🧊Thermodynamics II Unit 7 – Thermodynamic Relations & State Equations

Key Concepts and Definitions

• Thermodynamic state variables describe the equilibrium state of a system (pressure, temperature, volume, internal energy, enthalpy, entropy)
• Equation of state relates state variables for a specific substance under given conditions
• Ideal gas law assumes gas particles have negligible volume and no intermolecular forces ($PV = nRT$)
• Thermodynamic processes involve changes in state variables (isothermal, isobaric, isochoric, adiabatic)
• Thermodynamic potentials are functions that characterize the state of a system (internal energy, enthalpy, Helmholtz free energy, Gibbs free energy)
• Maxwell relations are derived from the equality of mixed partial derivatives of thermodynamic potentials
  • Connect different state variables and provide insights into system behavior
• Real gases deviate from ideal gas behavior due to intermolecular forces and finite particle volume

Fundamental Thermodynamic Relations

• First law of thermodynamics states that energy is conserved in a closed system ($dU = đQ - đW$)
  • Change in internal energy ($dU$) equals heat added ($đQ$) minus work done by the system ($đW$)
• Second law of thermodynamics introduces entropy as a measure of disorder or randomness in a system
  • Entropy of an isolated system never decreases ($dS \geq 0$)
• Fundamental thermodynamic relation combines the first and second laws ($dU = TdS - PdV$)
  • Relates changes in internal energy to changes in entropy and volume
• Enthalpy is defined as $H = U + PV$ and represents the total heat content of a system
• Gibbs free energy is defined as $G = H - TS$ and determines the spontaneity of processes at constant temperature and pressure
• Helmholtz free energy is defined as $F = U - TS$ and is useful for analyzing processes at constant temperature and volume

State Equations and Their Applications

• Ideal gas law is the simplest state equation and assumes no intermolecular forces or particle volume ($PV = nRT$)
• Van der Waals equation accounts for attractive forces and particle volume ($[P + a(n/V)^2](V - nb) = nRT$)
  • $a$ represents the attraction between particles, and $b$ represents the particle volume
• Virial equation of state is a power series expansion in terms of density or pressure ($PV = nRT(1 + B/V + C/V^2 + ...)$)
  • Virial coefficients ($B$, $C$, etc.) depend on temperature and account for deviations from ideal behavior
• Redlich-Kwong equation is a cubic equation of state that accurately describes the behavior of real gases ($P = RT/(V-b) - a/[T^{1/2}V(V+b)]$)
• State equations are used to calculate thermodynamic properties (compressibility factor, fugacity, enthalpy departures)
• Compressibility factor ($Z = PV/nRT$) quantifies the deviation of a real gas from ideal behavior

Maxwell Relations and Their Significance

• Maxwell relations are derived from the equality of mixed partial derivatives of thermodynamic potentials
• Four key Maxwell relations: $(\partial S/\partial V)_T = (\partial P/\partial T)_V$, $(\partial S/\partial P)_T = -(\partial V/\partial T)_P$, $(\partial T/\partial V)_S = -(\partial P/\partial S)_V$, $(\partial T/\partial P)_S = (\partial V/\partial S)_P$
  • Each relation connects different state variables and provides insights into system behavior
• Maxwell relations are useful for deriving other thermodynamic relationships and calculating properties that are difficult to measure directly
• Example: $(\partial S/\partial P)_T = -(\partial V/\partial T)_P$ relates entropy changes to volume expansivity and can be used to calculate entropy changes from volumetric data
• Maxwell relations are essential for developing equations of state and understanding the behavior of real substances

Thermodynamic Potentials and Free Energy

• Thermodynamic potentials are functions that characterize the state of a system and determine the direction of spontaneous processes
• Internal energy ($U$) is the total energy of a system, including kinetic and potential energy of particles
• Enthalpy ($H = U + PV$) represents the total heat content of a system and is useful for processes at constant pressure
• Helmholtz free energy ($F = U - TS$) is the maximum work a system can perform at constant temperature and volume
  • Minimized at equilibrium for processes at constant $T$ and $V$
• Gibbs free energy ($G = H - TS$) is the maximum non-expansion work a system can perform at constant temperature and pressure
  • Minimized at equilibrium for processes at constant $T$ and $P$
• Changes in free energy determine the spontaneity of processes: $\Delta G < 0$ for spontaneous processes, $\Delta G = 0$ at equilibrium, and $\Delta G > 0$ for non-spontaneous processes
• Free energy relationships are used to calculate equilibrium constants, phase transitions, and chemical reaction spontaneity

