Phase Equilibrium MCQ Quiz - Objective Question with Answer for Phase Equilibrium - Download Free PDF

CONCEPT:

Degrees of Freedom in Phase Diagrams

F = C - P + 2

• The degrees of freedom (F) of a system can be calculated using the Gibbs phase rule:
  • C is the number of components in the system (in this case, a single component system, so C = 1),
  • P is the number of phases in equilibrium at the point in the phase diagram.
• At different points in a phase diagram, the number of phases (P) can vary, affecting the degrees of freedom.

EXPLANATION:

• For example:
  • At point j, three phases meet (P = 3), so the system has zero degrees of freedom (F = 0).
  • At point i, two phases meet (P = 2), so the system has one degree of freedom (F = 1).
  • At point k, the system is in a single phase (P = 1), so the system has two degrees of freedom (F = 2).

• At point i (boundary between two phases), there is 1 degree of freedom because temperature and pressure can be varied independently.
• At point j (triple point where three phases meet), there are zero degrees of freedom because all variables (temperature and pressure) are fixed by the equilibrium of the three phases.
• At point k (single phase region), there are 2 degrees of freedom because both temperature and pressure can be varied independently.

Therefore, the degrees of freedom corresponding to points i, j, and k are 0, 1, and 2, respectively.

CONCEPT:

Triple Point Temperature and Vapor Pressure Equality

ln(p₁) = ln(p₂)

• The triple point of a substance is the unique temperature and pressure at which all three phases (solid, liquid, and gas) coexist in equilibrium.
• At the triple point, the vapor pressure of the solid phase (p₁) is equal to that of the liquid phase (p₂):
• Given expressions:
  • ln(p₁) = -2000/T + 5
  • ln(p₂) = -4000/T + 10

EXPLANATION:

• At the triple point:

  ln(p₁) = ln(p₂)
  ⇒ -2000/T + 5 = -4000/T + 10

• Rearranging terms:

  -2000/T + 4000/T = 10 - 5
  ⇒ 2000/T = 5

• Solving for T:

  T = 2000 / 5 = 400 K

Therefore, the triple point temperature of the substance is 400 K.

CONCEPT:

Variation of Molar Heat Capacity (C_V,m) with Temperature for a Diatomic Gas

• Diatomic molecules exhibit contributions to heat capacity from different degrees of freedom as temperature increases:
  1. Translational motion: Always active → contributes (3/2)R
  2. Rotational motion: Becomes active at moderate temperatures → adds R ⇒ Total = 2.5 R
  3. Vibrational motion: Activates at higher temperatures → adds another R (from both potential and kinetic energy) ⇒ Total = 3.5 R
• The sharp jump (discontinuity) in the graph represents dissociation, beyond which C_V can decrease due to bond breakage and energy redistribution.

EXPLANATION:

• At low T → only translational and rotational DOF contribute ⇒ X = 2.5 R
• At moderate T → vibrational modes begin to contribute ⇒ Y = 3.0 R
• At high T before dissociation is complete → full vibrational contribution ⇒ Z = 3.5 R

Therefore, the correct sequence is: X = 2.5 R, Y = 3.0 R, Z = 3.5 R — Option 2.

Concept:

Phase Rule and Calculation of Phases, Components, and Degrees of Freedom

• The phase rule (Gibbs' phase rule) is a fundamental principle in thermodynamics that describes the relationship between the number of phases, components, and degrees of freedom in a system at equilibrium.
• The phase rule is given by the equation:

  F = C - P + 2

  where:
  • F is the degrees of freedom (the number of independent variables, such as temperature and pressure, that can be changed independently).
  • C is the number of components (chemically independent substances in the system).
  • P is the number of phases (distinct forms of matter present, such as solid, liquid, and gas).

Explanation:

• The decomposition of calcium carbonate (CaCO₃) is represented by the reaction:

  CaCO₃(s) ⇌ CaO(s) + CO₂(g)

• For this reaction:
  • Number of phases (P): The system contains three phases—solid CaCO₃, solid CaO, and gaseous CO₂. Therefore, p = 3.
  • Number of components (C): The components are CaCO₃, CaO, and CO₂. Thus, C = 2, because CaCO₃ decomposes into CaO and CO₂, making it two chemically independent components.
  • Degrees of freedom (F): Using the phase rule equation:

    F = C - P + 2

    F = 2 - 3 + 2 = 1

Hence, the correct answer is P = 3; C = 2; F = 1.

