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

[tex]dE_X=C_{V,X}dT_X=-C_{V,Y}dT_Y=-dE_Y[/itex]

which can be integrated to give your original expression.

 

[astrokreng]

I can provide a mathematical proof for the final temperature of two identical objects in thermal contact. Let's consider the first law of thermodynamics, which states that energy cannot be created or destroyed, only transferred from one form to another. In this case, the energy being transferred is heat.

Initially, object 1 has a temperature of X and object 2 has a temperature of Y. When they are placed in thermal contact, heat will flow from the object with a higher temperature (X or Y) to the object with a lower temperature. This will continue until both objects reach a state of thermal equilibrium, where the heat transfer between them stops.

Now, let's assume that the final temperature of the two objects is T. This means that the amount of heat transferred from object 1 to object 2 is equal to the amount of heat transferred from object 2 to object 1. Using the equation for heat transfer, Q = mcΔT, where Q is the heat transferred, m is the mass of the object, c is the specific heat capacity, and ΔT is the change in temperature, we can set up the following equation:

m1c1(T-X) = m2c2(Y-T)

Where m1 and m2 are the masses of objects 1 and 2, and c1 and c2 are their specific heat capacities.

Since the two objects are identical, we can assume that their masses and specific heat capacities are equal, so we can simplify the equation to:

c(T-X) = c(Y-T)

Expanding the brackets, we get:

cT - cX = cY - cT

Rearranging the terms, we get:

2cT = cX + cY

Dividing both sides by 2c, we get:

T = (X + Y)/2

This is the final temperature of the two objects in thermal contact, which is the average of their initial temperatures. Therefore, the mathematical proof for the final temperature of two identical objects in thermal contact is T = (X + Y)/2.