Study Notes on The Second Law of Thermodynamics

Chapter 3: The Second Law - The Entropy of the Universe Increases

Concepts

• Natural Processes and the Second Law

  • Water does not flow spontaneously up a waterfall to enlarge a pool at the top.
  • A candle never assembles a column of wax out of thin air by burning in reverse.
  • The Second Law of Thermodynamics formalizes the intuition about natural processes, stating their directional behavior.

• Entropy (S)

  • Definition: Entropy is a state function that measures the disorder of a system.
  • The terms "disorder" and "order" alone do not fully capture the meaning of entropy as described by Ludwig Boltzmann.
  • Formula: $S = k ln W$
  • Where ( S ) is the entropy, ( k ) is the Boltzmann constant, and ( W ) is the number of different configurations available to a system.
  • In an isolated system, any change results in an increase of entropy.
  • If the system is open to surroundings, its entropy can decrease if the increase in entropy of the surroundings is greater.
  • Thus: The entropy of the universe (system + surroundings) always increases.

• Gibbs Free Energy (G)

  • Defined as: $G = H - TS$
  • Where ( H ) is the enthalpy, ( T ) is temperature, and ( S ) is entropy.
  • A decrease in Gibbs free energy indicates a spontaneous process at constant temperature and pressure.

Applications

• First and Second Laws of Thermodynamics

  • The First Law states that energy is conserved, but does not indicate if a reaction will occur spontaneously.
  • The Second Law predicts the direction of spontaneous change in a system.

• Homeostasis in Living Organisms

  • Organisms maintain stable conditions by controlling the direction of chemical reactions.
  • Example: Voltage across a cell membrane drives ATP synthesis; the dephosphorylation of ATP couples to a reaction requiring a minimum concentration of ATP for spontaneity.
  • Knowledge of thermodynamics enables the alteration of conditions making impossible reactions feasible.

• Limitations of Thermodynamics

  • While thermodynamics indicates spontaneity, it does not provide rates of reactions or dependency on conditions.
  • Example: Diamond can convert to graphite spontaneously but is kinetically slow.

Toward the Second Law: The Carnot Cycle

• Historical Context

  • In the early 1800s, steam engines were prevalent, but not fully understood until Sadi Carnot's analysis in 1824 of an idealized heat engine underwent a four-step cycle.

• Carnot Cycle Description

  • The Carnot cycle involves four reversible processes returning the engine to its original state.
  1. Isothermal Expansion: Ideal gas expands isothermally at temperature ( T_{hot} ).
     • Work done: ( w1 ) and heat absorbed: ( q1 ).
  2. Adiabatic Expansion: Ideal gas expands adiabatically.
     • No heat exchange: ( q_2 = 0 );
     • Work done is negative: ( w2 < 0 ), resulting in temperature drop to ( T{cold} ).
  3. Isothermal Compression: Gas is compressed isothermally at ( T_{cold} );
     • Work done: ( w3 ) and heat expelled: ( q3 ) (negative).
  4. Adiabatic Compression: Gas is compressed adiabatically back to its original state.
     • Work done: ( w4 ), ( q4 = 0 ).

• Work and Heat Relations

  • Total work for one cycle: $W<em>{cycle} = w</em>1 + w<em>2 + w</em>3 + w_4$
  • Total heat absorbed: $Q = q<em>1 + q</em>2 + q<em>3 + q</em>4$
  • Based on the First Law: $riangle U = Q + W = 0 \\Rightarrow Q = -W$

• Equations for Each Step

  • At constant temperature for ideal gas:
  • Step I: $q<em>1 - w</em>1 = 0 \Rightarrow q<em>1 = w</em>1 = nRT<em>{hot} ext{ln} \left(\frac{V</em>2}{V_1}\right)$
  • For step II, $w<em>2 = C</em>y (T<em>{cold} - T</em>{hot}) \text{ and } C_y = \text{constant specific heat at constant volume}$
  • For steps III and IV, similar equations are derived that incorporate changes in work, heat, and temperature.

Key Findings from Carnot Cycle

• Efficiency of Carnot Engine
  • Efficiency defined as: $Efficiency = \frac{-W}{q_1}$
  • Efficiency in terms of temperatures: $Efficiency = 1 - \frac{T<em>{cold}}{T</em>{hot}}$
  • Example calculation with given temperatures.
• Reversible Engines
  • When run in reverse, heat engines can function as refrigerators, transferring heat from cold to hot reservoirs.
• Perpetual Motion and Efficiency
  • Carnot concluded that all heat engines operating reversibly between the same temperatures have the same efficiency. If different heat engines operate between the same temperatures, one must be less efficient than another.
  • Discussed fallacy of perpetual motion and implications of efficiency.

Conclusion

• Carnot's work, although initially unrecognized, laid the groundwork for the Second Law of Thermodynamics, emphasizing that heat cannot be converted completely into work without some loss.