2 Identical Objects in Thermal Contact

[Master J]

If 2 identical objects with initial temperatures X and Y are placed together in thermal contact, then the final temperature is T= (X + Y)/2, X and Y actually being equal and excluding any heatloss to surroundings.

I understand this intuitively, but is there a precise mathematical proof? It's perhaps really simple, but the situation seems so straightforward to me I can't even think of a proof.

 

[Mapes]

The proof is based on law of conservation of energy (the energy lost by object X must be gained by object Y) and a way of writing energy in differential form, dE=C_V dT, where E is energy, C_V is heat capacity, and T is temperature.

Putting these together, we have

dE_X=C_{V,X}dT_X=-C_{V,Y}dT_Y=-dE_Y[/itex]<br /> <br /> which can be integrated to give your original expression.