Phase rule reasoning for a binary system (non-reacting): For which case does the number of degrees of freedom equal the number of components in a two-component system?

Introduction / Context:
Gibbs’ phase rule, F = C − P + 2 for non-reacting systems, relates the degrees of freedom (independent intensive variables), the number of components C, and the number of phases P. Understanding when F equals C is useful for interpreting phase diagrams and designing separation processes involving partially miscible liquids.

Given Data / Assumptions:

• Non-reacting system.
• Binary (C = 2) unless otherwise stated.
• Pressure and temperature are the standard intensive variables considered.

Concept / Approach:
Apply the phase rule F = C − P + 2. For a binary system (C = 2):
If P = 1 (single phase), F = 2 − 1 + 2 = 3, which does not equal C.If P = 2 (two phases), F = 2 − 2 + 2 = 2, which equals C.Therefore, for a binary system, F equals the number of components when two phases coexist (e.g., two partially miscible liquids in equilibrium), giving two degrees of freedom (commonly T and overall composition at fixed pressure, or T and P with composition constrained).

Step-by-Step Solution:
Identify that we want F = C with C = 2.Solve F = 2 − P + 2 = 2 ⇒ −P + 4 = 2 ⇒ P = 2.Recognize that a partially miscible binary liquid system often exhibits two liquid phases at certain T.Hence, the case “partially miscible two-liquid system with two phases present” satisfies the condition.

Verification / Alternative check:
At a fixed pressure (e.g., 1 atm), the condensed-phase rule (F = C − P + 1) might be used; for P = 2 and C = 2, F = 1 (equal to C − 1). The original question assumes the full form, consistent with including pressure as a variable.

Why Other Options Are Wrong:
Only soluble liquid components: P = 1 → F = 3 ≠ 2.Two liquids plus an extra solute: this is C = 3, not a binary system; it changes the equality sought.None of these: incorrect because the two-liquid, two-phase case satisfies F = C for C = 2.

Common Pitfalls:
Forgetting which version of the phase rule is applied; ignoring whether the system is binary; confusing “components” with “species present in each phase.”

Final Answer:
Partially miscible two-liquid system with two phases present.