Spontaneity, Entropy, and Gibbs Energy

Spontaneity

Spontaneity: A spontaneous process is a process that occurs naturally under a given set of conditions without continuous external intervention. It proceeds in a particular direction without the need for external energy input to keep it going.

Non-Spontaneous Process: A non-spontaneous process requires a continuous input of energy from the surroundings to occur. The reverse of a spontaneous process is always non-spontaneous.

Driving Force: While enthalpy change ($\Delta H$) is often associated with spontaneity, it is not the sole determinant.

Is Decrease In Enthalpy A Criterion For Spontaneity?

Observation: Many spontaneous processes are exothermic (release heat, $\Delta H < 0$). For example, the combustion of fuels, the rusting of iron, and the freezing of water below 0°C are all spontaneous and exothermic.

Counterexamples: However, not all spontaneous processes are exothermic. For example:

Melting of ice above 0°C: This process is spontaneous but endothermic ($\Delta H > 0$). Heat is absorbed from the surroundings to convert solid water to liquid water.
Dissolving of some salts: For instance, dissolving ammonium nitrate ($NH_4NO_3$) in water is spontaneous but endothermic, causing a cooling effect.
Expansion of a gas into a vacuum: This is spontaneous and involves no enthalpy change ($\Delta H = 0$).

Conclusion: Therefore, a decrease in enthalpy (exothermicity) is not the sole criterion for spontaneity. While many spontaneous reactions are exothermic, some are endothermic. There must be another factor involved.

Entropy And Spontaneity

Entropy ($S$): Entropy is a thermodynamic state function that measures the degree of randomness, disorder, or dispersal of energy within a system. A system with higher entropy is more disordered or has its energy spread out over more possible microscopic states.

Second Law of Thermodynamics: The second law of thermodynamics states that for any spontaneous process, the entropy of the universe (system + surroundings) increases.

$$\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings} > 0 \quad \text{(for a spontaneous process)}$$

Entropy Change of the System ($\Delta S_{sys}$):

Phase Transitions: $S_{gas} > S_{liquid} > S_{solid}$.
Dissolution: The dissolution of a solid or liquid into a solution generally increases entropy if the solute particles are dispersed.
Increase in Number of Particles: Reactions that increase the number of moles of gas generally increase entropy.
Temperature Increase: Entropy increases with temperature.

Entropy Change of the Surroundings ($\Delta S_{surr}$):

The entropy change of the surroundings is primarily related to the heat exchanged with the system at a given temperature. At constant pressure:

$$\Delta S_{surr} = \frac{q_{surr}}{T}$$

Since the heat exchanged by the surroundings is equal in magnitude but opposite in sign to the heat exchanged by the system ($q_{surr} = -q_{sys}$), and for processes at constant pressure $q_{sys} = \Delta H_{sys}$:

$$\Delta S_{surr} = -\frac{\Delta H_{sys}}{T}$$

Entropy and Spontaneity Criterion: The increase in the total entropy of the universe ($\Delta S_{universe}$) is the true thermodynamic criterion for spontaneity. However, calculating $\Delta S_{surr}$ can be inconvenient.

Gibbs Energy And Spontaneity

Gibbs Free Energy ($G$): Willard Gibbs introduced a thermodynamic potential called Gibbs free energy ($G$), which allows us to determine spontaneity by considering only the system's properties (at constant temperature and pressure). It is defined as:

$$G = H - TS$$

Where:

$G$ is Gibbs free energy.
$H$ is enthalpy.
$T$ is the absolute temperature.
$S$ is entropy.

Change in Gibbs Free Energy ($\Delta G$): For a process at constant temperature and pressure, the change in Gibbs free energy is:

$$\Delta G = \Delta H - T\Delta S$$

Relationship to $\Delta S_{universe}$ (Derivation):

We know $\Delta S_{universe} = \Delta S_{sys} + \Delta S_{surr}$.

At constant T and P, $\Delta S_{surr} = -\frac{\Delta H_{sys}}{T}$.

