Two others from (dU) and (dH). These are for converting unmeasurable quantities (entropy change) into measurable ones (volume, pressure, temperature). 5. Chemical Potential & Phase Equilibria The chemical potential of species (i): [ \mu_i = \left(\frac\partial G\partial N_i\right) T,P,N j\neq i ] Phase Equilibrium Condition (MIT Classic Derivation) For two phases (\alpha) and (\beta) in contact: [ T^\alpha = T^\beta,\quad P^\alpha = P^\beta,\quad \mu_i^\alpha = \mu_i^\beta ] Clausius-Clapeyron Equation [ \fracdPdT = \frac\Delta H_\textvapT \Delta V ] Used for calculating vapor pressure vs. temperature. 6. Mixtures & Partial Molar Quantities Partial molar Gibbs free energy = chemical potential (\mu_i).
Equilibrium condition: [ \sum_i \nu_i \mu_i = 0 ] chemical thermodynamics mit
From (dG = -SdT + VdP): [ -\left(\frac\partial S\partial P\right)_T = \left(\frac\partial V\partial T\right)_P ] Two others from (dU) and (dH)
For an ideal gas mixture: [ \mu_i(T,P) = \mu_i^\circ(T) + RT \ln\left(\fracP_iP^\circ\right) ] where (P_i = y_i P) (partial pressure). Chemical Potential & Phase Equilibria The chemical potential
For ideal gases: [ \sum_i \nu_i \mu_i^\circ(T) = -RT \ln K_P ] where [ K_P = \prod_i \left(\fracP_iP^\circ\right)^\nu_i ]