Lecture Notes on Thermodynamics and Entropy

Second Law of Thermodynamics (10.3, 10.5)

• It is impossible to obtain an equal amount of work out of a system relative to the energy put into it under isothermal reversible conditions.
  • W_{rev}
• For a real system:
  • Any spontaneous process occurring without outside intervention must increase the entropy of the Universe.
  • \Delta S{univ} = \Delta S{sys} + \Delta S_{surr}
    • \Delta S_{univ} will always increase for a real process until it reaches a max: Heat death of the Universe
  • Reversible:
    • \Delta S{sys} = -\Delta S{surr}

Enthalpy

• Real processes that are both spontaneous
• H₂O(g) ⇌ H₂O(l)
  • \Delta H_{vap}: H₂O(l) → H₂O(g)
  • \Delta H{cond}: H₂O(g) → H₂O(l) = -\Delta H{vap}
  • \Delta S{sys} > 0: \Delta S{vap}
    • Heat is stolen from the surroundings, increasing disorder, the system gains more order
    • Endothermic
  • H₂O(g) → H₂O(l)
    • \Delta S{sys} < 0: \Delta S{cond}
    • Heat is dumped into the surroundings in the form of disordered thermal motion
    • Exothermic

Entropy Problem

• Methane (CH₄) has a heat of vaporization of 8.17 kJ/mole and a boiling point of -162 °C.
• Calculate the entropy change methane undergoes associated with going from a gas to liquid.
  • Condensation: gas → liquid
  • T{surr} = T{B.P.}
  • \Delta S{sys} = \frac{q{rev}}{T{surr}} = \frac{\Delta H{cond}}{T_{B.P.}}
  • \Delta S{sys} = \frac{-\Delta H{vap}}{T_{B.P.}}
  • \Delta S_{sys} = \frac{-8170 \frac{J}{mol}}{111 K} = -73.6 \frac{J}{mol \cdot K}

Microscopic View of Entropy

• Assumed Kinetic Molecular Theory (KMT):
  • Atoms exist, they have mass + velocity
  • KE → velocity
• Result:
  • Energies of a gas follows a Gaussian Distribution
• Ludwig Boltzmann showed over time any distribution of gas energies will evolve to a gaussian
  • Entropy on a molecular level evolves over time to a max value

Microscopic View of Entropy (10.1)

• Boltzmann realized the more "positions" (Energy + locations) available to a system, the more likely that state is.
• Boltzmann Equation:
  • S = k_B \ln \Omega
    • S = Entropy
    • \Omega = # of microstates representing that state
    • kB= Boltzmann Constant: \frac{R}{NA} = \frac{8.314 \frac{J}{K \cdot mol}}{6.022 \times 10^{23} \frac{molecules}{mol}}
  • Microstate: Arrangement of position & Energy of a system
  • \Delta S = S2 - S1 = kB \ln \frac{\Omega2}{\Omega_1}

Microscopic View of Entropy (10.3)

• Example 1: 1 atom in an expanding Box
  • Ideal gas expands thermally
  • Double the volume
    • V2 = 2 \cdot V1
    • \Omega2 = 2 \cdot \Omega1
  • \Delta S = kB \ln (\frac{2\Omega1}{\Omega1}) = kB \ln (2)
  • This is why gases expand to fill their containers; \Delta S increases

Microscopic View of Entropy (Table 10.2)

• Example 2
• State I: AB CD
• State II: A B C D
• Microstates (\Omega)
  • I: [AB, CD]
  • II: [AC, BD], [AD, BC], [A,B,C,D]
• \Delta S{I \rightarrow II} = kB \ln (4)

Microscopic View of Entropy and 2nd Law (10.5)

• Entropy is a measure of the # of Equivalent Configurations (Energy & location) of a State
  • More configurations, higher S → More configurations, more likely
• Over time, \Delta S becomes more positive
• 2nd Law:
  • \Delta S{univ} = \Delta S{sys} + \Delta S_{surr}
  • \Delta S_{univ} increases

Entropy Changes At the Molecular Level (10.8)

• H₂O(g) → H₂O(s) : \Delta S -
• NaCl(s) → Na⁺(aq) + Cl⁻(aq) : \Delta S +
• N₂(g) + 3H₂(g) → 2NH₃(g) : \Delta S -
• Need to know something about the molecule

Degrees of Freedom and Entropy

• # degrees of freedom = #ways a molecule can move
  • translate
  • rotate
  • vibrate
• Linear: 3N-5
• Non-Linear: 3N-6
• N = # nuclei in molecule
• More degrees of freedom → higher S
  • H₂O larger S than H₂