Thermodynamics: A Few Selected Important Relations

Fundamental property relations or GIBBS equations:

d(nU)=Td(nS)−Pd(nV)+∑_iμ_idn_i

          ↓   LEGENDRE transformation

d(nG)=−(nS)dT+(nV)dP+∑_iμ_idn_i

General GIBBS-DUHEM equation:

M...molar thermodynamic quantity (homogeneous phase)

M_i≡[δ(nM)/δn_i]_{T,P,n_j≠i} ... partial molar thermodynamic quantity

(δM/δT)_{P,{x_i}}dT+(δM/δP)−∑_ix_idM_i=0

LEWIS-RANDALL (LR) rule and HENRY’s law (HL): for liquid solutions, two components i and j, we have the measurable quantities:

f_i(T,P,{x_i}) ... fugacity of component i in solution

f_i^*(T,P) ... fugacity of pure component i

h_i,j(T,P) ... HENRY fugacity (or HENRY's law constant) of i dissolved in j

Solubility of methane in water: temperature dependence of the Henry fugacity from the triple point of water to ca. 525 K

Plot of ln[h_2,1(T,P_s,1)/GPa] against temperature T for methane dissolved in liquid water. h_2,1(T,P_s,1) denotes the Henry fugacity (Henry's law constant) at temperature T and vapor pressure P_σ,1(T).

• expermental results of Rettich, Handa, Battino and Wilhelm, in J.Phys.Chem. 85, 3230-3237 (1981): the average percentage deviation of the Henry fugacity from the value calculated via the correlating BK function is about ±0.05%;
• experimental results of Crovetto et al. J.Chem.Phys. 76, 1077-1086 (1982): the average percentage deviation of the Henry fugacity from the value calculated via the correlating BK function is about ±2% after anchoring their data set onto our high-precision results.

Summary

Various intellectually attractive topics concerning fluid nonelectrolyte solutions (liquids, vapors, gases) are investigated experimentally and theoretically, frequently leading to practical applications. Paraphrasing Mr. Spock, I find the field fascinating, and hope you agree!

⇒ cf. the list of publications (> 190 papers) ⇐

• Motivation: theoretical and practical roots (chemical engineering aspects).
• Thermodynamics: exact and/or asymptotically exact relations are used, together with theory-based approximations.
• Data reduction: general approaches, including the determination or estimation of important auxiliary quantities, such as partial molar volumes or second virial coefficients.
• Experimental results & scientific profit: For instance, high-precision Henry fugacities h_2,1(T,P_σ,1), partial molar enthalpy changes ΔH₂^∞(T,P_σ,1) and heat capacity changes ΔC_P,2^∞(T,P_σ,1) on solution (infinite dilution) of gases in liquid water establish equivalency of

van’t Hoff analysis    ⇔    calorimetry

• Models: based on Molecular Thermodynamics ⇒ frequently also of relevance for biophysical chemistry

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Der einzige sichere Führer auf dem Weg der weiteren Entwicklung bleibt stets die Messung…

 Max Planck, 1926

Acknowledgments: I thank all collaborators (from A, F, I, E, USA, Japan,…) for many years of agreeable collaboration, but three in particular: Profs. R. Battino (USA), J.-P. E. Grolier (F) and A. Asenbaum (A); and, of course, the funding agencies for providing monetary support: Fulbright Commission, FWF Austria, Austrian-French Scientific-Technical Cooperation Programs (for more than 25 years!), USA Public Health Service, NSF,…

The complexity of the liquid state and the large differences between our simple models and the actual solutions of molecules of a variety of sizes, shapes and force fields leaves ample room for further study.

The entire history of chemistry bears witness to the extraordinary importance of the phenomena of solubility.

Joel H. Hildebrand and Robert L. Scott
The Solubility of Nonelectrolytes
3^rd edition, Reinhold, New York, 1950

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Cutting-edge topical books supported by IUPAC and IACT (International Association of Chemical Thermodynamics, successor of IUPAC Commission I.2, Thermodynamics)

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