Spontaneity, Entropy, and Free Energy

Shaun Williams, PhD

- Domain of Kinetics
- Rate of a reaction depends on the pathway from reactants to products.
- Thermodynamics tells us whether a reaction is spontaneous based only on the properties of reactants and products.

- Thermodynamics lets us predict the direction in which a process will occur but gives no information about the speed of the process.
- A spontaneous process is one that occurs without outside intervention.

Consider \(2.4\,\mathrm{moles}\) of a gas contained in a \(4.0\,\mathrm{L}\) bulb at a constant temperature of \(32^\circ \mathrm{C}\). This bulb is connected by a valve to an evacuated \(20.0\,\mathrm{L}\) bulb. Assume the temperature is constant.

What should happen to the gas when you open the value?

The gas should spread evenly throughout the two bulbs.

- The driving force for a spontaneous process is an increase in the entropy of the universe.
- A measure of molecular randomness or disorder.
- Thermodynamic function that describes the number of arrangements that are available to a system existing in a given state.
- Nature spontaneously proceeds toward the states that have the highest probabilities of existing.

- A gas expands into a vacuum to give a uniform distribution because the expanded state has the highest positional probability of states available to the system.
- Therefore: \(S_{solid} < S_{liquid} \ll S_{gas}\)

Predict the sign of \(\Delta S\) for each of the following:

- The evaporation of alcohol
- The freezing of water
- Compressing an ideal gas at constant temperature
- Heating an ideal gas at constant pressure
- Dissolving NaCl in water

positive

negative

negative

positive

positive

- In any spontaneous process there is always an increase in the entropy of the universe.
- The entropy of the universe is increasing.
- The total energy of the universe is constant, but the entropy is increasing. $$ S_{universe}=\Delta S_{system}+\Delta S_{surroundings} $$

- \(\Delta S_{surr} > 0\): entropy of the universe increases
- \(\Delta S_{surr} < 0\): process is spontaneous in opposite direction
- \(\Delta S_{surr} = 0\): process has no tendency to occur
- The sign of \(\Delta S_{surr}\) depends on the direction of the heat flow.
- The magnitude of \(\Delta S_{surr}\) depends on the temperature.
- Exothermic process: \( \Delta S_{surr} = +\frac{\text{quantity of heat (J)}}{\text{temperature (K)}} \)
- Endothermic process: \( \Delta S_{surr} = -\frac{\text{quantity of heat (J)}}{\text{temperature (K)}} \)
- Heat flow (constant \(P\)) = change in enthalpy = \(\Delta H\) $$\Delta S_{surr}=-\frac{\Delta H}{T}$$

Signs of Entropy Changes | |||
---|---|---|---|

\(\Delta S_{sys}\) | \(\Delta S_{surr}\) | \(\Delta S_{univ}\) | Process Spontaneous? |

+ | + | + | Yes |

- | - | - | No (reaction will occur in opposite direction) |

+ | - | ? | Yes, if \(\Delta S_{sys}\) has a larger magnitude than \(\Delta S_{surr}\) |

- | + | ? | Yes, if \(\Delta S_{surr}\) has a larger magnitude than \(\Delta S_{sys}\) |

$$ \Delta S_{univ} = -\frac{\Delta G}{T}\,\left( \text{at constant }T\text{ and }P \right) $$

- A process (at constant \(T\) and \(P\) is spontaneous in the direction in which the free energy decreases.
- Negative \(\Delta G\) means positive \(\Delta S_{univ}\).
- \( \Delta G = \Delta H - T\Delta S \) (at constant \(T\) and \(P\))

Case | Result |
---|---|

\(\Delta S > 0\), \(\Delta H <0\) | Spontaneous at all temperatures |

\(\Delta S > 0\), \(\Delta H > 0\) | Spontaneous at high temperatures (where exothermicity is relatively unimportant) |

\(\Delta S < 0\), \(\Delta H < 0\) | Spontaneous at low temperatures (where exothermicity is dominant) |

\(\Delta S < 0\), \(\Delta H > 0\) | Process not spontaneous at any temperatures (reverse process is spontaneous at all temperatures) |

- The entropy of a perfect crystal at \(0\,\mathrm{K}\) is zero.
- The entropy of a substance increases with temperature.

- Standard Entropy Values (\(S^\circ\))
- Represent the increase in entropy that occurs when a substance is heated from \(0\,\mathrm{K}\) to \(298\,\mathrm{K}\) at 1 atm pressure. $$ \Delta S^\circ_{rxn} = \sum n_pS^\circ_{products} - \sum n_r S^\circ_{reactants} $$
- Standard Free Energy Values (\(\Delta G^\circ\))
- The change in free energy that will occur if the reactants in their standard states are converted to the products in their standard states. $$ \Delta G^\circ = \Delta H^\circ - T\Delta S^\circ $$ $$ \Delta G^\circ_{rxn} = \sum n_p G^\circ_{products} - \sum n_r G^\circ_{reactants} $$

A stable diatomic molecule spontaneously forms from its atoms. Predict the sign of

- \(\Delta H^\circ\)
- \(\Delta S^\circ\)
- \(\Delta G^\circ\)

negative

negative

negative

- Free energy and pressure $$ \begin{align} G &= G^\circ + RT\ln (P) \\ &\mathrm{or} \\ \Delta G &= \Delta G^\circ + RT\ln (Q) \end{align} $$
- Free energy and equilibrium $$ \Delta G=0=\Delta G^\circ + RT\ln (K) $$ $$ \Delta G^\circ = -RT \ln (K) $$

- A system can achieve the lowest possible free energy by going to equilibrium, not by going to completion.

\(\Delta G^\circ\) | \(K\) |
---|---|

\(\Delta G^\circ = 0\) | \(K=1\) |

\(\Delta G^\circ < 0\) | \(K>1\) |

\(\Delta G^\circ > 0\) | \(K<1\) |

- Maximum possible useful work obtainable from a process at constant temperature and pressure is equal to the change in free energy. $$ w_{max} = \Delta G $$
- Achieving the maximum work available from a spontaneous process can occur only via a hypothetical pathway. Any real pathway wastes energy.
- All real processes are irreversible.
- First law: You can’t win, you can only break even.
- Second law: You can’t break even.
- As we use energy, we degrade its usefulness.

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