Figure 3.3 shows graphs of entropy vs. energy for two objects, A and B. Both graphs are on the same scale. The energies of these two objects initially have the values indicated; the objects are then brought into thermal contact with each other. Explain what happens subsequently and why, without using the word "temperature."

If two objects are in thermal equilibrium, their temperatures are said to be the same. It can be expressed in terms of the system's entropy and energy, the measure of randomness being the entropy. It is determined by the amount of energy that is not available for work.

Mathematically, temperature can be defined as:

$\frac{1}{T}=\left(\frac{\partial S}{\partial U}\right)$

Where,

$\partial S$is the change in entropy and $\partial U$is the change in the internal energy of the body.

Hence, a graph reflecting these values can be used to study the temperature of that body.

Conversely, it can also be said that if the slope $\frac{\partial S}{\partial U}$of bodies are same, their temperatures would be same.

From the figure, it can be seen that the slope $\frac{\partial S}{\partial U}$of A is greater than that of B. The system will evolve through energy exchange until the slopes are equal. Assume that $U_A+U_B=U_{\text{total}}$, i.e. that energy is exchanged in both directions. As a result, if $U_A$is increased, $U_B$must be reduced in equal measure, and vice versa. In this situation, one option to make the slope equal is to decrease the slope for B and increase the slope for A, which may be accomplished by reducing $U_B$and raising by the same amount. System B will transfer some energy to A until the slopes are equal. The temperature is referred to as the slope in this case.

As a result, the slope $\frac{\partial S}{\partial U}$of A is greater than that of B, and the system will evolve by exchanging energy until the slopes are equal.