Thermodynamics is principally concerned with energy flows and energetic barriers to reactions may prevent them from occurring for periods longer than the age of the universe, which may as well be the same as saying they never happen. Additionally, thermodynamics says nothing about whether or not two things can undergo a chemical reaction or not. For this, you need to know some structural details about the actual chemicals; so you can't talk about an abstract chemical reaction between "A" and "B" anymore, you have to talk about real compounds like water or oxygen. Such considerations, however, provide a more solid thermodynamic foundation to the clear utility of carbon-based molecules for life.
No other element really even comes close to carbon in terms of the diversity of possible structures it can form. Life on earth has been busy creating a wider and wider variey of organic molecules for the past 4 billion years, and there's no evidence yet that it has even scratched the surface of the variety of organic molecules that might be possible (to a chemist "organic" just means "has carbon in it"). So once the randomizing directives of the second law found an endless playground in the structural possibilities of carbon molecules, all it needed was a sustained energy flux through a materially closed system (provided by the sun, earth, and our mother the expanding sink of space) to begin generating actual variety out of these configurational possibilities.