Can you suggest some references for rigorous treatment of thermodynamics? I want things like reversibility, equilibrium to be clearly defined in terms of the basic assumptions of the framework.
The pioneer of the rigorous treatment of thermodynamics is Constantin Carathéodory. His article (Carathéodory, C., Untersuchung über die Grundlagen der Thermodynamik, Math. Annalen 67, 355-386) is cited everywhere in this context, but probably you want some newer and more modern things.
Buchdahl wrote a lot of papers about this subject in the 40's, 50's and 60's. He summarized these in the book: H.A. Buchdahl, The Concepts of Classical Thermodynamics (Cambridge Monographs on Physics), 1966.
Finally, I have come across the book T. Matolcsi, "Ordinary Thermodynamics" (since a few friends of mine went to the class of the author), which treats thermodynamics in a mathematically very rigorous way.
I hope some of these will help you.
I think this book is what you are looking for: (although this question is already answered a long time ago)
Thermodynamics and an Introduction to Thermostatistics, by Herbert B. Callen
It is sort of a modern version of the Gibbs formulation. Entropy is postulated at the beginning, from which all Thermodynamics is formally derived. The scheme is somewhat abstract, but the book is easy to read. It points in the direction of statistical mechanics (specially near the end of the book).
As some people point out, it depends on what you mean by "rigorous". If you mean rigorous, in the sense of a mathematician the books by David Ruelle are very nice.
- Statistical mechanics: rigorous results by David Ruelle. This is actually a very nice books, which takes the statistical mechanics point of view (instead of a phenomenological/geometric approach). Everything is defined very rigorously, the existence of the thermodynamic limit is proven in both classical/quantum and lattice/continuous systems (for well-behaved interactions) and the equivalence of different ensembles shown. Also phase transitions are covered where very general versions of the Lee-Yang and the Mermin-Wagner theorems are proven. And much more. The book can be rather dry and formal, so make sure to find the old papers by Michael Fisher which can be helpful.
- Thermodynamic formalism: the mathematical structures of equilibrium statistical mechanics by David Ruelle. I don't know this book very well, but it is also a very rigorous (and more advanced than 1.) treatment of thermodynamics.
Hope you will find this useful.
My paper ''Phenomenological thermodynamics'' http://www.mat.univie.ac.at/~neum/ms/phenTherm.pdf rigorously derives the core of thermodynamics in 18 pages, starting from a few simple axioms. No physics background is required; the main mathematical took is convexity (and of course calculus).
The exposition is far simpler than Caratheodory or Lieb & Yngvason.
A slightly different version - mainly augmented by cross references - is also available as Chapter 7 of my online book http://lanl.arxiv.org/abs/0810.1019 , but can be read independent of the remainder of the book. Chapters 8-10 of the book rigorously deduce the phenomenological axioms of Chapter 7 from statistical mechanics.
David R Owen, A first course in the Mathematical foundations of thermodynamics.
C Truesdell , The origins of rational thermodynamics. Well written, Truesdell sets thermodynamics on a stronger foot than all the mumbo-jumbo usualy found in books written by people that do not master math. The intro is a must be read.
And of course, for a correct derivation of macroscopic thermodynamics from kinetic theory, Radu Ballescu's numerous books.
There are many presentations that are mathematically rigorous – but they have different underlying philosophies and conceptions of what thermal physics is or should do. So it'd be best if you read as many of them as possible to find those that are closer to your own philosophy, and maybe use them to be critical about your own philosophy, too.
Besides the texts given in the other answers, I'd add these:
C. A. Truesdell: Rational Thermodynamics (2nd ed., Springer 1984). It is very rigorous in presenting thermodynamics in terms of primitive notions and general principles, both from a physical and from a purely mathematical points of view. It also discusses different ways thermodynamics can be presented, with their relative merits. And it gives plenty of historical remarks. Even if you don't agree with Truesdell's points of view, his insights can help you better understand and sharpen your own point of view.
G. Astarita: Thermodynamics: An Advanced Textbook for Chemical Engineers (Springer 1990). As the title says, this is applied and gives plenty of concrete experimental examples, but at the same time is mathematically rigorous and draws from Truesdell's book above.
I. Müller: Thermodynamics (Pitman 1985). Similar in spirit to Truesdell's, but with different assumptions motivated by different conceptions about physics.
I. Müller, W. H. Müller: Fundamentals of Thermodynamics and Applications: With Historical Annotations and Many Citations from Avogadro to Zermelo (Springer 2009). Same philosophy as the previous book, but with additional historical remarks.
1977 Chemistry Nobel Prize laureate Ilya Prigogine's
Introduction to Thermodynamics of Irreversible Processes. New York: Interscience Publishers, 1968
is "a concise exposition of the equations of the thermodynamics of irreversible processes" (D. Falkoff).