Flotation performance metric: if the concentrate grade equals the tailings grade, what is the selectivity index?

Introduction / Context:
The selectivity index (also called separation factor) is used in flotation and gravity separation to compare the enrichment of valuable mineral in concentrate versus tailings.

Given Data / Assumptions:

• C = grade (mass fraction of valuable) in concentrate.
• T = grade in tailings.
• Given C = T.

Concept / Approach:
One common form is SI = (C * (1 − T)) / (T * (1 − C)). This ratio equals 1 when C = T because numerator and denominator are identical. An SI greater than 1 indicates enrichment; less than 1 indicates poor separation relative to random.

Step-by-Step Solution:
Start from SI = (C * (1 − T)) / (T * (1 − C)).Set C = T → SI = (C * (1 − C)) / (C * (1 − C)) = 1.Therefore, when concentrate and tailings have identical grade, the process exhibits SI = 1 (no selectivity).

Verification / Alternative check:
Other equivalent forms of SI reduce to unity for C = T as well, reflecting zero separation power.

Why Other Options Are Wrong:
0 or ∞: would imply perfect reverse or perfect separation, not the case.0.5: arbitrary and unsupported by the relation.

Common Pitfalls:
Confusing selectivity index with enrichment ratio or recovery; mixing units—grades must be fractions, not percentages, inside the formula.

Final Answer:
1