Thermodynamic property relations are the backbone of fluid behavior analysis. They connect state variables like pressure and temperature to energy functions like enthalpy and Gibbs free energy, helping us predict how fluids will react under different conditions.

These fundamental relations are crucial for understanding fluid systems. By linking measurable properties to abstract concepts like entropy, we can solve real-world engineering problems and design more efficient processes in industries from power generation to chemical manufacturing.

Thermodynamic State Variables

Internal Energy, Enthalpy, and Entropy

• Internal energy represents the total energy of a thermodynamic system, including kinetic and potential energy of particles, as well as intermolecular forces
• Enthalpy is a state function that measures the total heat content of a system at constant pressure, defined as $H = U + PV$
• Entropy is a measure of the disorder or randomness of a system, representing the amount of energy unavailable for useful work (second law of thermodynamics)

Pressure, Volume, and Temperature

• Pressure is the force per unit area exerted by a fluid on its surroundings, measured in pascals (Pa) or atmospheres (atm)
• Volume is the amount of space occupied by a substance, typically measured in cubic meters ($m^3$) or liters (L)
• Temperature is a measure of the average kinetic energy of particles in a system, expressed in Kelvin (K) or degrees Celsius (°C)

Thermodynamic Potentials

Gibbs and Helmholtz Free Energy

• Gibbs free energy is a thermodynamic potential that measures the maximum reversible work that can be performed by a system at constant temperature and pressure, defined as $G = H - TS$
• Helmholtz free energy is a thermodynamic potential that measures the maximum reversible work that can be performed by a system at constant temperature and volume, defined as $A = U - TS$
• Both Gibbs and Helmholtz free energy are useful in determining the spontaneity and equilibrium conditions of chemical reactions and phase transitions

Thermodynamic Potentials and Their Relationships

• Thermodynamic potentials are state functions that describe the energy content and work potential of a system under specific constraints (constant temperature, pressure, volume, or entropy)
• The four primary thermodynamic potentials are internal energy (U), enthalpy (H), Gibbs free energy (G), and Helmholtz free energy (A)
• These potentials are related to each other through Legendre transformations, which involve expressing one potential in terms of another by adding or subtracting the product of an intensive and extensive variable (e.g., $H = U + PV$)

Fundamental Property Relations

Maxwell Relations

• Maxwell relations are a set of four equations that describe the relationships between partial derivatives of thermodynamic potentials with respect to their natural variables
• These relations are derived from the equality of mixed second partial derivatives of thermodynamic potentials (e.g., $(\partial S/\partial P)_T = -(\partial V/\partial T)_P$)
• Maxwell relations are useful in determining hard-to-measure properties from more easily measured ones, such as calculating the change in entropy from measurements of volume and temperature

Partial Derivatives and Their Applications

• Partial derivatives are used to describe how a thermodynamic property changes with respect to one variable while holding other variables constant
• Common partial derivatives in thermodynamics include $(\partial P/\partial V)_T$ (isothermal compressibility), $(\partial V/\partial T)_P$ (isobaric thermal expansion), and $(\partial S/\partial T)_V$ (isochoric heat capacity)
• Partial derivatives are essential for deriving equations of state, such as the ideal gas law ($PV = nRT$), and for understanding the behavior of materials under different conditions (e.g., phase transitions, thermal expansion, and compressibility)