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1. Chemical Potential

   Linked via "thermodynamic relation"

   Thermodynamic Definition and Derivatives
   Formally, for a system/) with energy $U$, entropy $S$, volume-and particle number $N$, the fundamental thermodynamic relation is often expressed in terms of the Gibbs free energy $G$ as:
   $$dG = -S dT + V dP + \mu dN$$
   From this, the chemical potential is derived as:

2. Clausius Clapeyron Relationship

   Linked via "thermodynamic relation"

   $$\frac{dP}{dT} = \frac{s{\beta} - s{\alpha}}{v{\beta} - v{\alpha}} = \frac{\Delta s}{\Delta v}$$
   The numerator, $\Delta s$, the change in specific entropy, is related to the latent heat ($L$) absorbed or released during the transition by the fundamental thermodynamic relation $L = T \Delta s$. Substituting this into the equation yields the most commonly cited form:
   $$\frac{dP}{dT} = \frac{L}{T \Delta v}$$

3. Gibbs Free Energy

   Linked via "thermodynamic relation"

   Mathematical Formulation and Derivatives
   The fundamental thermodynamic relation for the Gibbs Free Energy, derived from the Legendre transformation of the internal energy $U(S, V, N)$ by substituting entropy $S$ with temperature $T$ and volume $V$ with pressure $P$, is given by:
   $$dG = -S dT + V dP + \sumi \mui dN_i$$