One high- temperature superconductor has the general formula YBa2Cu3Ox. The copper is a mixture of Cu(II) and Cu(III) oxidation states. This mixture of oxidation states appears vital for high temperature superconductivity to occur. A simple method for determining the average copper oxidation state has been reported [D. C. Harris, M. E. Hillis, and T. A. Hewston, J. Chem. Educ. 64, 847(1987)]. The described analysis takes place in two steps:

i. One superconductor sample is treated directly with I₂:

Cu2+(aq) + I₂(aq) → CuI(s) + I₃^-(aq)_(Unbalanced)

Cu³⁺(aq) + I₂(aq) → CuI(s) + I₃^-(aq) _(Unbalanced)

ii. A second superconductor sample is dissolved in acid, converting all copper to Cu(II). This solution is then treated with I2:

Cu²⁺(aq) + I₂(aq) → CuI(s) + I₃^-(aq)_(Unbalanced)

In both steps the I₃^- is determined by titrating with a standard sodium thiosulfate (Na₂S₂O₃) solution:

I₃^–(aq) + S₂O₃^2-(aq) → S₄O₆^2-(aq) + I₂(aq) _(Unbalanced)

a. Calculate the average copper oxidation states for materials with the formulas YBa₂Cu₃O₆.5, YBa₂Cu₃O₇, and YBa₂Cu₃O₈. Interpret your results in terms of a mixture of Cu(II) and Cu(III) ions, assuming that only Y³⁺, Ba^2+, and O^2- are present in addition to the copper ions.

b. Balance the equations involved in the copper analysis.

c. A superconductor sample was analyzed by the above procedure. In step i, it took 37.77 mL of 0.1000 M Na₂S₂O₃ to react completely with the I₃^– generated from a 562.5- mg sample. In step ii, it took 2-.57 mL of 0.1000 M Na₂S₂O₃ to react with the I₃^- generated by a 504.2- mg sample. Determine the formula of this superconductor sample (that is, find the value of x in YBa₂Cu₃O_x). Calculate the average oxidation state of copper in this material.