Question

The phase diagram of a single component system is given below. \begin{center} \includegraphics[width=0.5width]{46.png} \end{center} The option with the correct number of degrees of freedom corresponding to the labelled points i, j, and k, respectively, is

• A. 0, 1, 2
• B. 3, 2, 1
• C. 2, 0, 1
• D. 0, 2, 1

Hint:

Remember the Gibbs phase rule \(F = C - P + 2\). For a single-component system (\(C=1\)), it simplifies to \(F = 3 - P\). - At a triple point (three phases coexist), \(F = 0\). - Along a phase boundary (two phases coexist), \(F = 1\). - In a single-phase region, \(F = 2\).

Answer 1

The Correct Option is A

The number of degrees of freedom (\(F\)) in a single-component system is given by the Gibbs phase rule: \[ F = C - P + 2 \] where \(C\) is the number of components and \(P\) is the number of phases in equilibrium. For a single-component system, \(C = 1\), so the phase rule becomes: \[ F = 1 - P + 2 = 3 - P \] Now, let's analyze the degrees of freedom at each labeled point: Point i: Point i lies at the intersection of three phase boundaries. This indicates that three phases are in equilibrium at this point (e.g., solid, liquid, and gas at the triple point). Number of phases, \(P = 3\). Degrees of freedom, \(F = 3 - P = 3 - 3 = 0\). Thus, point i is invariant; both temperature and pressure are fixed. Point j: Point j lies on a phase boundary, which represents the equilibrium between two phases (e.g., solid-liquid, liquid-gas, or solid-gas). Number of phases, \(P = 2\). Degrees of freedom, \(F = 3 - P = 3 - 2 = 1\). Thus, point j is univariant; either temperature or pressure can be independently varied, and the other will be fixed by the equilibrium condition. Point k: Point k lies in a region where only a single phase exists (e.g., solid, liquid, or gas). Number of phases, \(P = 1\). Degrees of freedom, \(F = 3 - P = 3 - 1 = 2\). Thus, point k is bivariant; both temperature and pressure can be independently varied within the region where the single phase is stable. Therefore, the number of degrees of freedom at points i, j, and k are 0, 1, and 2, respectively. This corresponds to option (A). \bigskip