Thermodynamics and Entropy in General Chemistry: Study Notes from CHEM1320 Midterm Exam 3

Thermodynamics and Entropy in General Chemistry

Thermodynamic Processes and Work

Thermodynamics studies the energy changes that accompany physical and chemical processes. In particular, the work done by or on a gas during expansion or compression is a key concept.

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

• Work Done by a Gas: For a reversible isothermal compression or expansion, the work is given by: $w = -nRT \ln\left(\frac{V_f}{V_i}\right)$ where $n$ is the number of moles, $R$ is the gas constant, $T$ is temperature, $V_f$ and $V_i$ are final and initial volumes.

• Irreversible Work: For an irreversible process against a constant external pressure ($P_{ext}$): $w = -P_{ext} (V_f - V_i)$

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

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

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

• Example: Compressing 2.0 mol of an ideal gas isothermally from 5.0 L to 1.0 L in two steps (against 10.0 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 of thermodynamics states that the total entropy of the universe increases in a spontaneous process.

• Entropy Change for Phase Transitions: For melting or vaporization at constant temperature: $\Delta S = \frac{q_{rev}}{T}$ where $q_{rev}$ is the heat absorbed or released reversibly.

• Entropy Change for Heating: For heating a substance at constant pressure: $\Delta S = nC_p \ln\left(\frac{T_f}{T_i}\right)$ 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 each step (warming, melting, further warming).

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, $\Omega$ is the number of microstates.

• Microstates for Solids: For a solid with $N$ molecules, each with $W$ possible orientations: $\Omega = W^N$

• Example: For 1.0 mol of a solid (Avogadro's number of molecules), each with 6 orientations: $S = R \ln(6)$ where $R$ is the gas constant.

Gibbs Free Energy and Spontaneity

The Gibbs free energy ($G$) determines the spontaneity of a process at constant temperature and pressure.

• Gibbs Free Energy Change: $\Delta G = \Delta H - T\Delta S$

• Standard Free Energy Change ($\Delta G^\circ$): Calculated from standard enthalpy and entropy values for reactants and products.

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

• Example: Calculating $\Delta G$ for the reaction $N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)$ at 25°C using standard enthalpy and entropy 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 fundamental.

• Relationship between $\Delta G^\circ$ and $K$: $\Delta G^\circ = -RT \ln K$

• Non-Standard Conditions: For reactions not at standard conditions: $\Delta G = \Delta G^\circ + RT \ln Q$ where $Q$ is the reaction quotient.

• Example: For the reaction $COCl_2(g) + 2H_2(g) \rightarrow CH_4(g) + 2HCl(g)$, calculating $\Delta G$ under standard and non-standard conditions using $K$ and $Q$.

Thermodynamic Stability and Kinetic Stability

Thermodynamic stability refers to the favorability of a process based on $\Delta G$, while kinetic stability refers to the rate at which a process occurs.

• Thermodynamic Stability: A substance is thermodynamically stable if $\Delta G > 0$ for its decomposition or reaction.

• Kinetic Stability: Even if a reaction is thermodynamically favorable ($\Delta G < 0$), it may not occur if the activation energy is high (slow kinetics).

• Example: Water formation 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	Units
Work (reversible, isothermal)	w	$-nRT \ln\left(\frac{V_f}{V_i}\right)$	J
Entropy Change (isothermal)	$\Delta S$	$nR \ln\left(\frac{V_f}{V_i}\right)$	J/K
Gibbs Free Energy Change	$\Delta G$	$\Delta H - T\Delta S$	J
Standard Free Energy and Equilibrium	$\Delta G^\circ$	$-RT \ln K$	J
Statistical Entropy	S	$k_B \ln \Omega$	J/K

Additional info: These notes synthesize the main thermodynamic concepts and calculations as presented in the CHEM1320 General Chemistry Midterm Exam 3, focusing on entropy, free energy, and equilibrium. The examples and equations are directly relevant to Ch. 16 (Thermodynamics: Entropy, Free Energy & Equilibrium) and related chapters in a standard General Chemistry curriculum.