Chapter 20: Spontaneity, Entropy, and Gibbs Free Energy

Learning Objectives

• Define spontaneity and entropy; predict the sign of \,Delta S associated with phase changes and chemical reactions.
• Calculate entropy changes, \,Delta S, for reactions using standard entropies or combining entropies for other reactions.
• Calculate entropy change in a reaction, \,Delta S, using stoichiometry.
• Define Gibbs Free Energy by relating it to work and spontaneity.
• Predict the sign of Gibbs Free Energy, \,Delta G, for a reaction and calculate \,Delta G from \,Delta S and \,Delta H.
• Calculate the temperature at which a reaction switches spontaneity.
• Calculate Gibbs Free Energy, \,Delta G, from standard Gibbs Free Energies of formation and by combining Gibbs Free Energies of other reactions.
• Calculate the Gibbs Free Energy of a reaction, \,Delta G, by proportionality (using stoichiometry).

Spontaneity

• Spontaneous Process: Occurs without ongoing external intervention; can be fast or slow.
• Exothermic Reactions: Many, but not all, spontaneous reactions are exothermic; some are endothermic.
• Temperature Dependence: Spontaneity can depend on temperature.
  • Example: Ice melting at room temperature is spontaneous but endothermic.
• Recrystallization: Used to purify reaction mixtures by exploiting temperature-dependent solubility.
• Phase Changes:
  • Melting: Not spontaneous at \,T < 0
  • Freezing: Not spontaneous at \,T > 0

Energy, Spontaneity, and Entropy

• Entropy (S): Dispersal of energy through available microstates at a given temperature.
  • Formula: \,S = k_B \,ln W
    • \,k_B = 1.38 \times 10^{-23}
    • \,W = # of microstates
  • Microstates: Possible locations of atoms or molecules; different locations = different ways to spread out kinetic energy (KE).
• Entropy Change (\,\Delta S): Quantifies the dispersion of energy.
  • Formula: \,\Delta S = \frac{q_{rev}}{T}
  • \,\Delta S_{system} > 0 indicates increasing molecular disorder.
• Second Law of Thermodynamics: The entropy of the universe is always increasing.
  • More precisely, processes occur spontaneously when \,\Delta S_{universe} > 0.
  • Formula: \,\Delta S{universe} = \Delta S{system} + \Delta S_{surroundings}
  • \,\Delta S_{universe} > 0 for spontaneous processes.

Predicting the Sign of \,\Delta S: States of Matter

• For \,s \,\,\rightarrow \,\, l \,\,\rightarrow \,\, g, \,\Delta S_{sys} > 0 (molecular disorder increases).
  • Liquids are more disordered than solids.
  • Gases are much more disordered due to high kinetic energy.
  • Changes in gas moles in a reaction indicate the sign of \,\Delta S.
  • \,S{gas} >> S{sol} and \,S_{liq} because gases have high KE.

Entropy and Disorder: A Macroscopic View

• \,\Delta S_{system} > 0: Molecular disorder is increasing.
• Dissolution: More disorder as solvated ions than in a well-ordered solid or liquid.
  • Putting more things in solution increases entropy.

Predicting the Sign of \,\Delta S: Molecular Structure

• \,\Delta S_{system} > 0: Molecular disorder is increasing.
• Molecular Complexity: More complex molecules (more atoms) have more potential for disorder (more possible configurations or microstates).
  • Propane has more configurations.
  • More atoms = more rotations or vibrations.
• Atomic Structure: Atoms with more electrons usually have higher entropy (more E transitions, e.g., \,Na \,S < K \,S).

Entropy Factors

• Moles of Gas: Increasing moles of gas increases entropy (for reaction in the forward direction).
• Temperature: Increasing T = increasing S.
• Volume: Increasing V = increasing S. Increasing volume allows more degrees of freedom/motion for the particles (more possibilities for different locations of particles = more miscrostates), which results in increasing entropy. Incr. T also incr. entropy because faster molecules move through more microstates

Entropy Practice

• Gas entropy > solid entropy (more motion).
• More moles of gas = more entropy (more possibilities of where the molecules can be = more disorder).

Predicting the Sign of \,\Delta S for Phase Changes

• For the sublimation of dry ice at room temperature:
  • \,\Delta S{CO2} is (+) because entropy increases when \,s \,\,\rightarrow g
  • \,\Delta S_{universe} is (+) because sublimation is spontaneous at room temp.
  • Below the sublimation temperature of dry ice, \,\Delta S_{universe} is (-) because sublimation is not spontaneous.

