Thermodynamics and Entropy in General Chemistry: Study Notes and Worked Examples

Thermodynamics and Entropy in General Chemistry

Thermodynamic Processes and Work

Thermodynamics studies the energy changes that accompany physical and chemical processes. Key concepts include work, heat, and internal energy, especially for gases undergoing expansion or compression.

• Isothermal Process: A process that occurs at constant temperature. For an ideal gas, the internal energy change ($\Delta U$) is zero.

• Work Done by/on a Gas: For a reversible isothermal compression or expansion, work is calculated as: $w = -nRT \ln\left(\frac{V_f}{V_i}\right)$

• Irreversible Work: When a gas expands or contracts against a constant external pressure: $w = -P_{ext} (V_f - V_i)$

• Enthalpy Change ($\Delta H$): For isothermal processes of ideal gases, $\Delta H = 0$.

• Free Energy Change ($\Delta G$): Indicates spontaneity; $\Delta G = \Delta H - T\Delta S$.

• Entropy Change ($\Delta S$): For isothermal expansion/compression: $\Delta S = nR \ln\left(\frac{V_f}{V_i}\right)$

Example: Compressing 2.0 mol of an ideal gas isothermally from 5.0 L to 1.0 L in two steps (against 100 atm, then 25.0 atm) involves calculating $w$, $q$, $\Delta U$, and $\Delta H$ for each step using the above formulas.

Entropy and the Second Law of Thermodynamics

Entropy ($S$) is a measure of the disorder or randomness of a system. The second law states that the total entropy of the universe increases in a spontaneous process.

• Standard Molar Entropy ($S^\circ$): The entropy content of 1 mol of a substance at standard conditions (usually 1 bar, 298 K).

• Entropy Change for Phase Transitions: $\Delta S = \frac{q_{rev}}{T}$ For melting (fusion) or vaporization, $q_{rev}$ is the enthalpy of transition.

• Entropy Change for Heating: $\Delta S = nC_p \ln\left(\frac{T_f}{T_i}\right)$ where $C_p$ is the molar heat capacity at constant pressure.

Example: Calculating the total entropy change when 2.0 mol of superheated ice at -42°C melts and warms to 0°C, using the sum of entropy changes for heating the ice, melting, and heating the liquid water.

Statistical Entropy and Microstates

Statistical mechanics relates entropy to the number of possible microstates ($\Omega$) of a system.

• Boltzmann's Entropy Formula: $S = k_B \ln \Omega$ where $k_B$ is Boltzmann's constant ($1.38 \times 10^{-23}$ J/K).

• Microstates: The number of ways the particles in a system can be arranged while maintaining the same energy.

Example: For 1.0 mol of a solid with 6 possible orientations per molecule, $\Omega = 6^{N_A}$, and $S = R \ln 6$ per mole.

Gibbs Free Energy and Spontaneity

The Gibbs free energy ($\Delta G$) determines whether a process is spontaneous at constant temperature and pressure.

• Standard Free Energy Change ($\Delta G^\circ$): Calculated from standard enthalpy and entropy changes: $\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$

• Temperature Dependence: The sign of $\Delta G$ can change with temperature, affecting spontaneity.

Example: Calculating $\Delta G$ for the reaction $N_2(g) + 3 H_2(g) \rightarrow 2 NH_3(g)$ at 25°C using tabulated $\Delta H^\circ$ and $S^\circ$ values.

Equilibrium and Free Energy

At equilibrium, the free energy change is zero, and the relationship between $\Delta G^\circ$ and the equilibrium constant $K$ is given by:

• $\Delta G = \Delta G^\circ + RT \ln Q$

• At equilibrium ($Q = K$): $\Delta G = 0$, so $\Delta G^\circ = -RT \ln K$

Example: For the reaction $COCl_2(g) + 2 H_2(g) \rightarrow CH_4(g) + 2 HCl(g)$, $K_c$ is given, and $\Delta G$ is calculated under standard and non-standard conditions using the above relationships.

Thermodynamic Stability and Kinetic Stability

Thermodynamic stability refers to whether a reaction is favored energetically, while kinetic stability refers to the rate at which a reaction occurs.

• Thermodynamic Stability: A reaction is thermodynamically favorable if $\Delta G < 0$.

• Kinetic Stability: Even if a reaction is thermodynamically favorable, it may not occur if the activation energy is very high (i.e., the reaction is kinetically stable).

Example: The formation of water from $H_2$ and $O_2$ is thermodynamically favorable at room temperature, but does not occur spontaneously due to kinetic stability (high activation energy).

Summary Table: Key Thermodynamic Quantities

Quantity	Symbol	Formula	Meaning
Internal Energy Change	$\Delta U$	$q + w$	Total energy change of the system
Enthalpy Change	$\Delta H$	$\Delta U + P\Delta V$	Heat exchanged at constant pressure
Entropy Change	$\Delta S$	$\frac{q_{rev}}{T}$	Change in disorder/randomness
Gibbs Free Energy Change	$\Delta G$	$\Delta H - T\Delta S$	Predicts spontaneity at constant T, P
Boltzmann Entropy	$S$	$k_B \ln \Omega$	Statistical definition of entropy

Additional info: These notes are based on worked solutions to midterm exam questions covering thermodynamics, entropy, and free energy, with calculations for isothermal processes, entropy of phase changes, statistical entropy, and equilibrium thermodynamics. All equations and relationships are standard for a General Chemistry course (Chapters 8, 13, 16).