Exam Study Notes: Spontaneity, Entropy, and Free Energy

Spontaneous Processes

• A spontaneous process occurs "naturally" at a given temperature and pressure without external force.
• Thermodynamics predicts whether a process will occur under specific conditions.
• Spontaneous processes occur naturally.
• Nonspontaneous processes do not occur naturally under specific conditions.
• Spontaneity is determined by comparing the chemical potential energy before and after the reaction.
• A reaction is thermodynamically favorable (spontaneous) if the system has less potential energy after the reaction (release of energy).
• Spontaneity ≠ fast or slow.

Spontaneity and Energy Release

• Spontaneous processes release energy from the system, typically proceeding from higher to lower potential energy (exothermic).
• Some spontaneous processes proceed from lower to higher potential energy (endothermic).

Entropy

• Spontaneity is favored by an increase in entropy ($S$).
• $S = k \ln W$
  • $k$ is the Boltzmann constant ($1.38 × 10^{-23} J/K$).
  • $W$ is the number of microstates possible.
• Microstate: A specific configuration of the locations and energies of particles in a system ("ENERGETIC POSSIBILITIES").
• The number of microstates possible is given by: $W = n^N$
  • $n$ is the number of boxes.
  • $N$ is the number of particles.

Entropy and Microstates

• The most probable distribution has the largest number of microstates.
• The most probable distribution is the one of greatest entropy.
• States of high entropy are favored because they are the most probable.

Entropy Changes

• Entropy is a state function.
• The change in entropy for a process is the difference in entropy between the final and initial states.
  • $\Delta S<em>{sys} = S</em>{final} – S_{initial}$
• Entropy change is favorable when the result is a more dispersed system (more energetic possibilities).
  • $\Delta S_{sys}$ is positive.

Factors That Increase Entropy

1. Phase of the substance: Solid to liquid to gas.
2. Temperature of the substance:
   • Temperature is proportional to the average kinetic energy of the particles.
   • Higher temperature means greater freedom of movement for particles.
3. Size of particles and complexity of arrangement.
4. Increase in the number of moles of product (gas in particular).
5. Variations in the type of particles: Pure substances vs. mixture?

Entropy and Phase Transitions

• The entropy of a substance increases (\Delta S > 0) as it transforms from a relatively ordered solid to a less-ordered liquid, and then to a still less-ordered gas.
• The entropy decreases (\Delta S < 0) as the substance transforms from a gas to a liquid and then to a solid.

The Second Law of Thermodynamics

• The entropy change of the universe is the sum of the entropy changes for the system and surroundings.
• The second law of thermodynamics states that all spontaneous changes cause an increase in the entropy of the universe.
• For a spontaneous process, $\Delta S_{universe}$ must be positive.
  • $\Delta S<em>{universe} = \Delta S</em>{system} + \Delta S_{surroundings}$
• A process with –$\Delta S<em>{system}$ can be spontaneous if \Delta S{universe} > 0
• \Delta S_{universe} < 0 for a nonspontaneous process (spontaneous in the reverse direction).
• $\Delta S_{universe} = 0$ for a process at equilibrium.
  • $\Delta S<em>{universe} = \Delta S</em>{system} + \Delta S_{surroundings}$

Change in Entropy of Surroundings

• The change in entropy of the surroundings ($\Delta S_{surr}$) is directly proportional to the change in enthalpy of the system.
• $\Delta S_{surr}$ is also inversely proportional to temperature.
  • $\Delta S<em>{surr} = -\frac{\Delta H</em>{sys}}{T}$

The Third Law of Thermodynamics

• The third law of thermodynamics: The entropy of a pure perfect crystalline substance at zero Kelvin is zero.
• Zero Kelvin is called absolute zero.
• There is no lower temperature than zero Kelvin.
• At zero Kelvin, all molecular movement completely stops.
• There is only one possible way to arrange the molecules.
  • $W=1$
  • $S = k \ln W$
• Third law of thermodynamics – everything has entropy!

Standard Entropies

• It is possible to determine the absolute entropy of a substance.
• Standard Entropies, $S°$
  • These values are for 1 mole of a substance at a pressure of 1 bar and a temperature of 298 K.
  • Aqueous species at 1 M concentration.
• Standard entropy values can be used to calculate the standard entropy change ($\Delta S°$) for a process.

