A manufacturer of high-voltage power lines must choose an aluminum-based material…
MATHEMATICS · Multi-part · CBSE · Mathematics · Linear Inequalities · Linear Inequalities
Problem
A manufacturer of high-voltage power lines must choose an aluminum-based material that balances mechanical strength with minimal loss of electrical conductivity. The company is considering an interstitial alloy formed by introducing a small amount of oxygen atoms into the aluminum lattice.
Laboratory tests provide the following empirical relationships at 298 K:
• The Vickers hardness H (dimensionless) increases linearly with mole fraction x of oxygen according to H=0.80+12x.
• The electrical resistivity ρ (in μΩcm) increases with mole fraction according to ρ=2.7+90x.
Industry specifications require: H≥2.0 and ρ≤7.0μΩcm.
(a) Determine the allowable range of mole fraction x of oxygen that simultaneously satisfies both specifications. Show your calculations clearly.
(b) If the plant wishes to maximize mechanical strength while meeting the conductivity requirement, what mole fraction of oxygen should be chosen? State the corresponding hardness and resistivity values for that composition.
(c) Explain, in chemical terms, why further increasing the oxygen content beyond the value found in part (b) would continue to strengthen the alloy but adversely affect its usefulness in power lines.