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Why the thermal capacity of two mixed system is equal to the sum of individual thermal capacities?
Let thermal capacity to be the specific heat multiplied by the mass of the thermodynamic system.
It is a experimental fact? Or there is some fundamentl principle guaranteeing this?
Heat capacity of an object is defined as the required amount of heat to change the object's temperature one unit. If you mix two systems, which don't interact with each other, then to increase their temperature you need to add the sum of the heats that must be added to each one separately. (If there are some kind of interaction between them, which can store energy, then the new heat capacity is not necessarily equal to sum of the heat capacities of the older systems)
The heat capacity of a system tells you the change in internal energy for a given change in temperature. So to answer your question we have to look at the temperature and energy of a pair of systems separately and to together.
The first one is easy. If the 2 systems are in thermal equilibrium then the two subsystems must have the same temperature by the zeroth law of thermodynamics. Quite simply the only sensible way to talk about the temperature of the combined system is if it is equal to the temperature of its subsystems, which we can only do if they have the same temperature.
Now for the energy we have to make an extra assumption. If the state of one subsystem does not effect the energy of the other then the energy of the combined system is simply the sum of the energies of the subsystems. From these two ideas it is pretty clear that the heat capacity of the combined system is the sum of the heat capacity of its component parts.
Why is it that we can normally assume that the energies of macroscopic object do not effect each other. Well most macroscopic thermodynamic systems are made up of billions of microscopic components, which generally interact with each other over microscopically short distances. This means that only those parts of the subsystems which are in direct contact with the other system will be effected by it, and this is (at least if one of the subsystems is a solid) normally a negligibly small fraction of the total. There will however be exceptions where the subsystems do interact strongly, for instance if you mix two liquids you would not generally expect the heat capacities to add.
