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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?

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    $\begingroup$ For alloys, the Neumann-Kopp "rule" is often invoked, where the heat capacity of a phase is equal to the weighted average of the individual elemental heat capacities. However, this is not a rule or a law, just an often (but not always!) useful value to use if you have not explicitly measured the heat capacity. Many alloys violate this "rule". $\endgroup$ – Jon Custer Sep 29 '16 at 20:02
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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)

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  • $\begingroup$ The if i know for example the heat capacity of water and and chayote, if i cook the chyote in water it is a good aproximation to the heat capacity of this system to be the sum of each heat capacity, otherwise if i know the heat capacity of two different gases, the heat capacity is not complete well described by the sum, is this right? $\endgroup$ – Yassin Rany Sep 29 '16 at 21:04
  • $\begingroup$ If the gases are ideal, which means there are no interactions at all, then their heat capacities simply add up, too. [addendum: the situation is a little different for the "specific heat"s, which are the required heat to change the temperature of a unit amount (mass, mole, etc) of object one degree. For the specific heat of a combined system, one should calculate a weighted average of the components specific heats (with weights corresponding to the objects amounts present in the system.] $\endgroup$ – SaMaSo Sep 29 '16 at 21:20
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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.

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