Free Energy, Spontaneity, and Entropy: Foundations of Thermodynamics

Spontaneous Processes and Chemical Potential

Understanding Spontaneity in Chemistry

Spontaneous processes are those that occur without ongoing external intervention. In chemistry, predicting whether a reaction or process will occur spontaneously is fundamental to understanding chemical change. The concept of chemical potential is analogous to mechanical potential energy, which determines the direction of spontaneous change in physical systems.

• Spontaneous process: Occurs naturally without continuous external energy input (e.g., a book falling off a table).

• Nonspontaneous process: Requires continuous energy input to proceed (e.g., lifting a book onto a table).

• For chemical systems, we seek a chemical potential that predicts the direction of spontaneous change.

Spontaneity vs. Speed: Thermodynamics and Kinetics

Distinguishing Thermodynamics from Kinetics

It is crucial to distinguish between whether a reaction can occur (thermodynamics) and how fast it occurs (kinetics). Spontaneity does not imply rapid change; some spontaneous processes are extremely slow.

• Thermodynamics: Determines if a reaction is possible and to what extent it will proceed (spontaneity and equilibrium).

• Kinetics: Studies the rate and mechanism of a reaction (how fast and by what pathway).

• A catalyst can speed up a spontaneous reaction but cannot make a nonspontaneous reaction spontaneous.

• Example: The conversion of diamond to graphite is spontaneous but occurs at an imperceptibly slow rate at room temperature.

Entropy and the Second Law of Thermodynamics

Key Definitions

• Entropy (S): A thermodynamic state function that measures the number of energetically equivalent ways to arrange a system. Units: J/(mol·K).

• Microstate (W): A specific microscopic arrangement of the components of a system at one instant.

• Macrostate: The macroscopic description of a system (e.g., defined by pressure, volume, temperature); one macrostate corresponds to many possible microstates.

• Boltzmann Equation: $S = k \ln W$, where $k = 1.38 \times 10^{-23}$ J/K (Boltzmann constant) and $W$ is the number of microstates.

Entropy as Energy Dispersal

Entropy quantifies the dispersal of energy and matter. Processes that increase disorder or randomness (i.e., increase the number of microstates) are generally spontaneous.

• Ice melting: $\mathrm{H_2O(s) \rightarrow H_2O(l)}$ (ordered to disordered; $\Delta S > 0$)

• Evaporation: $\mathrm{H_2O(l) \rightarrow H_2O(g)}$ (more disordered; $\Delta S \gg 0$)

• NaCl dissolution: $\mathrm{NaCl(s) \rightarrow Na^+(aq) + Cl^-(aq)}$ (ordered crystal to random ions; $\Delta S > 0$)

Boltzmann's Equation and Microstates

The number of microstates ($W$) available to a system determines its entropy. More microstates mean higher entropy.

• System A (one energy level): $W = 1$

• System B (two energy levels): $W = 2$

• $S_A = k \ln 1 = 0$; $S_B = k \ln 2 > 0$

Statistical Argument for Gas Expansion

When a gas expands into a vacuum, the number of possible microstates increases dramatically, making the expansion spontaneous. The most probable macrostate is the one with the greatest number of microstates.

• For $N = 4$ atoms, the equally distributed state has 6 microstates versus 1 for all-in-one-flask.

• For $N = 20$ atoms, the equally distributed state has 184,756 microstates versus 1 for all-in-one-flask.

The Second Law of Thermodynamics

The second law states that for any spontaneous process, the total entropy of the universe increases:

• $\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings} > 0$ (spontaneous process)

• $\Delta S_{universe} = 0$ (equilibrium, reversible process)

• $\Delta S_{universe} < 0$ (nonspontaneous process; reverse is spontaneous)

This law explains the unidirectional flow of time and the tendency for energy to disperse, leading to the concept of 'heat death' of the universe.

Entropy Changes with State Changes

Entropy and Phase Transitions

Entropy increases as a substance changes from solid to liquid to gas, reflecting greater energy dispersal and more available microstates.

• $S_{solid} < S_{liquid} \ll S_{gas}$

• Vaporization adds translational and rotational energy modes, dramatically increasing microstates.

Practice: Predicting Entropy Changes

For each process, predict whether $\Delta S_{system}$ is positive, negative, or approximately zero:

• H2O(l) → H2O(g) at 100°C: $\Delta S > 0$ (liquid to gas, more disorder)

• AgCl(s) → Ag+(aq) + Cl-(aq): $\Delta S > 0$ (solid to ions in solution, more disorder)

• 2 H2(g) + O2(g) → 2 H2O(l): $\Delta S < 0$ (gases to liquid, less disorder)

• CO2(g) → CO2(s): $\Delta S < 0$ (gas to solid, less disorder)

Calculating Entropy Change for Phase Transitions

For a reversible phase change at constant temperature:

$\Delta S = \frac{q_{rev}}{T}$

• $q_{rev}$ = heat exchanged at the phase transition (J)

• $T$ = temperature in Kelvin

• For melting: $q_{rev} = \Delta H_{fus}$ (positive, heat absorbed)

• For condensation: $q_{rev} = \Delta H_{cond}$ (negative, heat released)

• Temperature must be in Kelvin: $T(K) = T(°C) + 273.15$

Practice: Calculating $\Delta S$ for Phase Changes

• 1.00 mol of liquid water freezes at 0°C ($\Delta H_{fus} = 6.02$ kJ/mol): $T = 273.15$ K, $q_{rev} = -6020$ J/mol $\Delta S = \frac{-6020}{273.15} = -22.0$ J/(mol·K)

• 1.00 mol of ethanol vaporizes at 78.4°C ($\Delta H_{vap} = 38.6$ kJ/mol): $T = 351.55$ K, $q_{rev} = 38,600$ J/mol $\Delta S = \frac{38,600}{351.55} = +109.8$ J/(mol·K)

Reversible Processes

A reversible process is one that can be reversed without leaving any net change in the system or its surroundings. An example is the isothermal expansion or compression of a gas under infinitesimal changes in external force.

Key Takeaways

1. Spontaneous processes occur without ongoing external energy input; spontaneity does not imply speed.

2. Entropy ($S = k \ln W$) measures the number of energetically equivalent microstates; more microstates mean higher entropy.

3. Second Law: $\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings} > 0$ for all spontaneous processes.

4. Entropy increases: solid → liquid → gas; dissolution of solids; expansion of gases.

5. For reversible phase transitions: $\Delta S = q_{rev}/T$ (T in Kelvin).

6. The most probable macrostate is the one with the greatest number of microstates—this drives spontaneous change.