Truth that Contacting of Different Qualities Releases Energy (2)

In the previous column, I explained why the temperature of the mixed water is 56°C instead of 60°C. This time, as an appendix, I will present a calculation using the thermodynamic concept of entropy for those who are curious about the temperature calculation. At this stage, you do not need to try to understand the appendix. However, I have described it in a way that should be intuitively understandable.

Appendix 1 presents an overview of the entire calculation.
Appendix 2 provides an intuitive explanation of the fundamental concept of reversible heat transfer, where the entropy change is zero.
Appendix 3 offers a conceptual explanation to intuitively justify that thermal energy can be expressed as the product of temperature and entropy change.

Appendix 1: Calculation of Water Temperature When High-Temperature Water (1) and Low-Temperature Water (2) Mix Reversibly

The energy of the mixed water is smaller than the total energy of water 1 and water 2 before they came into contact and mix. If the mixing process is carried out reversibly through a mechanism that store energy externally, the energy of the mixed water is reduced by the amount of energy stored.

If the energy stored externally in a reversible manner is used to reversibly return heat, energy is conserved, and water 1 and water 2 can be restored to their original states. However, if energy is transferred externally in an irreversible (nonequilibrium) manner, water 1 and water 2 cannot be restored to their original states.

The maximum possible energy that can be released when mixing is done reversibly can be calculated. This corresponds to the maximum extractable energy from the water 1 and 2. Therefore, we can determine the final temperature of the mixed water when mixed in a fully reversible manner.

Figure 2-1 illustrates the mixing of 80°C water and 40°C water through contact. However, the figure does not depict how the mixing occurs. Since it does not specify the method of heat transfer, the diagram presents an ambiguous setup. Given that this was a question of elementary school, it may have been sufficiently good as a conceptual question for educators teaching elementary school students.

The method taught in elementary school for obtaining the answer was straightforward. If we denote the temperature after mixing as Tm, the following equation gave the answer:

　　　　　　Tm＝(80℃xCt＋40℃xCt）/2Ct=60℃

In the elementary school, heat capacity Ct might have been given simply as mass or weight. The principle behind the answer was the conservation of total energy.

[ Author : Y. F. ]