Usually I can see what the "aim" is of a certain field of physics. For classical mechanics it is to find the position as a function of time through Newton's law (or similar equations), for quantum mechanics it is to find the wave function by using the Schrodinger equation and for electrodynamics we seek to find electric and magnetic fields through the Maxwell equations. However, I do not yet see such a description for thermodynamics. What is it we hope to achieve by doing thermodynamics and can this be formulated in a way I did for other branches of physics?
You are assuming that branches of physics have discrete aims or goals? The goal of any science is to better understand the world around us. Classical mechanics, quantum mechanics, thermodynamics, electromagnetism are all inter-related. They do not have separate goals or aims. In any event, the purposes that you ascribe to them would be much too narrow. Rather than trying to come up with an abstract "aim" of thermodynamics, I would suggest that you read as much as you can about the subject of thermodynamics and then tell us what you think it is all about. AM
In industry, thermodynamics is essential to designing and operating equipment such as distillation columns, absorbers, compressors, turbines, chemical reactors, ion exchange columns, heat exchangers, cooling towers, air conditioning equipment, reverse osmosis separators, engines, pumps, motors, and a zillion other types of equipment.
One aim of chemical thermodynamics is to predict whether a chemical reaction will proceed spontaneously (without added energy) or whether energy must be added to make it proceed in the forward direction. In this, the progress of a chemical reaction is identified with one of the energy functions, eg at constant T and P, the Gibbs free energy. Chemistry is mostly about stuff, that is matter, so you also want to know about states, fluidity and other properties of matter.
Thermo is one of the first areas of physics that you study that deals with the statistical nature of the world we live in! Look around you. You seldom, very seldom, deal with just ONE particle, with just one interaction, with just one external conditions. Our macroscopic world is made up of a gazillion objects with a gazillion interactions. Thermodynamics is one of the first principles you deal with that deals with this scale of interactions. In other words, this is where real world, rather than the idealized situation, comes into your lessons. Zz.
When the field was orginally started by the likes of Carnot, Clausius, etc, the aim was to understand the relationships between heat, work and energy. Of course, later, the field expanded and was unified with other branches of physics, in particular mechanics and statistical mechanics. This was due to Boltzmann, Maxwell, etc. The ultimate aim of thermodynamics, as well as physics, is to describe reality (in this way, it is similar to art, religion, etc.)
In the same vein as your examples for CM, QM and EM I'm tempted to answer that your goal in thermodynamics could be to obtain the fundamental equation for a system. It contains all the information (like the Lagrangian/Hamiltonian in CM provided you've the initial conditions) of the system. For example take a monoatomic ideal gas. The fundamental relation is ##S(U,V,n)= S_0 +NR \ln \left [ \left ( \frac {U}{U_0} \right ) ^{3/2} \left ( \frac{V}{V_0} \right ) \left ( \frac {N}{N_0} \right ) ^{-5/2} \right ]##. From that equation you can derivate the famous ##PV=nrT## and ##U=(3/2)nrT##.
Interestingly, I think Einstein saw it as a way of testing atomic theory. The method, of course, was to invent statistical mechanics. In the 1905 paper, though, he used different phenomena, Brownian motion, osmotic pressure to calculate Avogadro's number and when they gave the same number that was finally accepted. Of course, the practical aim of thermo was to build a more efficient steam engine.
Paraphrasing my thermo lecturer: Understand and predict the behavior of -mostly- classical systems without full knowledge of their internal constituents (ie: all their positions and momenta). Predicting the future, if you will. Much like you would predict the trajectory of a single particle -massive and/or charged- in the presence (or absence) of force fields of many kinds, you can work out how much energy yield you can expect to get from a combustion reaction, or how to build the best balloon for carrying an experiment into the upper atmosphere, or work out what the chance of rainfall there is in some region of the country given a few initial conditions (ie: will a phase transition occur in an air parcel with this much water, pressure and temperature?), or predict wind speeds and directions with a knowledge of the pressure field in a region of the globe (with a little help from fluid dynamics, which ties in very tightly with classical thermo, in fact the main equations of fluid dynamics involve equations for the internal energy and entropy). For some systems, predicting 'trajectories' or quantities like heat/work/energy with any reasonable level of confidence failed when classical thermodynamics was applied. Ie: the huge theory-experiment disparity of the internal energy of a crystal lattice's dependence on temperature, which was initially solved by Einstein and later refined by Debye (I think?) by incorporating the ideas of quantum physics into the thermo/stat-mech calculations that went into it. So adding QM where necessary, I think the aim of thermo is whatever you want it to be.