Concept:

Phases in a System during Thermal Decomposition

The reaction is:

CaCO₃(s) → CaO(s) + CO₂(g)

• The number of phases in a system refers to the distinct states of matter (solid, liquid, gas) present in the system during a chemical process.
• When calcium carbonate undergoes thermal decomposition, it breaks down into calcium oxide and carbon dioxide gas.
• In this reaction, calcium carbonate (CaCO₃) is solid, calcium oxide (CaO) is also solid, and carbon dioxide (CO₂) is a gas. Therefore, the system contains three distinct phases: solid, solid, and gas.

Explanation:

• The decomposition of calcium carbonate involves solid calcium carbonate (CaCO₃) turning into solid calcium oxide (CaO) and gaseous carbon dioxide (CO₂).
• Thus, there are three phases in the system: solid (CaCO₃), solid (CaO), and gas (CO₂).

Therefore, the correct answer is: 3 phases.

Concept:

Phase Rule and Calculation of Phases, Components, and Degrees of Freedom

• The phase rule (Gibbs' phase rule) is a fundamental principle in thermodynamics that describes the relationship between the number of phases, components, and degrees of freedom in a system at equilibrium.
• The phase rule is given by the equation:

  F = C - P + 2

  where:
  • F is the degrees of freedom (the number of independent variables, such as temperature and pressure, that can be changed independently).
  • C is the number of components (chemically independent substances in the system).
  • P is the number of phases (distinct forms of matter present, such as solid, liquid, and gas).

Explanation:

• The decomposition of calcium carbonate (CaCO₃) is represented by the reaction:

  CaCO₃(s) ⇌ CaO(s) + CO₂(g)

• For this reaction:
  • Number of phases (P): The system contains three phases—solid CaCO₃, solid CaO, and gaseous CO₂. Therefore, p = 3.
  • Number of components (C): The components are CaCO₃, CaO, and CO₂. Thus, C = 2, because CaCO₃ decomposes into CaO and CO₂, making it two chemically independent components.
  • Degrees of freedom (F): Using the phase rule equation:

    F = C - P + 2

    F = 2 - 3 + 2 = 1

Hence, the correct answer is P = 3; C = 2; F = 1.

CONCEPT:

Degrees of Freedom in Phase Diagrams

F = C - P + 2

• The degrees of freedom (F) of a system can be calculated using the Gibbs phase rule:
  • C is the number of components in the system (in this case, a single component system, so C = 1),
  • P is the number of phases in equilibrium at the point in the phase diagram.
• At different points in a phase diagram, the number of phases (P) can vary, affecting the degrees of freedom.

EXPLANATION:

• For example:
  • At point j, three phases meet (P = 3), so the system has zero degrees of freedom (F = 0).
  • At point i, two phases meet (P = 2), so the system has one degree of freedom (F = 1).
  • At point k, the system is in a single phase (P = 1), so the system has two degrees of freedom (F = 2).

• At point i (boundary between two phases), there is 1 degree of freedom because temperature and pressure can be varied independently.
• At point j (triple point where three phases meet), there are zero degrees of freedom because all variables (temperature and pressure) are fixed by the equilibrium of the three phases.
• At point k (single phase region), there are 2 degrees of freedom because both temperature and pressure can be varied independently.

Therefore, the degrees of freedom corresponding to points i, j, and k are 0, 1, and 2, respectively.

CONCEPT:

Triple Point Temperature and Vapor Pressure Equality

ln(p₁) = ln(p₂)

• The triple point of a substance is the unique temperature and pressure at which all three phases (solid, liquid, and gas) coexist in equilibrium.
• At the triple point, the vapor pressure of the solid phase (p₁) is equal to that of the liquid phase (p₂):
• Given expressions:
  • ln(p₁) = -2000/T + 5
  • ln(p₂) = -4000/T + 10

EXPLANATION:

• At the triple point:

  ln(p₁) = ln(p₂)
  ⇒ -2000/T + 5 = -4000/T + 10

• Rearranging terms:

  -2000/T + 4000/T = 10 - 5
  ⇒ 2000/T = 5

• Solving for T:

  T = 2000 / 5 = 400 K

Therefore, the triple point temperature of the substance is 400 K.

