Fundamental equation

The fundamental equation of thermodynamics (also fundamental relation or Gibbs' fundamental equation according to Josiah Willard Gibbs ) is the starting point of formal thermodynamics . It is the most important characteristic function and describes the set of all equilibrium points of a thermodynamic system as a function of the state variable internal energy U of all extensive variables ${\displaystyle X_{i}:}$

${\displaystyle U=U(X_{i})}$

In non- magnetic single-material systems , the natural variables to entropy  S , volume  V and amount of substance  n are simplified :

${\displaystyle U=U(S,V,n)}$

This also applies analogously to non-magnetic multi-  substance systems with k different substances:

${\displaystyle U=U(S,V,n_{1},\dots ,n_{k})}$

The function can also be given equivalent in the form

${\displaystyle \Leftrightarrow S=S(U,V,n_{1},\dots ,n_{k})}$

Both functions each contain the entire thermodynamic information of the system under consideration. The mathematical structure of thermodynamics is thus established. Further, especially physical, content can be found by following the main clauses .

A differential notation is also often used:

${\displaystyle \mathrm {d} U=\left({\frac {\partial U}{\partial S}}\right)_{V,n_{i}}\cdot \mathrm {d} S+\left({\frac {\partial U}{\partial V}}\right)_{S,n_{i}}\cdot \mathrm {d} V+\sum _{i=1}^{k}\left({\frac {\partial U}{\partial n_{i}}}\right)_{V,S}\cdot \mathrm {d} n_{i}}$

With the definitions for the temperature  T , the pressure  p and the chemical potential it  follows: ${\displaystyle \mu }$

${\displaystyle \mathrm {d} U=T\cdot \mathrm {d} S-p\cdot \mathrm {d} V+\mu \cdot \mathrm {d} n}$

Assuming a constant amount of substance ( ), this is further simplified to: ${\displaystyle \mathrm {d} n=0}$

${\displaystyle \mathrm {d} U=T\cdot \mathrm {d} S-p\cdot \mathrm {d} V}$

This shows that the equations of state are in principle the first derivatives of the fundamental equation .

From mathematical theorems about differentiable functions of several variables, relationships of the second derivatives can be found: the Maxwell relationships . The experimentally important response coefficients can also be derived from the second derivatives, e.g. B. compressibility , specific heat capacity and coefficient of thermal expansion .

The Legendre transformation of the fundamental relation leads to the thermodynamic potentials : free energy , enthalpy and Gibbs energy .

literature

HB Callen : Thermodynamics and an Introduction to Thermostatistics. 2nd edition, John Wiley & Sons, New York / Chichester / Brisbane / Toronto / Singapore 1985, ISBN 978-0471862567