⚛Molecular Physics Unit 12 – Thermodynamics of Molecular Systems

Key Concepts and Definitions

• Thermodynamics studies the relationships between heat, work, temperature, and energy in a system
• A thermodynamic system is a specific portion of the universe that is being studied, which can be open, closed, or isolated
• The surroundings include everything outside the system that can interact with it through the exchange of energy or matter
• Thermal equilibrium occurs when two systems in contact have the same temperature and no net heat flow between them
• Internal energy ($U$) represents the total kinetic and potential energy of all particles within a system
  • Includes translational, rotational, vibrational, and electronic energy contributions
• Enthalpy ($H$) is a state function that equals the internal energy plus the product of pressure and volume ($H = U + PV$)
• Entropy ($S$) measures the degree of disorder or randomness in a system and always increases in spontaneous processes
• Gibbs free energy ($G$) predicts the spontaneity of a process at constant temperature and pressure ($G = H - TS$)

Fundamental Laws of Thermodynamics

• The zeroth law establishes the concept of thermal equilibrium and provides the basis for temperature measurement
• The first law states that energy cannot be created or destroyed, only converted from one form to another
  • Mathematically expressed as $\Delta U = Q - W$, where $Q$ is heat added and $W$ is work done by the system
• The second law introduces entropy and states that the total entropy of an isolated system always increases over time
  • Implies that heat flows spontaneously from hot to cold objects until thermal equilibrium is reached
• The third law states that the entropy of a perfect crystal at absolute zero is zero, providing a reference point for entropy calculations
• These laws govern the behavior of thermodynamic systems and set limits on the efficiency of energy conversion processes

Statistical Mechanics Basics

• Statistical mechanics connects the microscopic properties of individual atoms and molecules to the macroscopic thermodynamic properties of materials
• The Boltzmann distribution describes the probability of a system being in a particular microstate with energy $E_i$ at temperature $T$
  • Probability is proportional to $e^{-E_i/kT}$, where $k$ is the Boltzmann constant
• The partition function ($Z$) is a sum over all possible microstates and relates microscopic properties to macroscopic thermodynamic quantities
  • $Z = \sum_i e^{-E_i/kT}$ for a canonical ensemble (fixed $N$, $V$, and $T$)
• Thermodynamic properties can be derived from the partition function, such as internal energy ($U = -\partial \ln Z / \partial \beta$) and entropy ($S = k \ln Z + kT \partial \ln Z / \partial T$)
• The equipartition theorem states that each quadratic degree of freedom (translational, rotational, or vibrational) contributes $\frac{1}{2}kT$ to the average energy per molecule

Energy and Heat in Molecular Systems

• Heat is the transfer of energy between two systems due to a temperature difference, while work is the transfer of energy through organized motion
• The heat capacity ($C$) is the amount of heat required to raise the temperature of a substance by one degree
  • $C = \frac{dQ}{dT}$, and can be measured at constant volume ($C_V$) or constant pressure ($C_P$)
• The molar heat capacity is the heat capacity per mole of a substance and depends on the degrees of freedom available to the molecules
• The equipartition theorem predicts that the molar heat capacity of an ideal gas is $\frac{1}{2}R$ per quadratic degree of freedom, where $R$ is the gas constant
• The Dulong-Petit law states that the molar heat capacity of a solid at high temperatures is approximately $3R$, as each atom has three vibrational degrees of freedom
• The heat of vaporization and heat of fusion are the amounts of energy required to convert a substance from liquid to gas or solid to liquid, respectively

Entropy and the Second Law

• Entropy is a measure of the disorder or randomness in a system and always increases in spontaneous processes
• The second law of thermodynamics states that the total entropy of an isolated system always increases over time
  • Mathematically, $\Delta S_\text{total} = \Delta S_\text{system} + \Delta S_\text{surroundings} \geq 0$
• The Clausius inequality relates entropy change to heat transfer and temperature: $\Delta S \geq \int \frac{dQ}{T}$
• The Boltzmann equation defines entropy in terms of the number of microstates ($\Omega$) accessible to a system: $S = k \ln \Omega$
• The third law of thermodynamics provides an absolute reference point for entropy, stating that the entropy of a perfect crystal at absolute zero is zero
• Entropy drives many processes in molecular systems, such as the mixing of gases, the unfolding of proteins, and the formation of micelles in solution

Thermodynamic Potentials and Free Energy

• Thermodynamic potentials are state functions that describe the energy content of a system under different conditions
• The internal energy ($U$) is the total kinetic and potential energy of all particles within a system
• Enthalpy ($H$) is the sum of internal energy and the product of pressure and volume: $H = U + PV$
  • Represents the heat content of a system at constant pressure
• Helmholtz free energy ($A$) is the maximum amount of work a system can perform at constant temperature and volume: $A = U - TS$
• Gibbs free energy ($G$) is the maximum amount of non-expansion work a system can perform at constant temperature and pressure: $G = H - TS$
  • The change in Gibbs free energy determines the spontaneity of a process: $\Delta G < 0$ for spontaneous processes
• The chemical potential ($\mu$) is the change in Gibbs free energy per mole of substance added at constant temperature, pressure, and composition

Phase Transitions and Equilibrium

• Phase transitions occur when a substance changes from one state of matter (solid, liquid, or gas) to another
• The Gibbs phase rule relates the number of components ($C$), phases ($P$), and degrees of freedom ($F$) in a system at equilibrium: $F = C - P + 2$
• The Clapeyron equation describes the slope of a phase boundary in a pressure-temperature diagram: $\frac{dP}{dT} = \frac{\Delta H}{T \Delta V}$
• The Clausius-Clapeyron equation is a special case for the vapor-liquid equilibrium: $\ln \frac{P_2}{P_1} = -\frac{\Delta H_\text{vap}}{R} (\frac{1}{T_2} - \frac{1}{T_1})$
• The chemical potential of each component is equal in all phases at equilibrium: $\mu_i^\alpha = \mu_i^\beta = \mu_i^\gamma$
• The Gibbs-Duhem equation relates changes in chemical potential to changes in temperature and pressure at equilibrium: $SdT - VdP + \sum_i N_i d\mu_i = 0$

Applications in Molecular Physics

• Thermodynamics plays a crucial role in understanding the behavior of gases, liquids, and solids at the molecular level
• The ideal gas law ($PV = nRT$) describes the relationship between pressure, volume, temperature, and the number of moles for an ideal gas
• The van der Waals equation modifies the ideal gas law to account for intermolecular forces and the finite volume of molecules: $(P + \frac{an^2}{V^2})(V - nb) = nRT$
• The Langmuir adsorption isotherm describes the adsorption of molecules onto a surface as a function of pressure: $\frac{\theta}{1-\theta} = KP$
• The Gibbs adsorption isotherm relates the change in surface tension to the adsorption of solutes at an interface: $d\gamma = -\sum_i \Gamma_i d\mu_i$
• The Arrhenius equation describes the temperature dependence of reaction rates: $k = Ae^{-E_a/RT}$, where $E_a$ is the activation energy
• The Nernst equation relates the equilibrium potential of an electrochemical cell to the standard potential and the concentrations of reactants and products: $E = E^0 - \frac{RT}{nF} \ln Q$