Maxwell relations are powerful tools that link different thermodynamic properties. They're derived from the equality of mixed partial derivatives of thermodynamic potentials, allowing us to calculate hard-to-measure properties from easier ones.

Thermodynamic potentials, like Gibbs and Helmholtz free energies, help us understand system behavior at equilibrium. These functions, along with enthalpy and internal energy, characterize a system's state and predict its tendency to change.

Maxwell Relations and Thermodynamic Potentials

Maxwell Relations

• Derived from equality of mixed second partial derivatives of thermodynamic potentials
• Relate changes in thermodynamic properties to each other
• Four key Maxwell relations:
  • $(\frac{\partial T}{\partial V})_S = -(\frac{\partial P}{\partial S})_V$
  • $(\frac{\partial T}{\partial P})_S = (\frac{\partial V}{\partial S})_P$
  • $(\frac{\partial S}{\partial V})_T = (\frac{\partial P}{\partial T})_V$
  • $(\frac{\partial S}{\partial P})_T = -(\frac{\partial V}{\partial T})_P$
• Enable calculation of hard-to-measure properties from easier-to-measure ones (entropy from pressure-volume data)

Thermodynamic Potentials

• State functions that characterize thermodynamic systems at equilibrium
• Gibbs free energy ($G$)
  • Measures maximum reversible work obtainable from a system at constant temperature and pressure
  • Defined as $G = H - TS$, where $H$ is enthalpy, $T$ is temperature, and $S$ is entropy
  • Minimized at equilibrium for systems at constant temperature and pressure (chemical reactions, phase transitions)
• Helmholtz free energy ($A$)
  • Measures maximum reversible work obtainable from a system at constant temperature and volume
  • Defined as $A = U - TS$, where $U$ is internal energy
  • Minimized at equilibrium for systems at constant temperature and volume (elastic deformations)
• Enthalpy ($H$)
  • Measures heat content of a system at constant pressure
  • Defined as $H = U + PV$, where $P$ is pressure and $V$ is volume
  • Changes in enthalpy equal heat absorbed or released at constant pressure (calorimetry)
• Internal energy ($U$)
  • Total kinetic and potential energy of a system's particles
  • Changes in internal energy equal heat absorbed or work done on the system (first law of thermodynamics)

Mathematical Tools for Thermodynamics

Partial Derivatives and Jacobian Matrix

• Partial derivatives measure rates of change of a function with respect to one variable while holding others constant
• Jacobian matrix organizes partial derivatives of a vector-valued function
  • Enables coordinate transformations and change of variables in thermodynamic equations
  • Example: Jacobian determinant relates $(\frac{\partial P}{\partial V})_T$ to $(\frac{\partial V}{\partial P})_T$ via $(\frac{\partial P}{\partial V})_T = \frac{1}{(\frac{\partial V}{\partial P})_T}$
• Partial derivatives and Jacobians essential for manipulating thermodynamic equations and applying Maxwell relations

Thermodynamic Equations of State

• Relate thermodynamic properties of a system at equilibrium
• Ideal gas law: $PV = nRT$, where $n$ is number of moles and $R$ is the gas constant
  • Describes behavior of gases at low densities and high temperatures
• Van der Waals equation: $(P + \frac{an^2}{V^2})(V - nb) = nRT$, where $a$ and $b$ are constants specific to the gas
  • Accounts for molecular size and intermolecular attractions, improving upon ideal gas law
• Virial equation: $\frac{PV}{nRT} = 1 + \frac{B(T)}{V} + \frac{C(T)}{V^2} + ...$, where $B(T)$, $C(T)$, etc. are virial coefficients dependent on temperature
  • Expresses compressibility factor as a power series in reciprocal molar volume
  • Useful for describing real gas behavior at moderate densities
• Equations of state combined with Maxwell relations and thermodynamic potentials enable complete characterization of thermodynamic systems