Thallium, Calcium, Barium, Copper  and oxygen atoms are present in one cubic unite is calculated by the total number of atoms present in the one cube divided by total sharing is to give formula of the give superconductor material.

• The empirical formula of given high-temperature superconductors (a) is ${\text{TlBa}}_{\text{2}}{\text{CuO}}_{\text{5}}$.

• The empirical formula of given high-temperature superconductors (b) is ${\text{TlBa}}_{\text{2}}{\text{CaCu}}_{\text{2}}{\text{O}}_{\text{7}}$.

• The empirical formula of given high-temperature superconductors (c) is ${\text{TlBa}}_{\text{2}}{\text{Ca}}_{\text{3}}{\text{Cu}}_{\text{4}}{\text{O}}_{\text{11}}$.
• The empirical formula of given high-temperature superconductors (d) is ${\text{TlBa}}_{\text{2}}{\text{Ca}}_3{\text{Cu}}_4{\text{O}}_{11}$.

(b)

To explain the thermal conductivity orders given high-temperature superconductors a, b, c, d.

• The structure superconductors (a) plain sheet of Copper and Oxygen atoms by development of one by remaining each (b), (c), (d) structures.

• From the above reason, the thermal conductivity order given high-temperature superconductors a, b, c, d is $\text{(a)}\text{<}\text{(b)}\text{<}\text{(c)}\text{<}\text{(d)}$.

(c)

To calculate the oxidation states to copper in given -temperature superconductors a, b, c, d.

$\begin{array}{l}\text{For}\text{(a)}\\ {\text{TlBa}}_{\text{2}}{\text{CuO}}_{\text{5}}\text{=}\text{3+2(2)+x+5(-2)=0}\\ \text{x=}\text{3+}\\ \text{Only}{\text{Cu}}^{\text{3+}}\text{ion}\text{present}\text{in}\text{(a)}\\ \text{For}\text{(b)}\\ \text{T}{\text{lBa}}_{\text{2}}{\text{CaCu}}_{\text{2}}{\text{O}}_{\text{7}}\text{=}\text{3 + 2(2) + 2 + 2(x) + 7(-2)}\text{=}\text{0}\\ \text{x=}\text{+5/2}\\ \text{One}{\text{Cu}}^{\text{3+}}\text{and}{\text{Cu}}^{\text{2+}}\text{are}\text{present}\text{in}\text{(b)}\end{array}$ $\begin{array}{l}\text{For}\text{(c)}\\ {\text{ TlBa}}_{\text{2}}{\text{Ca}}_{\text{2}}{\text{Cu}}_{\text{3}}{\text{O}}_{\text{9}}\text{=}\text{3 + 2(2) + 2(2) + 3(x) + 9(-2)=0}\\ \text{x=}\text{+7/3}\\ \text{Two}{\text{Cu}}^{\text{2+}}\text{and}{\text{Cu}}^{\text{3+}}\text{ions}\text{are}\text{present}\text{in}\text{(c)}\\ \\ \text{For(d)}\\ {\text{TlBa}}_{\text{2}}{\text{Ca}}_{\text{3}}{\text{Cu}}_{\text{4}}{\text{O}}_{\text{11}}\text{=}\text{3 + 2(2) + 3(2) + 4(x) + 11(-2)=0}\\ \text{x=}\text{+9/4}\\ {\text{3 Cu}}^{\text{2+}}{\text{andCu}}^{\text{3+}}{\text{oneCu}}^{\text{3+}}\text{ions}\text{are}\text{present}\text{in}\text{(d)}\end{array}$

• The general oxidation states each element (Except Copper) present in the given superconductor materials are added and the given total oxidation states, this divided by number of Copper to give oxidation states to Copper in given -temperature superconductors a, b, c, d.

• The oxidation states of Cu in given high-temperature superconductors are,

• (a) ${\text{Only Cu}}^{\text{3+}}$

• (b) ${\text{1 Cu}}^{\text{2+}}{\text{ and 1 Cu}}^{\text{3+}}$

• (c) ${\text{2 Cu}}^{\text{2+}}{\text{ and 1 Cu}}^{\text{3+}}$

• (d) ${\text{3 Cu}}^{\text{2+}}{\text{ and 1 Cu}}^{\text{3+}}$

(d)

To give the Copper is having variable oxidation in given superconductor’s respect with the numbers of other ions in the superconductors.

• The Copper is having variable oxidation in given superconductor’s respect with the numbers of other ions in the superconductors.

• The given below superconductor ${\text{YBa}}_{\text{2}}{\text{Cu}}_{\text{3}}{\text{O}}_{\text{7}}$ achieves variable oxidation state Copper ions by omitting oxygen at different sites in the crystal lattice.

Conclusion

The formula, order of thermal conductivity and oxidation state Copper ions in given high-temperature superconductors was explained.