Second Law of Thermodynamics & Entropy

CHAPTER 8: SECOND LAW OF THERMODYNAMICS - ENTROPY

Entropy Overview

• Entropy (S) is introduced as a thermodynamic property.
• Defined using the Clausius inequality.
• Increase of Entropy Principle:
  • States that the entropy for an isolated system (includes the system and surroundings) will either increase or stay the same.
  • This principle serves as a formulation of the Second Law of Thermodynamics.

Entropy as a Thermodynamic Property

• Definition and Properties:
  • Entropy, denoted as S, measures disorder within a system.
  • A more disordered system has higher entropy.

Understanding Entropy in Relation to Order

• Diagrammatic representation comparing states:
  • Ice Cube (Crystalline Structure):
  • Characteristics: Minimum entropy, maximum order.
  • Puddle of Water:
  • Characteristics: Maximum entropy, minimum order.

Formal Definition of Entropy

• Clausius Inequality:
  • Introduces the definition of S through heat transfer (δQ) and absolute temperature (T) at the boundary of heat transfer.
  • Applicable for all types of cycles (both reversible and irreversible).
  • Reversible Carnot Cycle:
  • Efficiency relates to heat transferred between high and low temperatures using the relationship:
    ext{Efficiency} = 1 - rac{T_L}{T_H}
  • The heat transfer for reversible cycles does not change:
    dS = rac{ ext{Rev } Q}{T}

Entropy Changes in Thermodynamic Processes

• For any reversible process, the change in entropy (ΔS) can be defined as:
  ext{Change in entropy} = S_2 - S_1 = rac{Q}{T}
• For irreversible processes, the change will include entropy generation (S_gen) where:
  $ext{Total Change} = S_{ ext{Transfer}} + S_{ ext{Generation}}$
• Notation relates that the cyclic integral of the differential heat exchange divided by temperature is always less than or equal to zero:
  rac{dq}{T} ext{ must abide by } ext{Inequality: }rac{dq}{T} ext{ } ext{is } ext{≤0}

Clausius Inequality Detailed Study

• Engaged systems connected to a thermal reservoir (T) receiving heat (dQ) and producing work (dW_rev).
• Heat Interaction:
  • work is related to heat transfer through:
    $ext{dW}<em>{sys} = ext{dW}</em>{rev} + dW_c$,
• Clarification of the relationship for internal energy changes of the system and energy interactions.

Special Cases and Applications of Entropy

• Change in entropy values is fixed across thermodynamic states, allowing calculations for processes targeting known start and end states.
• Example of Internally Reversible Isothermal Heat Transfer Processes:
  • It simplifies calculations for thermal reservoirs where the temperature remains fixed.

Increase of Entropy Principle

• States that:
  • Entropy can be generated (
    $S_{gen} ext{ (where } S_{gen} ext{ must be } ext{≥0)}$
  • For isolated systems, entropy can only increase or stay the same:
    $ext{Entropy Change} herefore ext{ΔS} ext{ } ext{≥ 0}$
• Implications:
  • Entropy differs fundamentally from energy, as it does not conserve under processes; total entropy is always increasing.
• Entropy generation links to irreversible processes, impacting the system's efficiency.

Example Problem: Isothermal Process

• In a piston-cylinder device containing a mixture of water:
  • Given: Heat transfer of 750 kJ at 300 K leads to vaporization.
  • Task: Determine the entropy change under the assumption of no irreversibility.

Detailed Examples of Heat Transfer Process

• Entropy generation during heat transfer:
  • Given temperatures of 800 K for the heat source and various sinks (500 K and 750 K).
  • Required to analyze which process is more irreversible:
  • Case (a)
    • Heat exchange calculations showing:
      $ext{AS}<em>{source} = -2.5 kJ/K$$ext{AS}</em>{sink} = 2.7 kJ/K$
      $ext{Total AS} = ext{AS}<em>{total} + ext{AS}</em>{system}$
  • Case (b) follows similar steps.

Second Law Theorem in Entropy Balance

• The Increase of Entropy Principle encapsulates the Second Law.
• Entropy Balance Equation:$ext{Entropy Change}( ext{ΔS}) = ext{Entropy Transfer} + ext{Entropy Generation}$
  • Closed system formula:
    $ext{ΔS} = ext{summing terms of } Q/T$
  • Rate form for open systems involves careful observations of mass flow and analysis.

Entropy Change Examples

• Given cases of pure substances in closed systems:
  • Example: Calculate entropy change from known values (initial and final states) for Refrigerant 134.

Mechanisms of Entropy Transfer

• Heat Transfer:
  • Heat transfers, linked by temperature, relate to entropy change equations.
  • Relationships to work done and no entropy transfer during work crossing boundaries.
• Mass Flow:
  • Involves energy and entropy transfer through mass interchange.

Example Problem on Entropy Change in a Tank

• A rigid tank contains refrigerant subject to cooling.
  • Seek to establish the change of entropy through assessments from the first state (initial) and decreasing pressures to a final state.
  • Conclusively, calculate the change and derive its significance in system understanding.

Exam Questions from Final Exam – December 2016

• Question a: State the Increase of Entropy Principle.
• Question b: Write a mathematical expression for entropy generated in an isolated system.
• In refrigeration scenarios, assess the entropy changes reflecting on process reversibility.