Molecular Physics

Molecular Physics Unit 12 – Thermodynamics of Molecular Systems

Thermodynamics in molecular physics explores how heat, work, and energy interact in systems of particles. It covers key concepts like internal energy, enthalpy, entropy, and Gibbs free energy, which help us understand the behavior of gases, liquids, and solids at the molecular level. The laws of thermodynamics provide a framework for analyzing energy transfer and transformation in molecular systems. Statistical mechanics bridges the gap between microscopic properties and macroscopic behavior, while concepts like phase transitions and chemical equilibrium have practical applications in various fields of science and engineering.

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 (UU) represents the total kinetic and potential energy of all particles within a system
    • Includes translational, rotational, vibrational, and electronic energy contributions
  • Enthalpy (HH) is a state function that equals the internal energy plus the product of pressure and volume (H=U+PVH = U + PV)
  • Entropy (SS) measures the degree of disorder or randomness in a system and always increases in spontaneous processes
  • Gibbs free energy (GG) predicts the spontaneity of a process at constant temperature and pressure (G=HTSG = 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 ΔU=QW\Delta U = Q - W, where QQ is heat added and WW 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 EiE_i at temperature TT
    • Probability is proportional to eEi/kTe^{-E_i/kT}, where kk is the Boltzmann constant
  • The partition function (ZZ) is a sum over all possible microstates and relates microscopic properties to macroscopic thermodynamic quantities
    • Z=ieEi/kTZ = \sum_i e^{-E_i/kT} for a canonical ensemble (fixed NN, VV, and TT)
  • Thermodynamic properties can be derived from the partition function, such as internal energy (U=lnZ/βU = -\partial \ln Z / \partial \beta) and entropy (S=klnZ+kTlnZ/TS = k \ln Z + kT \partial \ln Z / \partial T)
  • The equipartition theorem states that each quadratic degree of freedom (translational, rotational, or vibrational) contributes 12kT\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 (CC) is the amount of heat required to raise the temperature of a substance by one degree
    • C=dQdTC = \frac{dQ}{dT}, and can be measured at constant volume (CVC_V) or constant pressure (CPC_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 12R\frac{1}{2}R per quadratic degree of freedom, where RR is the gas constant
  • The Dulong-Petit law states that the molar heat capacity of a solid at high temperatures is approximately 3R3R, 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, ΔStotal=ΔSsystem+ΔSsurroundings0\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: ΔSdQT\Delta S \geq \int \frac{dQ}{T}
  • The Boltzmann equation defines entropy in terms of the number of microstates (Ω\Omega) accessible to a system: S=klnΩS = 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 (UU) is the total kinetic and potential energy of all particles within a system
  • Enthalpy (HH) is the sum of internal energy and the product of pressure and volume: H=U+PVH = U + PV
    • Represents the heat content of a system at constant pressure
  • Helmholtz free energy (AA) is the maximum amount of work a system can perform at constant temperature and volume: A=UTSA = U - TS
  • Gibbs free energy (GG) is the maximum amount of non-expansion work a system can perform at constant temperature and pressure: G=HTSG = H - TS
    • The change in Gibbs free energy determines the spontaneity of a process: ΔG<0\Delta G < 0 for spontaneous processes
  • The chemical potential (μ\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 (CC), phases (PP), and degrees of freedom (FF) in a system at equilibrium: F=CP+2F = C - P + 2
  • The Clapeyron equation describes the slope of a phase boundary in a pressure-temperature diagram: dPdT=ΔHTΔV\frac{dP}{dT} = \frac{\Delta H}{T \Delta V}
  • The Clausius-Clapeyron equation is a special case for the vapor-liquid equilibrium: lnP2P1=ΔHvapR(1T21T1)\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: μiα=μiβ=μiγ\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: SdTVdP+iNidμi=0SdT - 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=nRTPV = 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+an2V2)(Vnb)=nRT(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: θ1θ=KP\frac{\theta}{1-\theta} = KP
  • The Gibbs adsorption isotherm relates the change in surface tension to the adsorption of solutes at an interface: dγ=iΓidμid\gamma = -\sum_i \Gamma_i d\mu_i
  • The Arrhenius equation describes the temperature dependence of reaction rates: k=AeEa/RTk = Ae^{-E_a/RT}, where EaE_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=E0RTnFlnQE = E^0 - \frac{RT}{nF} \ln Q


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.