Maxwell equations (thermodynamics)

From Knowino
Jump to: navigation, search

In thermodynamics, the Maxwell equations are a set of equations derived by application of Euler's reciprocity relation to the thermodynamic characteristic functions. The Maxwell relations, first derived by James Clerk Maxwell, are the following expressions between partial differential quotients:

 \begin{align} dU &= TdS - PdV &\Longrightarrow  \left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V\\ dA &= -SdT - PdV &\Longrightarrow    \left(\frac{\partial S}{\partial V}\right)_T =\left(\frac{\partial P}{\partial T}\right)_V\\ dH &= TdS +VdP &\Longrightarrow  \left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P\\ dG &= -SdT +VdP &\Longrightarrow  -\left(\frac{\partial S}{\partial P}\right)_T = \left(\frac{\partial V}{\partial T}\right)_P\\ \end{align}

The characteristic functions are: U (internal energy), A (Helmholtz free energy), H (enthalpy), and G (Gibbs free energy). The thermodynamic parameters are: T (temperature), S (entropy), P (pressure), and V (volume).

As an example of a derivation, consider

 dU = \underbrace{\left(\frac{\partial U}{\partial S}\right)_V}_{T} dS + \underbrace{\left(\frac{\partial U}{\partial V}\right)_S}_{-P} dV

Euler's reciprocity relation reads:

 \frac{\partial^2 U}{\partial V \partial S} = \frac{\partial^2 U}{\partial S \partial V},

which indeed gives

 \left(\frac{\partial T}{\partial V}\right)_S = - \left(\frac{\partial P}{\partial S}\right)_V.
Personal tools
Variants
Actions
Navigation
Community
Toolbox