Gibbs Phase Rule and EOS

Thermodynamics Properties and Relations

Overview of Thermodynamic Properties

• Focus on thermodynamic properties: pressure, volume, temperature.
• Aim: Relate properties to solve complex problems related to the first law of thermodynamics.

Degrees of Freedom and Phase Rule

• Group Change Group: Indicates the degrees of freedom in a system.
• Phase Rule Equation: $F = C - P + 2$
  • Where:
  • $F$ = degrees of freedom (number of independent variables),
  • $C$ = number of components,
  • $P$ = number of phases present.
• Once temperature is specified, either pressure or volume can be determined, leading to knowledge of the remaining properties.

States of Aggregation and Saturation Conditions

• States of Aggregation: Solid, liquid, gas.
• Use of Saturated Steam Tables to determine the phase of substance:
  • Higher Pressure: Indicates presence of liquid phase.
  • Lower Pressure: Indicates presence of vapor phase.
  • Saturation Conditions: At saturation temperature and saturation pressure, a mixture of liquid and vapor exists.
• Superheated Conditions: Indicated by both temperature and pressure being above saturation.

PVT Properties and Diagrams

• Pressure-Temperature (P-T) Diagrams: Visual representation of state changes and phases.
  • Important Points:
  • Triple Point: All three phases coexist,
  • Critical Point: The point at which the liquid and gas phases are indistinguishable.
  • Phase Transition Curves:
  • Vaporization Curve: Separates liquid and vapor regions.
  • Melting Curve: Separates solid and liquid regions.
  • Sublimation Curve: Separates solid and gas regions.
• Pressure-Volume (P-V) Diagrams:
  • Curve regions represent liquid-vapor mixtures, identified as the Phase Envelope.
  • Properties can be derived from points on each side of the phase envelope.

Equations of State

• Relationships between pressure, volume, temperature defined by the equation of state.
• Ideal Gas Law: For an ideal gas, valid under conditions of low pressure and high temperature.
  • Behavior: At high temperatures and low pressures, real gases behave similarly to ideal gases.
• Use of reference states for simplicity in calculations; reference state does not have to be uniform across all problems.

Internal Energy and Properties of Ideal Gas

• Internal Energy: Dependent solely on temperature for ideal gases.
• Heat Capacities:
  • At Constant Volume: $C<em>{V,IG} = \frac{ ext{d}U</em>{IG}}{ ext{d}T}$
  • At Constant Pressure: $C<em>{P,IG} = \frac{ ext{d}H</em>{IG}}{ ext{d}T}$
  • Relationship between capacities: $C<em>P = C</em>V + R$ where $R$ is the gas constant.

First Law of Thermodynamics for Ideal Gases

• Closed Systems: Examination of changes occurring from state 1 to state 2 for ideal gases under different processes.
• Use of the First Law: $ext{Δ}U = Q - W$
  • Where:
    • $Q$ = heat added to the system,
    • $W$ = work done by the system.
• Analyzing specific processes leads to expressions for the heat transfer and work.

Special Processes in Ideal Gases

• Isothermal Process (constant temperature):

  • $T<em>1 = T</em>2 = T$
  • Heat transfer equation: $Q = nRT ext{ln}\frac{V<em>2}{V</em>1}$
  • Thus, $Q = -W$ because internal energy change is zero.

• Isobaric Process (constant pressure):

  • Both initial and final states are at the same pressure.
  • Heat transfer expression derived relates to enthalpy change:
  • $Q = nC_P ext{Δ}T$

• Isochoric Process (constant volume):

  • Volume remains constant during the process.
  • Heat transfer is equal to the change in internal energy:
  • $Q = nC_V ext{Δ}T$

• Adiabatic Process (no heat transfer):

  • Changes are derived from First Law noting $Q=0$
  • Relationships established include:
  • $T<em>2 = T</em>1 imes \frac{V<em>1}{V</em>2}^{\frac{R}{C_V}}$
  • Expressions for work derived, including those linking pressure and volume:
    $W = \frac{P<em>1 V</em>1}{ ext{(γ-1)}} imes ext{(P2/P1)}^{ ext{(γ-1)}}$

Adiabatic Expansion Coefficient

• Define: $ext{γ} = \frac{C<em>P}{C</em>V}$, relates heat capacities.
• Derived relationships between pressure, volume, and temperature based on initial and final states:
  • $T<em>2 / T</em>1 = (V<em>1 / V</em>2)^{(γ - 1)}$
  • $P<em>2 / P</em>1 = (V<em>1 / V</em>2)^{γ}$
• Derived final relationship linking pressure and temperature through adiabatic processes and state changes.

Conclusion

• Illustrated relationships through processes while emphasizing ideal gas behaviors and equations of state.
• Transition between states analyzed through various defined processes such as isothermal, isobaric, isochoric, and adiabatic, applying First Law principles to compute work and heat transfer efficiently.