CONCEPT:

Variation of Molar Heat Capacity (C_V,m) with Temperature for a Diatomic Gas

• Diatomic molecules exhibit contributions to heat capacity from different degrees of freedom as temperature increases:
  1. Translational motion: Always active → contributes (3/2)R
  2. Rotational motion: Becomes active at moderate temperatures → adds R ⇒ Total = 2.5 R
  3. Vibrational motion: Activates at higher temperatures → adds another R (from both potential and kinetic energy) ⇒ Total = 3.5 R
• The sharp jump (discontinuity) in the graph represents dissociation, beyond which C_V can decrease due to bond breakage and energy redistribution.

EXPLANATION:

• At low T → only translational and rotational DOF contribute ⇒ X = 2.5 R
• At moderate T → vibrational modes begin to contribute ⇒ Y = 3.0 R
• At high T before dissociation is complete → full vibrational contribution ⇒ Z = 3.5 R

Therefore, the correct sequence is: X = 2.5 R, Y = 3.0 R, Z = 3.5 R — Option 2.

Concept:

Phases in a System during Thermal Decomposition

The reaction is:

CaCO₃(s) → CaO(s) + CO₂(g)

• The number of phases in a system refers to the distinct states of matter (solid, liquid, gas) present in the system during a chemical process.
• When calcium carbonate undergoes thermal decomposition, it breaks down into calcium oxide and carbon dioxide gas.
• In this reaction, calcium carbonate (CaCO₃) is solid, calcium oxide (CaO) is also solid, and carbon dioxide (CO₂) is a gas. Therefore, the system contains three distinct phases: solid, solid, and gas.

Explanation:

• The decomposition of calcium carbonate involves solid calcium carbonate (CaCO₃) turning into solid calcium oxide (CaO) and gaseous carbon dioxide (CO₂).
• Thus, there are three phases in the system: solid (CaCO₃), solid (CaO), and gas (CO₂).

Therefore, the correct answer is: 3 phases.

Concept:

Triangular Diagram for a Three-Component System

• A triangular diagram is used to represent the composition of a three-component system (A, B, and C) in terms of mole fractions.
• Each vertex of the triangle represents 100% of one component (x_A, x_B, or x_C), while the opposite side corresponds to 0% of that component.
• The sum of the mole fractions of all three components at any point inside the triangle is always equal to 1:

  x_A + x_B + x_C = 1

Explanation:

• For point e on the diagram:
  • The value of x_A is determined by measuring the perpendicular distance from point e to the side opposite to vertex A. This value is approximately 0.20.
  • The value of x_B is determined by the perpendicular distance from point e to the side opposite to vertex B. This value is approximately 0.40.
  • The value of x_C is determined by the perpendicular distance from point e to the side opposite to vertex C. This value is also approximately 0.40.
• Hence, the mole fractions of x_A, x_B, and x_C for point e are 0.20, 0.40, and 0.40, respectively.

Therefore, the correct answer is: 0.20, 0.40, 0.40.

Concept:

Degrees of Freedom and Reduced Phase Rule

• The degrees of freedom (F) in a system are determined using Reduced Gibbs' Phase Rule:

  F = C - P + 1

  where:
  • F = degrees of freedom
  • C = number of components
  • P = number of phases in equilibrium
• This applies when considering a condensed system, where the pressure is essentially constant and the only relevant variables are temperature and composition, effectively reducing the number of degrees of freedom by one. 

Explanation:

• System: Liquid water and water vapor are in equilibrium at 1 atm pressure.
• Number of components (C): The system has one component, H₂O.
• Number of phases (P): Two phases are present: liquid and vapor.
• Calculation:

  F = C - P + 1

  F = 1 - 2 + 1

  F = 0

Therefore, the number of degrees of freedom is: 0.