Substitute this into the entropy criterion:

$$\Delta S_{universe} = \Delta S_{sys} - \frac{\Delta H_{sys}}{T} > 0 \quad \text{(for spontaneity)}$$

Multiply by $T$ (since $T$ is always positive):

$$T \Delta S_{sys} - \Delta H_{sys} > 0$$

Multiply by -1 and reverse the inequality sign:

$$\Delta H_{sys} - T \Delta S_{sys} < 0$$

Comparing this to the definition of $\Delta G = \Delta H - T\Delta S$, we get:

$$\Delta G_{sys} < 0$$

Gibbs Free Energy as a Criterion for Spontaneity:

$\Delta G < 0$: The process is spontaneous (exergonic).
$\Delta G > 0$: The process is non-spontaneous (endergonic); the reverse process is spontaneous.
$\Delta G = 0$: The system is at equilibrium.

Key Factors Affecting Spontaneity: $\Delta G$ shows that spontaneity is determined by both the enthalpy change ($\Delta H$) and the entropy change ($\Delta S$), weighted by temperature ($T$).

Exothermic and Entropy Increasing ($\Delta H < 0, \Delta S > 0$): $\Delta G$ will always be negative. The reaction is spontaneous at all temperatures.
Endothermic and Entropy Decreasing ($\Delta H > 0, \Delta S < 0$): $\Delta G$ will always be positive. The reaction is non-spontaneous at all temperatures.
Exothermic and Entropy Decreasing ($\Delta H < 0, \Delta S < 0$): $\Delta G$ is negative at low temperatures (enthalpy dominates) and positive at high temperatures (entropy term $-T\Delta S$ becomes positive and larger). Spontaneous at low T.
Endothermic and Entropy Increasing ($\Delta H > 0, \Delta S > 0$): $\Delta G$ is positive at low temperatures (enthalpy dominates) and negative at high temperatures (entropy term $-T\Delta S$ becomes negative and larger). Spontaneous at high T.

Entropy And Second Law Of Thermodynamics

Second Law Restated: In its most general form, the second law of thermodynamics states that the total entropy of an isolated system can only increase over time, or remain constant in ideal cases where the system is in a steady state or undergoing a reversible process. For any spontaneous process in an isolated system, the entropy must increase.

Implications:

Direction of Natural Processes: The second law dictates the natural direction of processes. Systems tend to move towards states of higher entropy (greater disorder or energy dispersal).
Irreversibility: All real-world processes are irreversible to some extent, meaning they generate entropy.
Heat Flow: Heat naturally flows from hotter objects to colder objects because this process increases the total entropy of the universe.
Limitations of Engines: The second law places fundamental limits on the efficiency of heat engines. It's impossible to convert heat completely into work in a cyclic process.

Absolute Entropy And Third Law Of Thermodynamics

Absolute Entropy ($S$): Entropy is a state function, but unlike enthalpy or internal energy, it is possible to define an absolute scale for entropy. The absolute entropy of a substance at a specific temperature and pressure can be calculated from its heat capacity data and phase transition enthalpies.

Third Law of Thermodynamics: The third law of thermodynamics provides a reference point for entropy. It states that the entropy of a perfect crystal at absolute zero (0 Kelvin or -273.15°C) is exactly zero.

$$S(T=0 \text{ K, perfect crystal}) = 0$$

Basis of Calculation: This law allows us to calculate the absolute entropy of a substance at any temperature $T$ by integrating the heat capacity data:

$$S_T = S_{0 K} + \int_{0}^{T} \frac{dq_{rev}}{T'} = \int_{0}^{T} \frac{C_p}{T'} dT'$$

This calculation includes contributions from heating the solid from 0 K to its melting point, the enthalpy of fusion at the melting point, heating the liquid to the boiling point, the enthalpy of vaporization at the boiling point, and heating the gas to the final temperature $T$.

Significance:

Reference Point: Provides a zero point for entropy measurements.
Calculation of $\Delta S^\circ$ for Reactions: Allows for the calculation of standard entropy changes for chemical reactions ($\Delta S^\circ_{rxn}$) using absolute entropy values of reactants and products:
Predicting Spontaneity: Combined with enthalpy data, absolute entropy values allow for the calculation of $\Delta G$ and thus the prediction of spontaneity under various conditions.