$\Delta S°$ for Reactions

• The equation for calculating $\Delta S°$ is similar to that for $\Delta H°$:
  • $\Delta S° = \Sigma S<em>{products} - \Sigma S</em>{reactants}$
• When calculating $\Delta S°$ and $\Delta H°$, remember to multiply the standard entropies and standard enthalpies of formation by the coefficients of the balanced equation.
  • $\Delta H = \Sigma \Delta H<em>{f(products)} – \Sigma \Delta H</em>{f(reactants)}$

Gibbs Free Energy Change, $\Delta G$

• Remaking the Second Law of Thermodynamics - with SYSTEM terms – no surroundings
  • $\Delta G = \Delta H – T\Delta S$
  • $\Delta S<em>{universe} = \Delta S</em>{system} + \Delta S_{surroundings}$

Gibbs Free Energy Change, $\Delta G$

• The changes in Gibbs free energy ($\Delta G$) or simply change in free energy allow us to predict spontaneity by focusing on the system only.
  • $\Delta G = \Delta H – T\Delta S$
  • If \Delta G < 0, the reaction is spontaneous in the forward direction.
  • If \Delta G > 0, the reaction is nonspontaneous in the forward direction
  • If $\Delta G = 0$, the system is at equilibrium

Relationship among $\Delta G$, $\Delta H$, and $\Delta S$

• $\Delta G = \Delta H – T\Delta S$
• Spontaneous reactions, those with –$\Delta G$, generally have:
  • \Delta H < 0
    • Exothermic reaction.
    • A negative $\Delta H$ will contribute to a negative $\Delta G$.
  • \Delta S > 0
    • A positive $\Delta S$ will contribute to a negative $\Delta G$.
• Note that a reaction can still be spontaneous (have a –$\Delta G$) when $\Delta H$ is positive or $\Delta S$ is negative, but not both.
• Also note that there is a temperature dependence.

Direction of Spontaneity Change

• To calculate the temperature at which the spontaneity of a reaction changes from …
  • Spontaneous to nonspontaneous
  • Or nonspontaneous to spontaneous … find the temperature at which $\Delta G = 0$
    • $\Delta G = 0 = \Delta H – T\Delta S$
    • $T = \frac{\Delta H}{\Delta S}$
• This is the temperature at which $\Delta G = 0$ and, by definition, the system is at equilibrium.

The Standard Free Energy Change, $\Delta G°$

• Although the Change in Gibbs Free Energy equation is valid under all conditions, we will most often apply it at standard conditions.
• Standard conditions:
• Under standard conditions, $\Delta G° = \Delta H° – T\Delta S°$
• Pay attention to J vs. kJ in calculations!

Standard Free Energy of Formation, $\Delta G°_f$

• The standard free energy of formation ($\Delta G°_f$) for a compound is defined as the free energy change for the formation of one mole of a substance from its elements in their standard state at 1 bar and 25 °C.
• Analogous to the $\Delta H°_f$ discussed in Chapter 9.
• Example:
  • $H<em>2(g) + \frac{1}{2}O</em>2(g) \rightarrow H<em>2O(l) \quad \Delta G°</em>f = –237.2 \text{ kJ/mol}$

$\Delta G°_f$ Values Can Be Used to Calculate $\Delta G°$

• $\Delta G° = \Sigma n\Delta G°<em>{f(products)} – \Sigma n\Delta G°</em>{f(reactants)}$
• This equation only works for calculating $\Delta G°$ of a reaction at the temperature for which the values of $\Delta G°_f$ are tabulated, which is 298 K.
• $\Delta G°_f$ for any element in its most stable form at standard conditions is defined as zero.

Additivity of $\Delta G$; Coupled Reactions

• As with enthalpy, free energy changes for reactions are additive
  • if Reaction 3 = Reaction 1 + Reaction 2 Then, $\Delta G<em>3 = \Delta G</em>1 + \Delta G_2$
• Also keep in mind that if a reaction is reversed, then the sign on $\Delta G$ is also reversed.
• If a reaction is multiplied by a factor of “n,” then $\Delta G$ is also multiplied by a factor of “n.”

What’s “Free” About Free Energy?

• The free energy is the theoretical maximum amount of energy released from a system that is available to do work on the surroundings.
• For many exothermic reactions, some of the heat released as a result of the enthalpy change goes into increasing the entropy of the surroundings, so it is not available to do work.
• If the reaction achieves its theoretical limit, it is a reversible reaction. All real reactions lose heat and are therefore irreversible.