CONTENTS XV 



31. Condition with Respect to the Variation of the 



Pressure 156 



32. Conditions of Stability in Terms of the Functions 



x// and f 156 



33. Conditions with Respect to Temperature and the 



Potentials 159 



34. Limits of Stabihty 161 



XL Critical Phases 163 



35. Number of Degrees of Freedom of a Critical Phase . 163 



36. Conditions in Regard to Stability of Critical 



Phases 164 



XII. Generalized Conditions of Stability 166 



37. The Conditions 166 



38. Critical Phases 172 



XIIL Equilibrium of Two Components in Two Phases 175 



39. The Equilibrium 175 



40. Konowalow's Laws 177 



XIV. Phases of Dissipated Energy. Catalysis 178 



41. Dissipated Energy 178 



E. Osmotic and Membrane Equilibria, including 

 Electrochemical Systems (Gibbs I, pp. 83-85; 



413-417), E. A. Guggenheim 181 



1. Introduction 181 



2. Proof of General Conditions of Membrane Equi- 



librium 183 



3. Choice of Independent Components 185 



4. Choice of Independent Variables 186 



5. Mols and Mol Fractions 187 



6. Ideal Solutions 188 



7. Non-ideal Solutions 190 



8. Osmotic Equilibrium 192 



9. Incompressible Solutions 193 



10. Relation between Activity Coefficients 193 



11. Osmotic Coefficients 194 



12. Osmotic Equilibrium in Terms of Osmotic Coeffi- 



cient 196 



13. Extremely Dilute Solutions 197 



14. Electric Potential Difference between Two Identi- 



cal Phases 198 



15. Electric Potential Difference between Two Phases 



of Different Composition 199 



16. Combinations of Ions with Zero Net Electric 



Charge 200 



17. Ideal Solutions of Ions 201 



18. Non-ideal Solutions of Ions 201 



19. Mean Activity Coefficient of Electrolyte 202 



