Detailed Solution

To determine the degrees of freedom for the points indicated in the phase diagram, we use the Gibbs Phase Rule. For an isobaric (constant pressure) binary system, the phase rule is expressed as:

$$ F = C - P + 1 $$

Where:

• \(F\) is the number of degrees of freedom (variance).
• \(C\) is the number of components in the system. Since it is a binary system, \(C = 2\).
• \(P\) is the number of phases in equilibrium at that point.
• The "+ 1" represents the single variable of temperature (as pressure is kept constant).

Analysis of the Points:

• Point \(w\):

This point is located in the single-phase Liquid region.

• Number of phases, \(P = 1\)
• Calculation: \(F = 2 - 1 + 1 = 2\)
• Interpretation: Both temperature and composition can be varied independently within this field without changing the number of phases.
• Point \(x\):

This point lies on the liquidus curve (the boundary between the Liquid and the \(A + \text{Liquid}\) fields). At equilibrium on this line, two phases coexist: Solid \(A\) and Liquid.

• Number of phases, \(P = 2\)
• Calculation: \(F = 2 - 2 + 1 = 1\)
• Interpretation: The system is univariant; if the temperature is changed, the composition must change along the curve to maintain equilibrium.
• Point \(y\):

This point is located on the horizontal boundary representing the transition from the \(A + \text{Liquid}\) field. In the context of phase assemblage regions in such diagrams, points on boundaries or within two-phase fields represent an equilibrium between two phases (e.g., \(A\) and Liquid or \(A\) and \(B\)).

• Number of phases, \(P = 2\)
• Calculation: \(F = 2 - 2 + 1 = 1\)
• Point \(z\):

This point is located within the \(C + \text{Liquid}\) field. Any point inside a two-phase region of a binary isobaric diagram represents an equilibrium between two phases.

• Number of phases, \(P = 2\)
• Calculation: \(F = 2 - 2 + 1 = 1\)

Summary Table:

Therefore, the degrees of freedom are \(w = 2, x = 1, y = 1, z = 1\), which matches Option 1.