Third Law of Thermodynamics

• @ 0 K, no molecular motion = no disorder.
  • Only one configuration if no motion.
  • Microstates \,W = 1 and \,S = k_B \,ln W
  • \,ln 1 = 0, so \,S = 0.

Calculating \,\Delta S_{sys}^o

• Can use Hess’s Law with \,\Delta S from several equations:
  • \,\Delta S{rxn}^o = \,\Sigma \,ni \,S^o (products)i - \,\Sigma \,mj \,S^o (reactants)_j
  • n and m are the stoichiometric coefficients for each reactant/product from the balanced chemical equation.
  • Degree symbol (naught) means standard conditions (1 atm gas, 1 M soln).
• Absolute Entropies:
  • \,S^o is the entropy gained by a substance as it is heated from 0 K to 298 K.
  • These are absolute entropies (the entropy gained by heating from 0 K). Since we can’t get colder (less ordered) than 0 K, these are not changes in entropy but absolute because they go from no entropy to a known amt.

Entropy Calculation Example

• N2(g) + 3H2(g) → 2NH3(g) (4 gas moles → 2 gas moles, entropy decreases)
  • \,\Delta S_{rxn}^o = \,\Sigma \,n \,S^o (products) - \,\Sigma \,m \,S^o (reactants)
  • \,= [2(192.5)] – [191.6 + 3(130.7)] = -198.7 \,J/K
  • Stoichiometry: If the question said you have 0.232 moles of NH3 and asked what the entropy change was, you need to use stoichiometry: \,0.232 \,mol NH3 * (\frac{-198.7\,J/K}{2 \,mol NH3}) bc NH3 has 2 coeffic.
• \,\Delta S depends on stoichiometry (proportionality).

Determining Spontaneity: Calculating \,\Delta S_{universe}

• \,\Delta S{universe} = \,\Delta S{system} + \,\Delta S_{surroundings}
• @ Const. P, \, -q{sys} = -\,\Delta H{sys}
• If \,\Delta S{surroundings} \,\propto \,-q{sys} \,\propto \frac{1}{T}
• Then \,\Delta S{universe} = \,\Delta S{system} - \frac{\Delta H_{sys}}{T}
• Rearranging: \,-T\,\Delta S{universe} = -T\,\Delta S{sys} + \,\Delta H_{sys}
• Gibbs Free Energy (\,\Delta G{sys}): \,\Delta G{sys} = \,\Delta H{sys} - T\,\Delta S{sys}
  • Another state function

Gibbs Free Energy (\,\Delta G = \,\Delta H - T\,\Delta S)

• \,\Delta G < 0 (negative) → spontaneous and product favored (in forward direction) → work done by system (engines), work = product
• \,\Delta G > 0 (positive) → non-spontaneous and reactant favored (in forward direction) → no work done by system, work would need to be a reactant to make this happen

Spontaneity and Gibbs Free Energy

• Gibbs Free Energy (\,\Delta G) and work: \,\Delta G = -w_{max}
  • Maximum work: \,\Delta G tells us how much work the system could theoretically do (for a spontaneous reaction).
  • Work is a product for a spontaneous reaction.
  • Work is a reactant for a non-spontaneous reaction (needs input of work).
• Carnot cycle/engine: max efficiency process
  • Not possible in real reactions
  • All heat cannot be converted to work due to entropy

Gibbs Free Energy: Temperature Dependence

• \,\Delta G^o = \,\Delta H^o + (-T\,\Delta S^o)
  

• \,\Delta G < 0 (neg) = spont
• \,\Delta G > 0 (pos) = nonspont
• at equilibrium (not spont or nonspont), \,\Delta G = 0 → crossover temp between spont and non.
• \,\Delta G = 0 (equilibrium) examples: Boiling point, Melting point, Triple point, etc.
• Crossover \,T = \frac{\Delta H}{\Delta S}

Gibbs Free Energy: Reference Tables

• Can also do Hess’s Law with \,\Delta G from several equations
• Tables list \,\Delta G_f values
• Defined & used like \,\Delta Hf’s (\Delta Gf^o = 0 for elements)
  • refer to formation reactions
  • same standard state idea (1 atm(g), 1 M(aq))
• \Delta G{rxn}^o = \,\Sigma \,ni \,\Delta Gf^o (prod)i - \,\Sigma \,nj \,\Delta Gf^o (reactant)_j