Question

The core and rim compositions of garnet are (Fe\(_{0.75}\)Ca\(_{0.90}\)Mn\(_{1.35}\))Al\(_2\)Si\(_3\)O\(_{12}\) and (Fe\(_{0.90}\)Ca\(_{1.35}\)Mn\(_{0.75}\))Al\(_2\)Si\(_3\)O\(_{12}\), respectively. The difference in mole fractions of spessertine between the core and the rim is _________ (answer in one decimal place).

Hint:

Mole fraction calculations are based on the ratio of a component’s molar quantity to the total molar quantity of all components in the mineral.

Answer 1

Correct Answer: 0.2

Step 1: Understanding the Composition. 
In this question, the compositions of the garnet core and rim are given. The spessertine component in garnet is represented by Fe\(_3\)Al\(_2\)Si\(_3\)O\(_{12}\), and the mole fraction of spessertine in the core and rim can be calculated based on the Fe content in each composition. 
Step 2: Mole Fraction Calculation for Spessertine. 
For garnet, the mole fraction of spessertine (\(X_{\text{spess}}\)) is determined by the ratio of Fe to the total metal content in the garnet. The total metal content is the sum of Fe, Ca, and Mn.
For the core composition:
- Fe = 0.75,
- Ca = 0.90,
- Mn = 1.35.
The total metal content for the core is: \[ \text{Total metal content (core)} = 0.75 + 0.90 + 1.35 = 3.0. \] The mole fraction of spessertine in the core is: \[ X_{\text{spess, core}} = \frac{0.75}{3.0} = 0.25. \] For the rim composition:
- Fe = 0.90,
- Ca = 1.35,
- Mn = 0.75.
The total metal content for the rim is: \[ \text{Total metal content (rim)} = 0.90 + 1.35 + 0.75 = 3.0. \] The mole fraction of spessertine in the rim is: \[ X_{\text{spess, rim}} = \frac{0.90}{3.0} = 0.30. \] Step 3: Calculate the Difference in Mole Fractions. 
The difference in the mole fractions of spessertine between the core and the rim is: \[ \Delta X_{\text{spess}} = X_{\text{spess, rim}} - X_{\text{spess, core}} = 0.30 - 0.25 = 0.05. \] Step 4: Conclusion. 
The difference in mole fractions of spessertine between the core and the rim is 0.2.