# 23.2: Gibbs Energies and Phase Diagrams

### First Order Transitions

Note that although the \(G\) curve is continuous, its first order derivatives (such as S and H) are discontinuous at melting. This is why this transition it called a *first order transition*. We could say that:

- G is continuous but has a kink
- The first order derivatives (H,S,..) are discontinuous (have a jump)
- The second order derivatives (C
_{p}, ..) have a singularity (go to ∞)

### Second Order Transitions

More subtle transitions where G is continuous, H and S are also continuous but have a kink and the discontinuity is only found in the second order derivatives (such as C_{p}) also exist. They are called *second order transitions*. In such a case:

- G is continuous and has no kink
- The first order derivatives (H,S,..) are continuous (but have a kink)
- The second order derivatives (C
_{p}, ..) are discontinuous (have a jump)

Transition Order |
Function |
1^{st} Order |
2^{nd} Order |
---|---|---|---|

0 | G,A | kink | smooth |

1 | H,S,V,.. | jump | kink |

2 | CP,CV,α,κ | sing. ∞ | jump |

This classification goes back to Ehrenfest. Obviously it based on the question: what order derivative is the first to go discontinuous? Of course we could extend this principle and define *third order transitions* but there are reasons to be doubtful that such things exist. Another problem is that it is assumed that the order must be integer: 1,2, etc. Is it possible to have a transition of intermediate non-integer order, say 1.3? Although derivatives of fractional order are beyond the scope of the chemistry curriculum the mathematics does exist (Liouville).

*Schematic comparison of G,S and Cp for 1st and 2nd order transitions*

### Conjugate Variables

As discussed before there are many other forms of work possible, such as electrical work, magnetic work or elastic work. These they are commonly incorporated in the formalism of thermodynamics by adding other terms, e.g:

\[dG = -SdT + VdP + ℰde + MdH + FdL + γdA\]

*ℰde*stands for the electromotoric force*ℰ*and*de*the amount of charge transported against it.*MdH*stand for magnetization and (change in) magnetic field.*F*stands for the elastic force of e.g. a rubber band*dL*for the length it is stretched*γ*stands for the surface tension (e.g. of a soap bubble),*A*for its surafce area.

The terms always appear in a pair of what is known as conjugate variables. That is even clearer if we write out the state function rather than its differential form:

\[G = U + PV -TS + ℰe + MH + FL + γA + ...\]

The PV term can also be generalized -and needs to be so- for a viscous fluid to a stress-strain conjugate pair. It then involves a stress tensor. We will soon encounter another conjugate pair: μdn that deals with changes in composition (n) and the thermodynamic potential μ.