Extensive and intensive conjugate quantities

For a system described by N extensive quantities \(e_k\) and N intensive quantities \(i_k\).

The differential increase in energy \(U\) per unit volume of the system for a variation of \(e_k\) is:

\[dU = \sum_k i_k d e_k\]

The Gibbs potential of the system is defined as:

\[G = U - \sum_k i_k e_k\]

And, the differential increase of Gibbos potential can be expressed as:

\[dG = dU - d\sum_k i_k e_k = \sum_k i_k d e_k - \sum e_k d i_k - \sum i_k d e_k = \sum e_k d i_k\]

Therefore, the intensive quantities \(i_k\) can be defined as partial derivatives of the energy \(U\) with respect to their extensive conjugate quantities \(e_k\):

\[i_k = \frac{\partial U}{\partial e_k}\]

The extensive quantities \(e_k\) can be defined as partial derivatitives of the Gibbos potential \(G\) with respect to their intensive conjugate quantities \(i_k\):

\[e_k = - \frac{\partial G}{\partial i_k}\]