8.2 Maxwell relations and their applications

Maxwell relations connect thermodynamic properties in surprising ways. These equations link temperature, pressure, volume, and entropy, allowing us to calculate hard-to-measure quantities from easier ones. They're essential tools for solving real-world thermodynamic problems.

By relating different partial derivatives, Maxwell relations simplify complex equations. They help us understand how changing one property affects others, making it easier to analyze and predict the behavior of thermodynamic systems in various processes.

Maxwell Relations

Derivation of Maxwell relations

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• Begin with fundamental thermodynamic relation $dU = TdS - PdV$ relates internal energy $U$ to entropy $S$, temperature $T$, pressure $P$, and volume $V$
• Obtain differential forms of thermodynamic potentials
  • Enthalpy $H$: $dH = TdS + VdP$ considers system and surroundings (constant pressure processes)
  • Helmholtz free energy $A$: $dA = -SdT - PdV$ useful for isothermal processes
  • Gibbs free energy $G$: $dG = -SdT + VdP$ describes isothermal, isobaric processes (constant temperature and pressure)
• Equality of mixed partial derivatives for function $f(x, y)$: $\left(\frac{\partial^2 f}{\partial x \partial y}\right) = \left(\frac{\partial^2 f}{\partial y \partial x}\right)$ mathematical property
• Four Maxwell relations derived:
  • $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$ relates temperature, volume, pressure, and entropy
  • $\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P$ connects temperature, pressure, volume, and entropy
  • $\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$ associates entropy, volume, pressure, and temperature
  • $\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P$ links entropy, pressure, volume, and temperature

Physical interpretation of Maxwell relations

• $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$
  • Constant entropy: temperature change with volume equals negative pressure change with entropy at constant volume
  • Adiabatic processes (no heat exchange)
• $\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P$
  • Constant entropy: temperature change with pressure equals volume change with entropy at constant pressure
  • Isentropic processes (reversible adiabatic)
• $\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$
  • Constant temperature: entropy change with volume equals pressure change with temperature at constant volume
  • Isothermal processes
• $\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P$
  • Constant temperature: entropy change with pressure equals negative volume change with temperature at constant pressure
  • Isothermal processes

Application for thermodynamic calculations

• Express thermodynamic properties using measurable quantities
  • $\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P$ calculates entropy changes from volume data (thermal expansion)
• Derive property derivatives using Maxwell relations
  • $\left(\frac{\partial C_P}{\partial V}\right)_T = -T\left(\frac{\partial^2 P}{\partial T^2}\right)_V$ from $\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$ relates heat capacity $C_P$ to pressure-temperature behavior

Simplification of thermodynamic equations

• Substitute Maxwell relations to simplify equations
  • Clapeyron equation $\frac{dP}{dT} = \frac{\Delta S}{\Delta V}$ becomes $\frac{dP}{dT} = \frac{\Delta H}{T\Delta V}$ using $\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P$ (phase transitions)
• Apply Maxwell relations to solve problems
  • Isothermal entropy change from $\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P$ and volume data (gas expansion/compression)