Notes on the Second Law of Thermodynamics

THE SECOND LAW OF THERMODYNAMICS: CONCEPTS, MACHINERY & APPLICATIONS

SECOND LAW STATEMENTS

• Statement 1: A cyclic process must transfer heat from a hot to a cold reservoir if it is to convert heat into work.

• Statement 2: Work must be done to transfer heat from a cold to a hot reservoir.

• Statement 3: The Carnot engine is the most efficient engine there is.

• Statement 4: A perpetual motion machine of the second kind does not exist.

• Statement 5: The entropy of the universe is always increasing.

• Energy Conservation Equation: Energy Input = Energy Output

CARNOT ENGINE / CYCLE (IDEAL ENGINE)

• Definition: An ideal cycle that serves as the framework for representing the conversion of heat (q) into work (w).

• Characteristics:

  • Comprises 4 reversible steps.

  • Operates between two temperature reservoirs.

  • Consists of:

  • 2 expansion steps.

  • 2 compression steps.

• Steps Explained:

  • Isothermal Steps (1 and 3): Occur at a constant temperature; these are slow processes.

  • Adiabatic Steps (2 and 4): Occur without heat transfer; these are rapid processes.

• First Law of Thermodynamics Application:

  • The change in internal energy (DU) is given by:
    $DU = q - w$

• Connection to Second Law:

  • The Carnot Cycle introduces entropy (S) as a thermodynamic state function and establishes a basis for evaluating thermodynamic efficiency.

THERMODYNAMIC EFFICIENCY OF THE CARNOT CYCLE

• Efficiency Equation:

  • The thermodynamic efficiency $( ext{e})$ is defined as:
    $ext{e} = 1 - rac{T<em>C}{T</em>H}$

  • Where:

  • $T_C$ = Temperature of the cold reservoir

  • $T_H$ = Temperature of the hot reservoir

ENTROPY CHANGES

• Concept of Entropy:

  • A measure of disorder or randomness in a system.

  • Criterion for spontaneity: If ext{ΔS} > 0, the process is spontaneous.

• Entropy Change Calculations:

  • For various thermodynamic processes including phase changes:
    $S = mc imes ext{ln} rac{T<em>F}{T</em>0}$

  • Heat ( ext{q}) equations:
    $Q = M imes L$
    $Q = M imes C imes (T<em>F - T</em>0)$

• Entropy Addition from Reactions:

  • Entropy change for reactions is given by:
    $ΔS_ ext{rxn} = ΣS_ ext{products} - ΣS_ ext{reactants}$

• General Entropy Relationships:

  • Solids < Liquids < Gases in terms of entropy: as temperature increases, entropy also increases.

CALCULATING ENTROPY CHANGES UNDER VARIOUS THERMODYNAMIC PROCESSES

• Heat Transfer and Volume-Pressure-Temperature Changes:

  • Processes that result in entropy changes.

• Example Problems:

  • Problems involving phase changes, heat transfer, and mixing.

CONCLUSIONS FROM THE CARNOT CYCLE

• Analysis of first law applied to each path of the cycle.

• The relation of efficiency and entropy illustrates thermodynamic principles.

• Carnot's efficiency is the upper limit to the efficiency of any heat engine

THIRD LAW OF THERMODYNAMICS

• Definition:

  • The entropy, $S_0$, of a perfect crystalline substance at absolute zero (0 K) is defined as zero.

PHASE CHANGES AND ENTROPY CHANGE CALCULATIONS

• Heating Curve Analysis:

  • Use heating curve to analyze heat and enthalpy changes across phase changes.

  • Total heat energy and entropy change for each segment of the curve are to be calculated.

CHEMICAL REACTIONS: ENTROPY CHANGE AND DEPENDENCE ON TEMPERATURE

• Entropy Change Definition:

  • $ΔS = ΣS^ ext{°}{products} - ΣS^ ext{°}_ ext{reactants}$

  • Entropy is a state function.

  • Dependence of entropy on temperature illustrated through ΔS = C_p ext{dT}$.

CONSEQUENCES OF THE FIRST AND SECOND LAWS OF THERMODYNAMICS

• First Law:

  • dU = dq - dw, where dw = -PdV, establishes criteria for transformations.

• Second Law:

  • dq = TdS, representing the spontaneity criterion: TdS ≥ dq; TdS = dq in reversible processes.

ISOTHERMAL PROCESSES

• At Constant Volume:

  • dU ext{ and } dS relationships characterized by spontaneous conditions.

• At Constant Pressure:

  • Enthalpy change (dH ≤ 0) demonstrates spontaneity through the Gibbs Free Energy change.

FOUR FUNDAMENTAL EQUATIONS OF THERMODYNAMICS

• First Law Transition:

  • Equations transform to express relationships between internal energy (U), enthalpy (H), and free energies (G, A).

• Gibbs Free Energy: Relationship established for determining spontaneity at constant pressure and temperature:

  • $dG ≤ 0$

GIBBS FREE ENERGY DEPENDENCIES

• Pressure and Temperature Dependencies:

  • Gibbs Free Energy can be expressed as a function of both pressure and temperature:
    $dG = -SdT + PdV$

• Energy Mix of Gases:

  • Gibbs free energy change upon mixing ideal gases expressed as:
    ext{AG}{mix} = RT iggl[n1 ext{ln}(x1) + n2 ext{ln}(x_2) + … iggr]

  • Where $x = ext{mole fraction}$.

  • Illustrates spontaneous nature of mixing processes.