Energy, Entropy, and Free Energy

Thermodynamics

  • Provides information on whether a reaction is spontaneous based only on the properties of the reactants and products.
  • Spontaneous process: Occurs without external intervention.

Entropy (S)

  • Measure of molecular randomness or disorder.
  • Thermodynamic function that describes the number of combinations available to a system in a given state.
  • Nature spontaneously proceeds toward the states with the highest probabilities.
  • S{solid} < S{liquid} < S_{gas}
  • Entropy change when mixing two pure substances is expected to be positive.

Positional Probability

  • Depends on the number of configurations in space that yield a particular state.
  • Gas expands into a vacuum to give a uniform distribution.

Second Law of Thermodynamics

  • In any spontaneous process, there is always an increase in the entropy of the universe.
  • First law of thermodynamics: Energy of the universe is constant.
  • \Delta S_{univ} > 0: Process is spontaneous in the direction written
  • \Delta S_{univ} < 0: Process is spontaneous in the opposite direction.
  • ΔSuniv=0\Delta S_{univ} = 0: Process has no tendency to occur; system is at equilibrium.

Entropy Changes in the Surroundings (ΔSsurr)

  • Determined by the flow of energy as heat.
    • Exothermic process increases KE associated with the random motions of atoms in the surroundings so ΔSsurr\Delta S_{surr} is positive.
    • Endothermic process decreases KE associated with the random motions of atoms in the surroundings so ΔSsurr\Delta S_{surr} is negative.
  • Magnitude of ΔSsurr\Delta S_{surr} depends on the magnitude of the heat and the temperature.
  • ΔSsurr=−quantity of heat(J)temperature(K)=−ΔHT\Delta S_{surr} = -\frac{quantity \space of \space heat (J)}{temperature (K)} = -\frac{\Delta H}{T}

Interplay of ΔS<em>Sys\Delta S<em>{Sys} and ΔS</em>Surr\Delta S</em>{Surr} in Determining the Sign of ΔSuniv\Delta S_{univ}

  • ΔS<em>univ=ΔS</em>sys+ΔSsurr\Delta S<em>{univ} = \Delta S</em>{sys} + \Delta S_{surr}

Entropy Changes and Chemical Reactions

  • ΔS=S<em>product–S</em>reactant\Delta S = S<em>{product} – S</em>{reactant}
  • If the number of product molecules is greater than the number of reactant molecules, ΔS\Delta S is positive.

Entropy Values

  • Third law of thermodynamics: Entropy of a perfect crystal at 0 K is zero.
  • Standard entropy values (S∘S^\circ) represent the increase in entropy when a substance is heated from 0 K to 298 K at 1 atm.

Entropy Change for a Given Chemical Reaction

  • ΔS∘=∑n<em>pS∘</em>(products)−∑n<em>rS∘</em>(reactants)\Delta S^\circ = \sum n<em>pS^\circ</em>{(products)} - \sum n<em>rS^\circ</em>{(reactants)}

Free Energy (G)

  • G=H−TSG = H - TS
  • At constant temperature, processes are spontaneous in the direction in which free energy decreases.
  • Negative ΔG\Delta G means positive ΔSuniv\Delta S_{univ}

Various Possible Combinations of ΔH\Delta H and ΔS\Delta S

  • ΔS\Delta S positive, ΔH\Delta H negative: Spontaneous at all temperatures.
  • ΔS\Delta S positive, ΔH\Delta H positive: Spontaneous at high temperatures.
  • ΔS\Delta S negative, ΔH\Delta H negative: Spontaneous at low temperatures.
  • ΔS\Delta S negative, ΔH\Delta H positive: Process not spontaneous at any temperature.

Standard Free Energy Change (ΔG∘\Delta G^\circ)

  • Change in G that will occur if the reactants in their standard states are converted to the products in their standard states.
  • Standard state: 1 atm and 25°C.

Methods for Calculating ΔG∘\Delta G^\circ

  • ΔG∘=ΔH∘−TΔS∘\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ
  • Treat free energy as a state function and use Hess’s law.
  • Use standard free energy of formation.

Standard Free Energy of Formation (ΔGf∘\Delta G_f^\circ)

  • Change in free energy that accompanies the formation of 1 mole of a substance from its constituent elements.
  • All reactants and products are in their standard states.
  • ΔGf∘\Delta G_f^\circ of an element in its standard state = 0.

Free Energy and Pressure

  • G=G∘+RTln⁡(P)G = G^\circ + RT\ln(P)

Free Energy Change at non-standard conditions

  • ΔG=ΔG∘+RTln⁡(Q)\Delta G = \Delta G^\circ + RT\ln(Q)

Free Energy and Equilibrium

  • System under constant P and T proceeds spontaneously in the direction that lowers its free energy.
  • Equilibrium is the point where the free energy value is at its lowest.
  • ΔG∘=−RTln⁡(K)\Delta G^\circ = -RT\ln(K)
  • ΔG=ΔG∘+RTln⁡(Q)\Delta G = \Delta G^\circ + RT\ln(Q)

Qualitative Relationship between the ΔG and the K for a Given Reaction

  • ΔG∘=0\Delta G^\circ = 0, K = 1
  • ΔG∘<0\Delta G^\circ < 0, K > 1
  • \Delta G^\circ > 0, K < 1