Unit 7

20.1 Entropy

• Section Objectives
    - Define entropy
    - Recite the second law of thermodynamics
    - Calculate entropy
    - Recite the third law of thermodynamics

Introduction to Entropy

• Entropy is a concept that quantifies the level of disorder or randomness in a system, and it is fundamentally related to the idea of order and disorder in physical and chemical processes.   
• Example:
    - A neat room represents low entropy (order).
    - A messy room represents high entropy (disorder).

Spontaneous Processes

• Definition: A reaction is spontaneous if it occurs under specified conditions, favoring the formation of products without needing any further assistance.   - Key factors influencing spontaneity include temperature, pressure, and concentration.

• Examples:
    - Rusting of iron (slow spontaneous process).
    - Combustion of gasoline (fast spontaneous process).
    - Water freezing at below 0°C or melting above 0°C shows spontaneity as well.

• Thermodynamics & Reaction Rate:
    - Thermodynamics indicates the direction of spontaneity but does not dictate the speed of a reaction, which is examined through chemical kinetics (activation energy, temperature, catalysts).

Understanding Entropy

• Entropy (S): A measure of the number of arrangements of particles in a system.
    - Higher entropy indicates a greater number of possible states (or configurations) of a system.

• Probabilistic Nature of Entropy:
    - States with higher probabilities have higher entropy values. Example: Organizing a room (more arrangements lead to higher entropy).

• Phase Transitions as Examples of Entropy Change:
    - Vaporization (Liquid to Gas): Increases entropy - gas particles can move freely.
    - Freezing (Liquid to Solid): Decreases entropy - solid structure is ordered, limiting arrangements.

The Second Law of Thermodynamics

• Definition: Entropy in the universe always increases for spontaneous processes.
    - There are processes where entropy decreases locally, but overall entropy increases in the universe.

• Equation:
  $\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr}$
    - Where:
      - $\Delta S_{univ}$ = change in universal entropy
      - $\Delta S_{sys}$ = change in system entropy
      - $\Delta S_{surr}$ = change in surroundings entropy

• Spontaneity Prediction:
    - If \Delta S_{univ} > 0: Spontaneous in the given direction.
    - If \Delta S_{univ} < 0: Spontaneous in the reverse direction.
    - If $\Delta S_{univ} = 0$: System is in equilibrium.

Entropy of a System

• Example of Water Molecule:

1. When water vaporizes:
      - From $H_2O(l)$ to $H_2O(g)$, $\Delta S_{sys}$ is positive because of an increase in arrangements of gas molecules.
2. Predicting Signs of Entropy Change:
      1. More gaseous moles in products = higher entropy.
      2. Solid forming from liquid = decrease in entropy.

Entropy of the Surroundings

• Exothermic vs Endothermic:
    - Exothermic reactions result in positive $\Delta S_{surr}$ as heat is released.
    - Endothermic reactions lead to negative $\Delta S_{surr}$ as heat is absorbed.
• Equation:
  $\Delta S_{surr} = -\frac{\Delta H}{T}$ where $\Delta H$ = change in enthalpy, and $T$ is temperature in Kelvin.

The Third Law of Thermodynamics

• Definition: At absolute zero (0 K), the entropy of a perfect crystal is zero.
    - As temperature increases, the entropy of a substance increases due to the vibrational energies of its particles.

• Standard Entropy Values:
    - These values are measured at 1 atm and 25°C. Example representative values:
      - $H_2O(l): 69.95 \,J/K\cdot mol$
      - $H_2O(g): 188.84 \,J/K\cdot mol$

Calculation of Entropy Changes

1. Calculate $\Delta S_{rxn}$ using standard entropy values:
      - $\Delta S_{rxn} = \Sigma S_{products} - \Sigma S_{reactants}$
      - Multiply each standard entropy value by the corresponding stoichiometric coefficients in the balanced chemical equation.

• Example Calculation:
    - $N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)$
      - Given entropy values:
        - $N_2: 191.6 \ J/K \cdot mol$
        - $H_2: 130.7 \ J/K \cdot mol$
        - $NH_3: 192.5 \ J/K \cdot mol$
        - Calculate $\Delta S_{rxn}$.

1. Find Current Conditions: Check if system entropy increases or decreases based on stoichiometry.

20.2 Spontaneous Reactions and Free Energy

• Gibbs Free Energy:

• Defined as:
  $\Delta G = \Delta H - T \Delta S$
    - \Delta G < 0 indicates spontaneous process.

• The relationship between:
    \Delta S_{univ} > 0 = \Delta S_{sys} + \Delta S_{surr}
    - Substitute $\Delta S_{surr} = -\frac{\Delta H}{T}$ into equation.

• Rearranged form:
    \Delta G = \Delta H - T \Delta S < 0
    - Standard Free Energy Change:
    - $\Delta G°_{rxn} = \Sigma n \Delta G°_f(products) - \Sigma n \Delta G°_f(reactants)$

Example Gibbs Free Energy Calculation

• Combustion of Methane:
    - $CH_4(g) + 2O_2(g) \rightarrow CO_2(g) + 2H_2O(l)$
    - Use standard free energy of formation values for products and reactants.

20.3 Free Energy and Equilibrium

• Definitions:

1. Equilibrium Constant:
     - $K_{eq}$ describes the ratio of [products]/[reactants] at equilibrium.
2. Reaction Quotient (Q):
     - Used for non-equilibrium reactions.

• Relationship:
    - $Q = K_{eq}$ shows established equilibrium while $Q eq K_{eq}$ means reaction shifts direction.

Gibbs Free Energy Relation to Equilibrium Constant

• Calculation:
    - At equilibrium, $\Delta G = 0$ and $Q = K_{eq}$
• Equation:
    - $\Delta G° = -RT ext{ln}K$
• Titration Curve: Graph showing pH change in relation to titrant volume.

Summary

• The relationship between Gibbs free energy, entropy, and enthalpy is crucial for determining whether a reaction will occur spontaneously or not. Understanding and calculating changes in these thermodynamic properties provides insight into the direction and feasibility of chemical reactions.