Thermodynamics of micellisation

Micellisation is found to be spontaneous i.e. $$\Delta G < 0.$$ It's found that $$\Delta S > 0,$$ which was intuitive since, solvated molecules are released. And, it may be most loosely started that $$\Delta H > 0$$ i.e endothermic at low temperature and exothermic ($$\Delta H<0$$) at higher temperatures. Guess why?

Hint, derive that $$\Delta H^\circ=-RT^2 \dfrac{\mathrm d(\ln{\text{CMC}})}{\mathrm dT}$$. And CMC may be defined as CMC as the point corresponding to the maximum change in gradient in an ideal property–concentration $$(\phi$$ against $$S_\mathrm{tot})$$ relationship $$\dfrac{\mathrm d^3\phi}{\mathrm dS^3_\mathrm{tot}} = 0.$$

(emphasis mine)

Why was most loosely stated term used in the text? Isn't it true? Excluding non-dressed micelle model, $$\Delta G$$ can be shown to be approximately $$(1 + \beta)\ln{\text{CMC}}$$.

Writing $$\Delta H^\circ = \Delta G^\circ + T\Delta S^\circ$$ and plugging in the values I couldn't get the desired term. Is my process correct?

Also, what's the meaning of “ideal property–concentration”?

• 1. Deriving $∆H^\circ$ won't help in telling $∆H$ probably, author wanted that you deduce above inference during deriving process. 2. There's a typo it should be excluding dressed model, not non-dressed model. Aug 21 '20 at 10:41