The article “The Equilibrium Partitioning of Titanium

Chapter 7, Problem 11E

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The article “The Equilibrium Partitioning of Titanium Between \(\mathrm {Ti}^{3+}\) and \(\mathrm {Ti}^{4+}\) Valency States in \(\mathrm{CaO}\text-\mathrm{SiO_2}\text-\mathrm{TiO}_x\) Slags” (G. Tranell, O. Ostrovski, and S. Jahanshahi, Metallurgical and Materials Transactions B, 2002:61–66) discusses the relationship between the redox ratio \(\mathrm {Ti}^{3+}/\mathrm {Ti}^{4+}\) and oxygen partial pressure \(p_{\mathrm O_{2}}\) in \(\mathrm{CaO}\text-\mathrm{SiO_2}\text-\mathrm{TiO}_x\) melts. Several independent measurements of the redox ratio were made at each of five different partial pressures: \(10^{-7}\), \(10^{-8}\), \(10^{-9}\), \(10^{-10}\), and \(10^{-12}\) atmospheres. The results for the runs at 14 mass percent TiOx are presented in the following table.

a. Denoting the redox ratio by y and the partial pressure by x, theory states that y should be proportional to \(x^\beta\) for some \(\beta\). Express this theoretical relationship as a linear model.

b. Compute the least-squares line for this linear model. Plot the residuals versus the fitted values. Does the linear model hold?

c. Further theoretical considerations suggest that under the conditions of this experiment, y should be proportional to \(x^{-1/4}\). Are the data in the preceding table consistent with this theory? Explain.

Equation Transcription:

Text Transcription:

Ti^3+

Ti^4+

CaO-SiO_2-TiO_x

Ti^3+/Ti^4+

p_{O}_2

CaO-SiO_2-TiO_x

10^-7

10^-8

10^-9

10^-10

10^-12

10^-7

10^-8

10^-9

10^-10

10^-12

x

beta

x^{-¼}