Partial Derivatives and Their Physical Meaning

• Partial derivatives represent the change in one variable with respect to another while holding other variables constant
• Common partial derivatives in thermodynamics: $(\partial P/\partial V)_T$, $(\partial P/\partial T)_V$, $(\partial S/\partial T)_V$, $(\partial S/\partial V)_T$, $(\partial U/\partial V)_T$, $(\partial H/\partial P)_T$
• $(\partial P/\partial V)_T$ is related to the isothermal compressibility, which measures the volume change in response to pressure at constant temperature
• $(\partial P/\partial T)_V$ is related to the thermal pressure coefficient, which measures the pressure change with temperature at constant volume
• $(\partial S/\partial T)_V$ is related to the heat capacity at constant volume ($C_V$), which measures the heat required to raise the temperature at constant volume
• $(\partial S/\partial V)_T$ is related to the thermal expansion coefficient, which measures the volume change with temperature at constant pressure
• $(\partial U/\partial V)_T$ is related to the Joule-Thomson coefficient, which measures the temperature change during throttling (constant enthalpy expansion)
• $(\partial H/\partial P)_T$ is related to the isothermal Joule-Thomson coefficient, which measures the enthalpy change with pressure at constant temperature

Real Gas Behavior and Equations of State

• Real gases deviate from ideal gas behavior due to intermolecular forces and finite particle volume
• Compressibility factor ($Z = PV/nRT$) quantifies the deviation of a real gas from ideal behavior
  • $Z = 1$ for an ideal gas, $Z < 1$ for attractive forces dominating, and $Z > 1$ for repulsive forces dominating
• Van der Waals equation accounts for attractive forces and particle volume ($[P + a(n/V)^2](V - nb) = nRT$)
  • Predicts the existence of critical point and liquid-vapor phase transitions
• Virial equation of state is a power series expansion in terms of density or pressure ($PV = nRT(1 + B/V + C/V^2 + ...)$)
  • Virial coefficients ($B$, $C$, etc.) depend on temperature and account for deviations from ideal behavior
• Redlich-Kwong and Peng-Robinson equations are cubic equations of state that accurately describe the behavior of real gases and liquids
• Equations of state are used to calculate thermodynamic properties (compressibility factor, fugacity, enthalpy departures) and phase equilibria
• Principle of corresponding states suggests that all fluids, when compared at the same reduced temperature and pressure, have approximately the same compressibility factor

Practical Applications and Problem Solving

• Thermodynamic relations and state equations are essential for designing and optimizing various processes and systems
• Equations of state are used to predict the behavior of fluids in pipelines, storage tanks, and processing equipment
  • Help in determining the required compressor power, pipe sizes, and insulation thickness
• Maxwell relations are used to calculate properties that are difficult to measure directly, such as entropy changes and Joule-Thomson coefficients
• Free energy relationships are used to determine the spontaneity and equilibrium conditions of chemical reactions and phase transitions
  • Help in selecting optimal operating conditions and catalysts for chemical processes
• Partial derivatives are used to analyze the sensitivity of system properties to changes in variables, such as temperature and pressure
  • Help in designing control systems and safety measures for industrial processes
• Real gas behavior is crucial for accurate modeling and simulation of processes involving high pressures or low temperatures, such as natural gas processing and cryogenic applications
• Problem-solving in thermodynamics involves identifying the relevant equations, state variables, and boundary conditions
  • Requires a systematic approach and careful attention to units and sign conventions
• Common problem-solving strategies include:
  1. Drawing a clear diagram of the system and surroundings
  2. Listing the known and unknown variables
  3. Selecting the appropriate equations and relations
  4. Solving the equations and checking the units and reasonableness of the results