Thermodynamics and Chemical Equilibrium: Study Notes for General Chemistry

Thermodynamics and Chemical Equilibrium

Isothermal Compression of an Ideal Gas

Isothermal processes occur at constant temperature. For an ideal gas, compressing or expanding isothermally involves work, heat, and changes in thermodynamic state functions.

• Work Done by/on the Gas ($w$): For isothermal, reversible compression or expansion:

$$ w = -nRT \ln\left(\frac{V_f}{V_i}\right) $$

• Change in Internal Energy ($\Delta U$): For an ideal gas, $\Delta U$ depends only on temperature. For isothermal processes, $\Delta U = 0$.

• Change in Enthalpy ($\Delta H$): Also zero for isothermal processes of an ideal gas ($\Delta H = 0$).

• Heat Exchanged ($q$): By the First Law, $q = -w$ for isothermal processes.

• Example: Compressing 2.0 mol of an ideal gas from 5.0 L to 1.0 L at constant temperature, in two steps against different external pressures, requires calculating $w$, $q$, $\Delta U$, and $\Delta H$ for each step.

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 for a spontaneous process.

• Entropy Change for Isothermal Expansion/Compression:

$$ \Delta S = nR \ln\left(\frac{V_f}{V_i}\right) $$

• Entropy Change for the Surroundings ($\Delta S_{surr}$): For a process at constant temperature:

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

• Total Entropy Change ($\Delta S_{univ}$): $\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr}$

• Example: Calculating $\Delta S_{sys}$, $\Delta S_{surr}$, and $\Delta S_{univ}$ for the isothermal compression of an ideal gas.

Phase Changes and Entropy

Phase transitions (e.g., melting, freezing) involve changes in entropy and enthalpy. The entropy change for melting (fusion) at constant temperature is:

$$ \Delta S_{fus} = \frac{\Delta H_{fus}}{T} $$

• Heating Across a Phase Change: The total entropy change is the sum of entropy changes for heating, phase transition, and further heating.

• Example: Calculating $\Delta S$, $\Delta H$, and $\Delta G$ for the conversion of superheated ice to liquid water, including heating, melting, and further heating.

Statistical Entropy

Statistical mechanics relates entropy to the number of microstates ($\Omega$):

$$ S = k_B \ln \Omega $$

• Microstates: The number of ways the molecules can be arranged. For a solid with random orientations, $\Omega = X^N$ where $X$ is the number of orientations per molecule and $N$ is the number of molecules.

• Example: Calculating the entropy of 1.0 mol of a solid at 0 K with 6 possible orientations per molecule.

Gibbs Free Energy and Spontaneity

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

• Definition:

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

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

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

• Example: Calculating $\Delta G$ for the formation of ammonia from nitrogen and hydrogen at 25°C using standard thermodynamic data.

Chemical Equilibrium and Free Energy

At equilibrium, the free energy change is zero. The relationship between $\Delta G$, $\Delta G^\circ$, and the reaction quotient $Q$ is:

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

• Equilibrium Constant ($K$): At equilibrium, $\Delta G = 0$ and $Q = K$:

$$ \Delta G^\circ = -RT \ln K $$

• Example: Calculating $\Delta G$ for a reaction under standard conditions, at equilibrium, and with non-standard concentrations/pressures.

Thermodynamic Stability of Water

The formation of water from hydrogen and oxygen is highly exergonic, but water is kinetically stable under normal conditions due to a high activation energy barrier.

• Thermodynamic vs. Kinetic Stability: A reaction can be thermodynamically favorable ($\Delta G < 0$) but not occur rapidly if the activation energy is large.

• Example: Calculating the temperature at which water formation becomes thermodynamically unstable, and explaining why water does not spontaneously decompose under normal conditions.

Summary Table: Key Thermodynamic Quantities

Quantity	Symbol	Equation	Notes
Work (isothermal, reversible)	$w$	$-nRT \ln\left(\frac{V_f}{V_i}\right)$	For ideal gases
Change in Internal Energy	$\Delta U$	$nC_v\Delta T$	Zero for isothermal ideal gas
Change in Enthalpy	$\Delta H$	$nC_p\Delta T$	Zero for isothermal ideal gas
Entropy Change (isothermal)	$\Delta S$	$nR \ln\left(\frac{V_f}{V_i}\right)$	For ideal gases
Gibbs Free Energy	$\Delta G$	$\Delta H - T\Delta S$	Determines spontaneity
Standard Free Energy and $K$	$\Delta G^\circ$	$-RT \ln K$	At equilibrium
Statistical Entropy	$S$	$k_B \ln \Omega$	$\Omega$ = number of microstates

Additional info:

• These notes cover core concepts from General Chemistry chapters on thermodynamics, entropy, Gibbs free energy, and chemical equilibrium, as well as applications to phase changes and statistical mechanics.

• All equations are standard in undergraduate chemistry and are essential for solving problems involving energy, spontaneity, and equilibrium.